id sid tid token lemma pos cana-2728 1 1 communications communication NOUN cana-2728 1 2 on on ADP cana-2728 1 3 applied apply VERB cana-2728 1 4 nonlinear nonlinear ADJ cana-2728 1 5 analysis analysis NOUN cana-2728 1 6 issn issn NOUN cana-2728 1 7 : : PUNCT cana-2728 1 8 1074 1074 NUM cana-2728 1 9 - - PUNCT cana-2728 1 10 133x 133x NUM cana-2728 1 11 vol vol NOUN cana-2728 1 12 32 32 NUM cana-2728 1 13 no no NOUN cana-2728 1 14 . . PUNCT cana-2728 2 1 3s 3s NUM cana-2728 2 2 ( ( PUNCT cana-2728 2 3 2025 2025 NUM cana-2728 2 4 ) ) PUNCT cana-2728 2 5 699 699 NUM cana-2728 2 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 2 7 some some DET cana-2728 2 8 properties property NOUN cana-2728 2 9 of of ADP cana-2728 2 10 ricci ricci PROPN cana-2728 2 11 solitons soliton NOUN cana-2728 2 12 in in ADP cana-2728 2 13 lp lp ADJ cana-2728 2 14 - - PUNCT cana-2728 2 15 kenmotsu kenmotsu NOUN cana-2728 2 16 manifolds manifolds PROPN cana-2728 2 17 bidyabati bidyabati PROPN cana-2728 2 18 thangjam1 thangjam1 PROPN cana-2728 2 19 , , PUNCT cana-2728 2 20 m. m. PROPN cana-2728 2 21 s. s. PROPN cana-2728 2 22 devi*2 devi*2 PART cana-2728 2 23 1,2 1,2 NUM cana-2728 2 24 article article NOUN cana-2728 2 25 history history NOUN cana-2728 2 26 : : PUNCT cana-2728 2 27 received receive VERB cana-2728 2 28 : : PUNCT cana-2728 2 29 28 28 NUM cana-2728 2 30 - - SYM cana-2728 2 31 09 09 NUM cana-2728 2 32 - - PUNCT cana-2728 2 33 2024 2024 NUM cana-2728 2 34 revised revise VERB cana-2728 2 35 : : PUNCT cana-2728 2 36 18 18 NUM cana-2728 2 37 - - SYM cana-2728 2 38 11 11 NUM cana-2728 2 39 - - PUNCT cana-2728 2 40 2024 2024 NUM cana-2728 2 41 accepted accept VERB cana-2728 2 42 : : PUNCT cana-2728 2 43 30 30 NUM cana-2728 2 44 - - SYM cana-2728 2 45 11 11 NUM cana-2728 2 46 - - PUNCT cana-2728 2 47 2024 2024 NUM cana-2728 2 48 abstract abstract NOUN cana-2728 2 49 : : PUNCT cana-2728 2 50 examining examine VERB cana-2728 2 51 ricci ricci PROPN cana-2728 2 52 solitons soliton NOUN cana-2728 2 53 in in ADP cana-2728 2 54 lorentzian lorentzian ADJ cana-2728 2 55 para para PROPN cana-2728 2 56 - - PUNCT cana-2728 2 57 kenmotsu kenmotsu NOUN cana-2728 2 58 manifolds manifold NOUN cana-2728 2 59 is be AUX cana-2728 2 60 the the DET cana-2728 2 61 goal goal NOUN cana-2728 2 62 of of ADP cana-2728 2 63 this this DET cana-2728 2 64 paper paper NOUN cana-2728 2 65 . . PUNCT cana-2728 3 1 we we PRON cana-2728 3 2 have have AUX cana-2728 3 3 established establish VERB cana-2728 3 4 that that SCONJ cana-2728 3 5 a a DET cana-2728 3 6 symmetric symmetric ADJ cana-2728 3 7 parallel parallel ADJ cana-2728 3 8 second second ADJ cana-2728 3 9 - - PUNCT cana-2728 3 10 order order NOUN cana-2728 3 11 covariant covariant ADJ cana-2728 3 12 tensor tensor NOUN cana-2728 3 13 in in ADP cana-2728 3 14 a a DET cana-2728 3 15 lorentzian lorentzian ADJ cana-2728 3 16 para para NOUN cana-2728 3 17 - - PUNCT cana-2728 3 18 kenmotsu kenmotsu PROPN cana-2728 3 19 manifold manifold NOUN cana-2728 3 20 is be AUX cana-2728 3 21 a a DET cana-2728 3 22 constant constant ADJ cana-2728 3 23 multiple multiple NOUN cana-2728 3 24 of of ADP cana-2728 3 25 the the DET cana-2728 3 26 metric metric ADJ cana-2728 3 27 tensor tensor NOUN cana-2728 3 28 . . PUNCT cana-2728 4 1 we we PRON cana-2728 4 2 have have AUX cana-2728 4 3 shown show VERB cana-2728 4 4 that that SCONJ cana-2728 4 5 if if SCONJ cana-2728 4 6 l_v l_v PROPN cana-2728 4 7 g+2s g+2s PROPN cana-2728 4 8 is be AUX cana-2728 4 9 parallel parallel ADJ cana-2728 4 10 to to ADP cana-2728 4 11 the the DET cana-2728 4 12 levi levi PROPN cana-2728 4 13 - - PUNCT cana-2728 4 14 civita civita PROPN cana-2728 4 15 connection connection NOUN cana-2728 4 16 associated associate VERB cana-2728 4 17 with with ADP cana-2728 4 18 g g PROPN cana-2728 4 19 , , PUNCT cana-2728 4 20 where where SCONJ cana-2728 4 21 v v NOUN cana-2728 4 22 is be AUX cana-2728 4 23 a a DET cana-2728 4 24 given give VERB cana-2728 4 25 vector vector NOUN cana-2728 4 26 field field NOUN cana-2728 4 27 , , PUNCT cana-2728 4 28 then then ADV cana-2728 4 29 ( ( PUNCT cana-2728 4 30 g g NOUN cana-2728 4 31 , , PUNCT cana-2728 4 32 v v NOUN cana-2728 4 33 , , PUNCT cana-2728 4 34 Ξ» Ξ» NOUN cana-2728 4 35 ) ) PUNCT cana-2728 4 36 is be AUX cana-2728 4 37 a a DET cana-2728 4 38 ricci ricci PROPN cana-2728 4 39 soliton soliton NOUN cana-2728 4 40 . . PUNCT cana-2728 5 1 we we PRON cana-2728 5 2 have have AUX cana-2728 5 3 observed observe VERB cana-2728 5 4 that that SCONJ cana-2728 5 5 a a DET cana-2728 5 6 ricci ricci PROPN cana-2728 5 7 soliton soliton NOUN cana-2728 5 8 in in ADP cana-2728 5 9 a a DET cana-2728 5 10 w_2semi w_2semi NOUN cana-2728 5 11 - - ADJ cana-2728 5 12 symmetric symmetric ADJ cana-2728 5 13 lorentzian lorentzian ADJ cana-2728 5 14 para para NOUN cana-2728 5 15 - - PUNCT cana-2728 5 16 kenmotsu kenmotsu PROPN cana-2728 5 17 manifold manifold NOUN cana-2728 5 18 is be AUX cana-2728 5 19 shrinking shrink VERB cana-2728 5 20 . . PUNCT cana-2728 6 1 furthermore furthermore ADV cana-2728 6 2 , , PUNCT cana-2728 6 3 certain certain ADJ cana-2728 6 4 curvature curvature NOUN cana-2728 6 5 properties property NOUN cana-2728 6 6 of of ADP cana-2728 6 7 lorentzian lorentzian ADJ cana-2728 6 8 para para PROPN cana-2728 6 9 - - PUNCT cana-2728 6 10 kenmotsu kenmotsu NOUN cana-2728 6 11 manifolds manifold NOUN cana-2728 6 12 admitting admit VERB cana-2728 6 13 ricci ricci PROPN cana-2728 6 14 solitons soliton NOUN cana-2728 6 15 are be AUX cana-2728 6 16 studied study VERB cana-2728 6 17 . . PUNCT cana-2728 7 1 finally finally ADV cana-2728 7 2 , , PUNCT cana-2728 7 3 we we PRON cana-2728 7 4 have have AUX cana-2728 7 5 provided provide VERB cana-2728 7 6 an an DET cana-2728 7 7 example example NOUN cana-2728 7 8 of of ADP cana-2728 7 9 a a DET cana-2728 7 10 3 3 NUM cana-2728 7 11 - - PUNCT cana-2728 7 12 dimensional dimensional ADJ cana-2728 7 13 lorentzian lorentzian ADJ cana-2728 7 14 para para NOUN cana-2728 7 15 - - PUNCT cana-2728 7 16 kenmotsu kenmotsu NOUN cana-2728 7 17 manifold manifold ADJ cana-2728 7 18 . . PUNCT cana-2728 8 1 keywords keyword NOUN cana-2728 8 2 : : PUNCT cana-2728 8 3 lorentzian lorentzian ADJ cana-2728 8 4 para para NOUN cana-2728 8 5 - - PUNCT cana-2728 8 6 kenmotsu kenmotsu NOUN cana-2728 8 7 manifolds manifold NOUN cana-2728 8 8 , , PUNCT cana-2728 8 9 ricci ricci PROPN cana-2728 8 10 solitons soliton NOUN cana-2728 8 11 , , PUNCT cana-2728 8 12 symmetric symmetric ADJ cana-2728 8 13 second second ADJ cana-2728 8 14 order order NOUN cana-2728 8 15 tensors tensor NOUN cana-2728 8 16 . . PUNCT cana-2728 9 1 mathematics mathematic NOUN cana-2728 9 2 subject subject ADJ cana-2728 9 3 classification classification NOUN cana-2728 9 4 ( ( PUNCT cana-2728 9 5 2020 2020 NUM cana-2728 9 6 ): ): PUNCT cana-2728 9 7 53c21 53c21 NUM cana-2728 9 8 , , PUNCT cana-2728 9 9 53c25 53c25 NUM cana-2728 9 10 , , PUNCT cana-2728 9 11 53e20 53e20 NUM cana-2728 9 12 . . PUNCT cana-2728 10 1 introduction introduction NOUN cana-2728 10 2 hamilton hamilton PROPN cana-2728 11 1 [ [ X cana-2728 11 2 5 5 NUM cana-2728 11 3 ] ] PUNCT cana-2728 11 4 introduced introduce VERB cana-2728 11 5 the the DET cana-2728 11 6 concept concept NOUN cana-2728 11 7 of of ADP cana-2728 11 8 ricci ricci PROPN cana-2728 11 9 solitons soliton NOUN cana-2728 11 10 , , PUNCT cana-2728 11 11 which which PRON cana-2728 11 12 is be AUX cana-2728 11 13 a a DET cana-2728 11 14 natural natural ADJ cana-2728 11 15 generalization generalization NOUN cana-2728 11 16 of of ADP cana-2728 11 17 an an DET cana-2728 11 18 einstein einstein ADJ cana-2728 11 19 metric metric NOUN cana-2728 11 20 and and CCONJ cana-2728 11 21 is be AUX cana-2728 11 22 defined define VERB cana-2728 11 23 on on ADP cana-2728 11 24 a a DET cana-2728 11 25 riemannian riemannian ADJ cana-2728 11 26 manifold manifold ADJ cana-2728 11 27 𝑀. 𝑀. PROPN cana-2728 11 28 a a DET cana-2728 11 29 ricci ricci NOUN cana-2728 11 30 soliton soliton NOUN cana-2728 11 31 is be AUX cana-2728 11 32 a a DET cana-2728 11 33 tripled triple VERB cana-2728 11 34 ( ( PUNCT cana-2728 11 35 𝑔 𝑔 X cana-2728 11 36 , , PUNCT cana-2728 11 37 𝑉 𝑉 PROPN cana-2728 11 38 , , PUNCT cana-2728 11 39 Ξ» Ξ» NOUN cana-2728 11 40 ) ) PUNCT cana-2728 11 41 such such ADJ cana-2728 11 42 that that DET cana-2728 11 43 ℒ𝒱 ℒ𝒱 PROPN cana-2728 11 44 + + CCONJ cana-2728 11 45 2𝑆 2𝑆 NOUN cana-2728 11 46 + + CCONJ cana-2728 11 47 2πœ†π‘” 2πœ†π‘” NOUN cana-2728 11 48 = = SYM cana-2728 11 49 0 0 NUM cana-2728 11 50 , , PUNCT cana-2728 11 51 ( ( PUNCT cana-2728 11 52 1.1 1.1 NUM cana-2728 11 53 ) ) PUNCT cana-2728 11 54 where where SCONJ cana-2728 11 55 𝑔 𝑔 PROPN cana-2728 11 56 is be AUX cana-2728 11 57 a a DET cana-2728 11 58 riemannian riemannian ADJ cana-2728 11 59 metric metric NOUN cana-2728 11 60 , , PUNCT cana-2728 11 61 𝑉 𝑉 PROPN cana-2728 11 62 is be AUX cana-2728 11 63 a a DET cana-2728 11 64 vector vector NOUN cana-2728 11 65 field field NOUN cana-2728 11 66 , , PUNCT cana-2728 11 67 πœ† πœ† X cana-2728 11 68 is be AUX cana-2728 11 69 a a DET cana-2728 11 70 real real ADV cana-2728 11 71 scalar scalar NOUN cana-2728 11 72 , , PUNCT cana-2728 11 73 𝑆 𝑆 PROPN cana-2728 11 74 is be AUX cana-2728 11 75 a a DET cana-2728 11 76 ricci ricci PROPN cana-2728 11 77 tensor tensor NOUN cana-2728 11 78 of of ADP cana-2728 11 79 𝑀 𝑀 PROPN cana-2728 11 80 and and CCONJ cana-2728 11 81 ℒ𝒱 ℒ𝒱 PROPN cana-2728 11 82 denotes denote VERB cana-2728 11 83 the the DET cana-2728 11 84 lie lie NOUN cana-2728 11 85 derivative derivative ADJ cana-2728 11 86 operator operator NOUN cana-2728 11 87 along along ADP cana-2728 11 88 the the DET cana-2728 11 89 vector vector NOUN cana-2728 11 90 field field NOUN cana-2728 11 91 𝑉. 𝑉. NOUN cana-2728 11 92 a a DET cana-2728 11 93 ricci ricci PROPN cana-2728 11 94 soliton soliton NOUN cana-2728 11 95 is be AUX cana-2728 11 96 said say VERB cana-2728 11 97 to to PART cana-2728 11 98 be be AUX cana-2728 11 99 shrinking shrink VERB cana-2728 11 100 if if SCONJ cana-2728 11 101 πœ† πœ† NOUN cana-2728 11 102 is be AUX cana-2728 11 103 negative negative ADJ cana-2728 11 104 , , PUNCT cana-2728 11 105 steady steady ADJ cana-2728 11 106 if if SCONJ cana-2728 11 107 πœ† πœ† NOUN cana-2728 11 108 is be AUX cana-2728 11 109 zero zero NUM cana-2728 11 110 , , PUNCT cana-2728 11 111 and and CCONJ cana-2728 11 112 expanding expand VERB cana-2728 11 113 if if SCONJ cana-2728 11 114 πœ† πœ† NOUN cana-2728 11 115 is be AUX cana-2728 11 116 positive positive ADJ cana-2728 11 117 . . PUNCT cana-2728 12 1 in in ADP cana-2728 12 2 recent recent ADJ cana-2728 12 3 years year NOUN cana-2728 12 4 , , PUNCT cana-2728 12 5 many many ADJ cana-2728 12 6 geometers geometer NOUN cana-2728 12 7 have have AUX cana-2728 12 8 studied study VERB cana-2728 12 9 ricci ricci NOUN cana-2728 12 10 solitons soliton NOUN cana-2728 12 11 . . PUNCT cana-2728 13 1 ingalahalli ingalahalli PROPN cana-2728 13 2 and and CCONJ cana-2728 13 3 bagewadi bagewadi VERB cana-2728 13 4 [ [ X cana-2728 13 5 8 8 NUM cana-2728 13 6 ] ] PUNCT cana-2728 13 7 studied study VERB cana-2728 13 8 ricci ricci PROPN cana-2728 13 9 solitons soliton NOUN cana-2728 13 10 on on ADP cana-2728 13 11 𝛼-sasakian 𝛼-sasakian PROPN cana-2728 13 12 manifolds manifold NOUN cana-2728 13 13 . . PUNCT cana-2728 14 1 pokhariyal pokhariyal NOUN cana-2728 14 2 et et PROPN cana-2728 14 3 al al PROPN cana-2728 14 4 . . PUNCT cana-2728 15 1 [ [ X cana-2728 15 2 12 12 NUM cana-2728 15 3 ] ] PUNCT cana-2728 15 4 found find VERB cana-2728 15 5 some some DET cana-2728 15 6 results result NOUN cana-2728 15 7 on on ADP cana-2728 15 8 trans trans ADJ cana-2728 15 9 - - ADJ cana-2728 15 10 sasakian sasakian ADJ cana-2728 15 11 manifolds manifold NOUN cana-2728 15 12 . . PUNCT cana-2728 16 1 ayar ayar NOUN cana-2728 16 2 and and CCONJ cana-2728 16 3 demirhan demirhan ADV cana-2728 17 1 [ [ X cana-2728 17 2 1 1 X cana-2728 17 3 ] ] PUNCT cana-2728 17 4 provided provide VERB cana-2728 17 5 basic basic ADJ cana-2728 17 6 information information NOUN cana-2728 17 7 about about ADP cana-2728 17 8 ricci ricci PROPN cana-2728 17 9 solitons soliton NOUN cana-2728 17 10 on on ADP cana-2728 17 11 nearly nearly ADV cana-2728 17 12 kenmotsu kenmotsu NOUN cana-2728 17 13 manifolds manifold NOUN cana-2728 17 14 and and CCONJ cana-2728 17 15 obtained obtain VERB cana-2728 17 16 some some DET cana-2728 17 17 structures structure NOUN cana-2728 17 18 on on ADP cana-2728 17 19 this this DET cana-2728 17 20 manifold manifold ADJ cana-2728 17 21 satisfying satisfying NOUN cana-2728 17 22 a a DET cana-2728 17 23 semi semi ADJ cana-2728 17 24 - - ADJ cana-2728 17 25 symmetric symmetric ADJ cana-2728 17 26 metric metric ADJ cana-2728 17 27 connection connection NOUN cana-2728 17 28 . . PUNCT cana-2728 18 1 later later ADV cana-2728 18 2 , , PUNCT cana-2728 18 3 shah shah PROPN cana-2728 18 4 [ [ X cana-2728 18 5 15 15 NUM cana-2728 18 6 ] ] PUNCT cana-2728 18 7 studied study VERB cana-2728 18 8 ricci ricci PROPN cana-2728 18 9 solitons soliton NOUN cana-2728 18 10 in in ADP cana-2728 18 11 lorentzian lorentzian ADJ cana-2728 18 12 para para PROPN cana-2728 18 13 - - PUNCT cana-2728 18 14 sasakian sasakian ADJ cana-2728 18 15 manifolds manifold NOUN cana-2728 18 16 , , PUNCT cana-2728 18 17 while while SCONJ cana-2728 18 18 chen chen PROPN cana-2728 18 19 et et PROPN cana-2728 18 20 al al PROPN cana-2728 18 21 . . PUNCT cana-2728 19 1 [ [ X cana-2728 19 2 2 2 NUM cana-2728 19 3 ] ] PUNCT cana-2728 19 4 studied study VERB cana-2728 19 5 ricci ricci NOUN cana-2728 19 6 solitons soliton NOUN cana-2728 19 7 and and CCONJ cana-2728 19 8 certain certain ADJ cana-2728 19 9 related related ADJ cana-2728 19 10 metrics metric NOUN cana-2728 19 11 on on ADP cana-2728 19 12 a a DET cana-2728 19 13 three three NUM cana-2728 19 14 - - PUNCT cana-2728 19 15 dimensional dimensional ADJ cana-2728 19 16 trans trans ADJ cana-2728 19 17 - - ADJ cana-2728 19 18 sasakian sasakian ADJ cana-2728 19 19 manifolds manifold NOUN cana-2728 19 20 . . PUNCT cana-2728 20 1 many many ADJ cana-2728 20 2 other other ADJ cana-2728 20 3 geometers geometer NOUN cana-2728 20 4 had have AUX cana-2728 20 5 also also ADV cana-2728 20 6 studied study VERB cana-2728 20 7 ricci ricci NOUN cana-2728 20 8 solitons soliton NOUN cana-2728 20 9 on on ADP cana-2728 20 10 various various ADJ cana-2728 20 11 manifolds manifold NOUN cana-2728 20 12 . . PUNCT cana-2728 21 1 sato[14 sato[14 PROPN cana-2728 21 2 ] ] PUNCT cana-2728 21 3 introduced introduce VERB cana-2728 21 4 the the DET cana-2728 21 5 concept concept NOUN cana-2728 21 6 of of ADP cana-2728 21 7 an an DET cana-2728 21 8 almost almost ADV cana-2728 21 9 para para ADJ cana-2728 21 10 - - PUNCT cana-2728 21 11 contact contact NOUN cana-2728 21 12 riemannian riemannian NOUN cana-2728 21 13 manifold manifold NOUN cana-2728 21 14 . . PUNCT cana-2728 22 1 subsequently subsequently ADV cana-2728 22 2 , , PUNCT cana-2728 22 3 sinha sinha NOUN cana-2728 22 4 and and CCONJ cana-2728 22 5 prasad prasad PROPN cana-2728 23 1 [ [ X cana-2728 23 2 7 7 NUM cana-2728 23 3 ] ] PUNCT cana-2728 23 4 defined define VERB cana-2728 23 5 a a DET cana-2728 23 6 specific specific ADJ cana-2728 23 7 class class NOUN cana-2728 23 8 of of ADP cana-2728 23 9 almost almost ADV cana-2728 23 10 para para ADJ cana-2728 23 11 - - PUNCT cana-2728 23 12 contact contact NOUN cana-2728 23 13 metric metric ADJ cana-2728 23 14 manifolds manifold NOUN cana-2728 23 15 , , PUNCT cana-2728 23 16 namely namely ADV cana-2728 23 17 para para ADJ cana-2728 23 18 kenmotsu kenmotsu NOUN cana-2728 23 19 and and CCONJ cana-2728 23 20 special special ADJ cana-2728 23 21 para para NOUN cana-2728 23 22 kenmotsu kenmotsu NOUN cana-2728 23 23 manifolds manifold NOUN cana-2728 23 24 . . PUNCT cana-2728 24 1 another another DET cana-2728 24 2 related related ADJ cana-2728 24 3 structure structure NOUN cana-2728 24 4 is be AUX cana-2728 24 5 lorentzian lorentzian ADJ cana-2728 24 6 para para NOUN cana-2728 24 7 - - PUNCT cana-2728 24 8 sasakian sasakian NOUN cana-2728 24 9 manifold manifold NOUN cana-2728 24 10 , , PUNCT cana-2728 24 11 which which PRON cana-2728 24 12 were be AUX cana-2728 24 13 introduced introduce VERB cana-2728 24 14 by by ADP cana-2728 24 15 matsumoto matsumoto PROPN cana-2728 25 1 [ [ X cana-2728 25 2 9 9 NUM cana-2728 25 3 ] ] PUNCT cana-2728 25 4 . . PUNCT cana-2728 26 1 several several ADJ cana-2728 26 2 other other ADJ cana-2728 26 3 researchers researcher NOUN cana-2728 26 4 had have AUX cana-2728 26 5 also also ADV cana-2728 26 6 studied study VERB cana-2728 26 7 this this DET cana-2728 26 8 manifold manifold ADJ cana-2728 26 9 ( ( PUNCT cana-2728 26 10 [ [ X cana-2728 26 11 4 4 NUM cana-2728 26 12 ] ] PUNCT cana-2728 26 13 , , PUNCT cana-2728 26 14 [ [ X cana-2728 26 15 10 10 NUM cana-2728 26 16 ] ] PUNCT cana-2728 26 17 , , PUNCT cana-2728 26 18 [ [ X cana-2728 26 19 16 16 NUM cana-2728 26 20 ] ] PUNCT cana-2728 26 21 , , PUNCT cana-2728 26 22 [ [ X cana-2728 26 23 18 18 NUM cana-2728 26 24 ] ] NUM cana-2728 26 25 ) ) PUNCT cana-2728 26 26 . . PUNCT cana-2728 27 1 recently recently ADV cana-2728 27 2 , , PUNCT cana-2728 27 3 haseeb haseeb NOUN cana-2728 27 4 and and CCONJ cana-2728 27 5 prasad prasad PROPN cana-2728 28 1 [ [ X cana-2728 28 2 6 6 NUM cana-2728 28 3 , , PUNCT cana-2728 28 4 7 7 NUM cana-2728 28 5 ] ] PUNCT cana-2728 28 6 focused focus VERB cana-2728 28 7 on on ADP cana-2728 28 8 studying study VERB cana-2728 28 9 the the DET cana-2728 28 10 properties property NOUN cana-2728 28 11 of of ADP cana-2728 28 12 lorentzian lorentzian ADJ cana-2728 28 13 para para PROPN cana-2728 28 14 - - PUNCT cana-2728 28 15 kenmotsu kenmotsu NOUN cana-2728 28 16 manifolds manifold NOUN cana-2728 28 17 , , PUNCT cana-2728 28 18 particularly particularly ADV cana-2728 28 19 in in ADP cana-2728 28 20 terms term NOUN cana-2728 28 21 of of ADP cana-2728 28 22 ricci ricci NOUN cana-2728 28 23 - - PUNCT cana-2728 28 24 pseudosymmetricity pseudosymmetricity PROPN cana-2728 28 25 and and CCONJ cana-2728 28 26 ricci ricci PROPN cana-2728 28 27 department department PROPN cana-2728 28 28 of of ADP cana-2728 28 29 mathematics mathematics PROPN cana-2728 28 30 , , PUNCT cana-2728 28 31 mizoram mizoram PROPN cana-2728 28 32 university university PROPN cana-2728 28 33 , , PUNCT cana-2728 28 34 tanhril tanhril NOUN cana-2728 28 35 , , PUNCT cana-2728 28 36 aizawl-796004 aizawl-796004 NOUN cana-2728 28 37 , , PUNCT cana-2728 28 38 india india PROPN cana-2728 28 39 . . PUNCT cana-2728 29 1 devi_saroja@rediffmail.com devi_saroja@rediffmail.com VERB cana-2728 29 2 communications communication NOUN cana-2728 29 3 on on ADP cana-2728 29 4 applied apply VERB cana-2728 29 5 nonlinear nonlinear ADJ cana-2728 29 6 analysis analysis NOUN cana-2728 29 7 issn issn NOUN cana-2728 29 8 : : PUNCT cana-2728 29 9 1074 1074 NUM cana-2728 29 10 - - PUNCT cana-2728 29 11 133x 133x NUM cana-2728 29 12 vol vol NOUN cana-2728 29 13 32 32 NUM cana-2728 29 14 no no NOUN cana-2728 29 15 . . PUNCT cana-2728 30 1 3s 3s NUM cana-2728 30 2 ( ( PUNCT cana-2728 30 3 2025 2025 NUM cana-2728 30 4 ) ) PUNCT cana-2728 30 5 700 700 NUM cana-2728 30 6 https://internationalpubls.com https://internationalpubls.com NUM cana-2728 30 7 generalized generalized ADJ cana-2728 30 8 pseudosymmetricity pseudosymmetricity NOUN cana-2728 30 9 conditions condition NOUN cana-2728 30 10 . . PUNCT cana-2728 31 1 pandey pandey PROPN cana-2728 31 2 et et PROPN cana-2728 31 3 al al PROPN cana-2728 31 4 . . PUNCT cana-2728 32 1 [ [ X cana-2728 32 2 11 11 NUM cana-2728 32 3 ] ] PUNCT cana-2728 32 4 investigated investigate VERB cana-2728 32 5 the the DET cana-2728 32 6 geometric geometric ADJ cana-2728 32 7 properties property NOUN cana-2728 32 8 of of ADP cana-2728 32 9 πœ‚-ricci πœ‚-ricci NOUN cana-2728 32 10 solitons soliton NOUN cana-2728 32 11 on on ADP cana-2728 32 12 lorentzian lorentzian ADJ cana-2728 32 13 para para PROPN cana-2728 32 14 - - PUNCT cana-2728 32 15 kenmotsu kenmotsu NOUN cana-2728 32 16 manifolds manifold NOUN cana-2728 32 17 . . PUNCT cana-2728 33 1 based base VERB cana-2728 33 2 on on ADP cana-2728 33 3 these these DET cana-2728 33 4 studies study NOUN cana-2728 33 5 , , PUNCT cana-2728 33 6 our our PRON cana-2728 33 7 motivation motivation NOUN cana-2728 33 8 is be AUX cana-2728 33 9 to to PART cana-2728 33 10 investigate investigate VERB cana-2728 33 11 ricci ricci NOUN cana-2728 33 12 solitons soliton NOUN cana-2728 33 13 on on ADP cana-2728 33 14 lorentzian lorentzian ADJ cana-2728 33 15 para para PROPN cana-2728 33 16 - - PUNCT cana-2728 33 17 kenmotsu kenmotsu NOUN cana-2728 33 18 manifolds manifold NOUN cana-2728 33 19 . . PUNCT cana-2728 34 1 the the DET cana-2728 34 2 paper paper NOUN cana-2728 34 3 is be AUX cana-2728 34 4 structured structure VERB cana-2728 34 5 as as SCONJ cana-2728 34 6 follows follow VERB cana-2728 34 7 : : PUNCT cana-2728 34 8 section section NOUN cana-2728 34 9 1 1 NUM cana-2728 34 10 is be AUX cana-2728 34 11 the the DET cana-2728 34 12 introduction introduction NOUN cana-2728 34 13 , , PUNCT cana-2728 34 14 then then ADV cana-2728 34 15 there there PRON cana-2728 34 16 is be VERB cana-2728 34 17 a a DET cana-2728 34 18 preliminaries preliminary NOUN cana-2728 34 19 section section NOUN cana-2728 34 20 . . PUNCT cana-2728 35 1 in in ADP cana-2728 35 2 section section NOUN cana-2728 35 3 3 3 NUM cana-2728 35 4 , , PUNCT cana-2728 35 5 parallel parallel ADJ cana-2728 35 6 symmetric symmetric ADJ cana-2728 35 7 second second ADJ cana-2728 35 8 - - PUNCT cana-2728 35 9 order order NOUN cana-2728 35 10 tensors tensor NOUN cana-2728 35 11 and and CCONJ cana-2728 35 12 ricci ricci PROPN cana-2728 35 13 solitons soliton NOUN cana-2728 35 14 in in ADP cana-2728 35 15 lorentzian lorentzian ADJ cana-2728 35 16 para para PROPN cana-2728 35 17 - - PUNCT cana-2728 35 18 kenmotsu kenmotsu NOUN cana-2728 35 19 manifolds manifold NOUN cana-2728 35 20 are be AUX cana-2728 35 21 studied study VERB cana-2728 35 22 . . PUNCT cana-2728 36 1 the the DET cana-2728 36 2 next next ADJ cana-2728 36 3 section section NOUN cana-2728 36 4 investigates investigate VERB cana-2728 36 5 the the DET cana-2728 36 6 properties property NOUN cana-2728 36 7 of of ADP cana-2728 36 8 ricci ricci PROPN cana-2728 36 9 solitons soliton NOUN cana-2728 36 10 in in ADP cana-2728 36 11 π‘Š2 π‘Š2 PROPN cana-2728 36 12 - - ADJ cana-2728 36 13 semisymmetric semisymmetric ADJ cana-2728 36 14 lorentzian lorentzian ADJ cana-2728 36 15 para para NOUN cana-2728 36 16 - - PUNCT cana-2728 36 17 kenmotsu kenmotsu NOUN cana-2728 36 18 manifolds manifold NOUN cana-2728 36 19 of of ADP cana-2728 36 20 dimensions dimension NOUN cana-2728 36 21 ( ( PUNCT cana-2728 36 22 2𝑛 2𝑛 NOUN cana-2728 36 23 + + PROPN cana-2728 36 24 1 1 NUM cana-2728 36 25 ) ) PUNCT cana-2728 36 26 . . PUNCT cana-2728 37 1 in in ADP cana-2728 37 2 section section NOUN cana-2728 37 3 5 5 NUM cana-2728 37 4 , , PUNCT cana-2728 37 5 we we PRON cana-2728 37 6 study study VERB cana-2728 37 7 ricci ricci PROPN cana-2728 37 8 tensor tensor NOUN cana-2728 37 9 of of ADP cana-2728 37 10 a a DET cana-2728 37 11 lorentzian lorentzian ADJ cana-2728 37 12 para para NOUN cana-2728 37 13 - - PUNCT cana-2728 37 14 kenmotsu kenmotsu NOUN cana-2728 37 15 manifold manifold ADJ cana-2728 37 16 admitting admit VERB cana-2728 37 17 a a DET cana-2728 37 18 ricci ricci PROPN cana-2728 37 19 soliton soliton NOUN cana-2728 37 20 . . PUNCT cana-2728 38 1 then then ADV cana-2728 38 2 , , PUNCT cana-2728 38 3 in in ADP cana-2728 38 4 the the DET cana-2728 38 5 next next ADJ cana-2728 38 6 section section NOUN cana-2728 38 7 , , PUNCT cana-2728 38 8 curvature curvature NOUN cana-2728 38 9 properties property NOUN cana-2728 38 10 of of ADP cana-2728 38 11 lorentzian lorentzian ADJ cana-2728 38 12 para para PROPN cana-2728 38 13 - - PUNCT cana-2728 38 14 kenmotsu kenmotsu NOUN cana-2728 38 15 manifolds manifold NOUN cana-2728 38 16 admitting admit VERB cana-2728 38 17 ricci ricci PROPN cana-2728 38 18 solitons soliton NOUN cana-2728 38 19 are be AUX cana-2728 38 20 examined examine VERB cana-2728 38 21 . . PUNCT cana-2728 39 1 the the DET cana-2728 39 2 last last ADJ cana-2728 39 3 section section NOUN cana-2728 39 4 provides provide VERB cana-2728 39 5 an an DET cana-2728 39 6 example example NOUN cana-2728 39 7 of of ADP cana-2728 39 8 a a DET cana-2728 39 9 lorentzian lorentzian ADJ cana-2728 39 10 para para NOUN cana-2728 39 11 - - PUNCT cana-2728 39 12 kenmotsu kenmotsu NOUN cana-2728 39 13 manifold manifold ADJ cana-2728 39 14 . . PUNCT cana-2728 40 1 the the DET cana-2728 40 2 paper paper NOUN cana-2728 40 3 concludes conclude VERB cana-2728 40 4 with with ADP cana-2728 40 5 a a DET cana-2728 40 6 summary summary NOUN cana-2728 40 7 of of ADP cana-2728 40 8 the the DET cana-2728 40 9 findings finding NOUN cana-2728 40 10 . . PUNCT cana-2728 41 1 1 1 X cana-2728 41 2 . . X cana-2728 41 3 preliminaries preliminary NOUN cana-2728 41 4 a a DET cana-2728 41 5 ( ( PUNCT cana-2728 41 6 2𝑛 2𝑛 NOUN cana-2728 41 7 + + CCONJ cana-2728 41 8 1 1 X cana-2728 41 9 ) ) PUNCT cana-2728 41 10 βˆ’differentiable βˆ’differentiable ADJ cana-2728 41 11 manifold manifold ADJ cana-2728 41 12 𝑀 𝑀 PROPN cana-2728 41 13 with with ADP cana-2728 41 14 a a DET cana-2728 41 15 ( ( PUNCT cana-2728 41 16 1,1 1,1 NUM cana-2728 41 17 ) ) PUNCT cana-2728 41 18 tensor tensor NOUN cana-2728 41 19 field field NOUN cana-2728 41 20 πœ™ πœ™ NOUN cana-2728 41 21 , , PUNCT cana-2728 41 22 contravariant contravariant ADJ cana-2728 41 23 vector vector NOUN cana-2728 41 24 field field NOUN cana-2728 41 25 πœ‰ πœ‰ PROPN cana-2728 41 26 , , PUNCT cana-2728 41 27 a a DET cana-2728 41 28 1form 1form NUM cana-2728 41 29 πœ‚ πœ‚ NOUN cana-2728 41 30 , , PUNCT cana-2728 41 31 and and CCONJ cana-2728 41 32 a a DET cana-2728 41 33 lorentzian lorentzian ADJ cana-2728 41 34 metric metric ADJ cana-2728 41 35 𝑔 𝑔 PROPN cana-2728 41 36 is be AUX cana-2728 41 37 referred refer VERB cana-2728 41 38 as as ADP cana-2728 41 39 a a DET cana-2728 41 40 lorentzian lorentzian ADJ cana-2728 41 41 almost almost ADV cana-2728 41 42 para para ADJ cana-2728 41 43 - - PUNCT cana-2728 41 44 contact contact NOUN cana-2728 41 45 manifold[9 manifold[9 NOUN cana-2728 41 46 ] ] PUNCT cana-2728 41 47 if if SCONJ cana-2728 41 48 𝑔 𝑔 PROPN cana-2728 41 49 satisfies satisfy VERB cana-2728 41 50 the the DET cana-2728 41 51 following follow VERB cana-2728 41 52 conditions condition NOUN cana-2728 41 53 : : PUNCT cana-2728 41 54 πœ™2𝑋1 πœ™2𝑋1 SYM cana-2728 41 55 = = SYM cana-2728 41 56 𝑋1 𝑋1 PROPN cana-2728 41 57 + + CCONJ cana-2728 41 58 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 41 59 ) ) PUNCT cana-2728 41 60 , , PUNCT cana-2728 41 61 πœ™πœ‰ πœ™πœ‰ X cana-2728 41 62 = = SYM cana-2728 41 63 0 0 PROPN cana-2728 41 64 , , PUNCT cana-2728 41 65 ( ( PUNCT cana-2728 41 66 2.1 2.1 NUM cana-2728 41 67 ) ) PUNCT cana-2728 41 68 𝑔(πœ™π‘‹1 𝑔(πœ™π‘‹1 NUM cana-2728 41 69 , , PUNCT cana-2728 41 70 πœ™π‘‹2 πœ™π‘‹2 PROPN cana-2728 41 71 ) ) PUNCT cana-2728 41 72 = = SYM cana-2728 41 73 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 41 74 , , PUNCT cana-2728 41 75 𝑋2 𝑋2 VERB cana-2728 41 76 ) ) PUNCT cana-2728 42 1 + + CCONJ cana-2728 42 2 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 VERB cana-2728 42 3 ) ) PUNCT cana-2728 42 4 ( ( PUNCT cana-2728 42 5 2.2 2.2 NUM cana-2728 42 6 ) ) PUNCT cana-2728 42 7 and and CCONJ cana-2728 42 8 πœ‚(πœ‰ πœ‚(πœ‰ NUM cana-2728 42 9 ) ) PUNCT cana-2728 43 1 = = SYM cana-2728 43 2 βˆ’1 βˆ’1 NOUN cana-2728 43 3 , , PUNCT cana-2728 43 4 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 43 5 , , PUNCT cana-2728 43 6 πœ‰ πœ‰ NOUN cana-2728 43 7 ) ) PUNCT cana-2728 43 8 = = SYM cana-2728 43 9 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 43 10 ) ) PUNCT cana-2728 43 11 , , PUNCT cana-2728 43 12 ( ( PUNCT cana-2728 43 13 2.3 2.3 NUM cana-2728 43 14 ) ) PUNCT cana-2728 43 15 πœ‚(πœ™π‘‹1 πœ‚(πœ™π‘‹1 NUM cana-2728 43 16 ) ) PUNCT cana-2728 43 17 = = SYM cana-2728 43 18 0 0 X cana-2728 43 19 . . PUNCT cana-2728 44 1 ( ( PUNCT cana-2728 44 2 2.4 2.4 NUM cana-2728 44 3 ) ) PUNCT cana-2728 44 4 a a DET cana-2728 44 5 lorentzian lorentzian ADJ cana-2728 44 6 almost almost ADV cana-2728 44 7 para para ADJ cana-2728 44 8 - - PUNCT cana-2728 44 9 contact contact NOUN cana-2728 44 10 manifold manifold ADJ cana-2728 44 11 𝑀 𝑀 PROPN cana-2728 44 12 is be AUX cana-2728 44 13 a a DET cana-2728 44 14 lorentzian lorentzian ADJ cana-2728 44 15 para para NOUN cana-2728 44 16 - - PUNCT cana-2728 44 17 kenmotsu kenmotsu NOUN cana-2728 44 18 manifold manifold ADJ cana-2728 44 19 if if SCONJ cana-2728 44 20 it it PRON cana-2728 44 21 satisfies satisfy VERB cana-2728 44 22 [ [ X cana-2728 44 23 6 6 NUM cana-2728 44 24 ] ] PUNCT cana-2728 44 25 ( ( PUNCT cana-2728 44 26 βˆ‡π‘‹1 βˆ‡π‘‹1 NOUN cana-2728 44 27 πœ™ πœ™ X cana-2728 44 28 ) ) PUNCT cana-2728 44 29 = = SYM cana-2728 44 30 βˆ’π‘”(πœ™π‘‹1 βˆ’π‘”(πœ™π‘‹1 NOUN cana-2728 44 31 , , PUNCT cana-2728 44 32 𝑋2)πœ‰ 𝑋2)πœ‰ PROPN cana-2728 44 33 βˆ’ βˆ’ PROPN cana-2728 44 34 πœ‚(𝑋2)πœ™π‘‹1 πœ‚(𝑋2)πœ™π‘‹1 PROPN cana-2728 44 35 , , PUNCT cana-2728 44 36 ( ( PUNCT cana-2728 44 37 2.5 2.5 NUM cana-2728 44 38 ) ) PUNCT cana-2728 44 39 for for ADP cana-2728 44 40 any any DET cana-2728 44 41 vector vector NOUN cana-2728 44 42 fields field VERB cana-2728 44 43 𝑋1 𝑋1 PROPN cana-2728 44 44 and and CCONJ cana-2728 44 45 𝑋2 𝑋2 VERB cana-2728 44 46 on on ADP cana-2728 44 47 𝑀 𝑀 PROPN cana-2728 44 48 and and CCONJ cana-2728 44 49 βˆ‡ βˆ‡ PROPN cana-2728 44 50 is be AUX cana-2728 44 51 the the DET cana-2728 44 52 operator operator NOUN cana-2728 44 53 of of ADP cana-2728 44 54 covariant covariant ADJ cana-2728 44 55 differentiation differentiation NOUN cana-2728 44 56 with with ADP cana-2728 44 57 respect respect NOUN cana-2728 44 58 to to ADP cana-2728 44 59 the the DET cana-2728 44 60 lorentzian lorentzian ADJ cana-2728 44 61 metric metric ADJ cana-2728 44 62 𝑔. 𝑔. NOUN cana-2728 44 63 in in ADP cana-2728 44 64 lorentzian lorentzian ADJ cana-2728 44 65 para para PROPN cana-2728 44 66 - - PUNCT cana-2728 44 67 kenmotsu kenmotsu NOUN cana-2728 44 68 manifolds manifold NOUN cana-2728 44 69 , , PUNCT cana-2728 44 70 the the DET cana-2728 44 71 following follow VERB cana-2728 44 72 relations relation NOUN cana-2728 44 73 hold hold VERB cana-2728 44 74 [ [ X cana-2728 44 75 11 11 NUM cana-2728 44 76 ] ] PUNCT cana-2728 44 77 : : PUNCT cana-2728 44 78 βˆ‡π‘‹1 βˆ‡π‘‹1 NOUN cana-2728 44 79 πœ‰ πœ‰ ADP cana-2728 44 80 = = VERB cana-2728 44 81 βˆ’π‘‹1 βˆ’π‘‹1 PROPN cana-2728 44 82 βˆ’ βˆ’ PROPN cana-2728 45 1 πœ‚(𝑋1)πœ‰ πœ‚(𝑋1)πœ‰ NOUN cana-2728 45 2 , , PUNCT cana-2728 45 3 ( ( PUNCT cana-2728 45 4 2.6 2.6 NUM cana-2728 45 5 ) ) PUNCT cana-2728 45 6 and and CCONJ cana-2728 45 7 ( ( PUNCT cana-2728 45 8 βˆ‡π‘‹1 βˆ‡π‘‹1 NOUN cana-2728 45 9 πœ‚)𝑋2 πœ‚)𝑋2 NOUN cana-2728 45 10 = = PUNCT cana-2728 45 11 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 X cana-2728 45 12 , , PUNCT cana-2728 45 13 𝑋2 𝑋2 VERB cana-2728 45 14 ) ) PUNCT cana-2728 45 15 βˆ’ βˆ’ NUM cana-2728 46 1 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 NOUN cana-2728 46 2 ) ) PUNCT cana-2728 46 3 . . PUNCT cana-2728 47 1 ( ( PUNCT cana-2728 47 2 2.7 2.7 NUM cana-2728 47 3 ) ) PUNCT cana-2728 47 4 in in ADP cana-2728 47 5 addition addition NOUN cana-2728 47 6 to to ADP cana-2728 47 7 these these PRON cana-2728 47 8 , , PUNCT cana-2728 47 9 the the DET cana-2728 47 10 following follow VERB cana-2728 47 11 relations relation NOUN cana-2728 47 12 also also ADV cana-2728 47 13 hold hold VERB cana-2728 47 14 [ [ X cana-2728 47 15 6 6 NUM cana-2728 47 16 ] ] X cana-2728 47 17 : : PUNCT cana-2728 47 18 πœ‚(𝑅(𝑋1 πœ‚(𝑅(𝑋1 NOUN cana-2728 47 19 , , PUNCT cana-2728 47 20 𝑋2 𝑋2 VERB cana-2728 47 21 ) ) PUNCT cana-2728 47 22 𝑋3 𝑋3 NOUN cana-2728 47 23 ) ) PUNCT cana-2728 48 1 = = SYM cana-2728 48 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 48 3 , , PUNCT cana-2728 48 4 𝑋3 𝑋3 NOUN cana-2728 48 5 ) ) PUNCT cana-2728 48 6 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 48 7 ) ) PUNCT cana-2728 49 1 βˆ’ βˆ’ PROPN cana-2728 49 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 49 3 , , PUNCT cana-2728 49 4 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 49 5 ) ) PUNCT cana-2728 49 6 , , PUNCT cana-2728 49 7 ( ( PUNCT cana-2728 49 8 2.8 2.8 NUM cana-2728 49 9 ) ) PUNCT cana-2728 49 10 𝑅(πœ‰ 𝑅(πœ‰ PROPN cana-2728 49 11 , , PUNCT cana-2728 49 12 𝑋1 𝑋1 PROPN cana-2728 49 13 ) ) PUNCT cana-2728 49 14 𝑋2 𝑋2 VERB cana-2728 49 15 = = SYM cana-2728 49 16 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 49 17 , , PUNCT cana-2728 49 18 𝑋2 𝑋2 VERB cana-2728 49 19 ) ) PUNCT cana-2728 49 20 πœ‰ πœ‰ PART cana-2728 49 21 βˆ’ βˆ’ PROPN cana-2728 49 22 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 49 23 ) ) PUNCT cana-2728 50 1 𝑋2 𝑋2 VERB cana-2728 50 2 , , PUNCT cana-2728 50 3 ( ( PUNCT cana-2728 50 4 2.9 2.9 NUM cana-2728 50 5 ) ) PUNCT cana-2728 50 6 𝑅(𝑋1 𝑅(𝑋1 NUM cana-2728 50 7 , , PUNCT cana-2728 50 8 𝑋2 𝑋2 VERB cana-2728 50 9 ) ) PUNCT cana-2728 50 10 πœ‰ πœ‰ PROPN cana-2728 50 11 = = SYM cana-2728 50 12 πœ‚(𝑋2 πœ‚(𝑋2 PROPN cana-2728 50 13 ) ) PUNCT cana-2728 50 14 𝑋1 𝑋1 PROPN cana-2728 50 15 βˆ’ βˆ’ PROPN cana-2728 50 16 πœ‚(𝑋1)𝑋2 πœ‚(𝑋1)𝑋2 PROPN cana-2728 50 17 , , PUNCT cana-2728 50 18 ( ( PUNCT cana-2728 50 19 2.10 2.10 NUM cana-2728 50 20 ) ) PUNCT cana-2728 50 21 𝑆(𝑋1 𝑆(𝑋1 X cana-2728 50 22 , , PUNCT cana-2728 50 23 πœ‰ πœ‰ NOUN cana-2728 50 24 ) ) PUNCT cana-2728 50 25 = = SYM cana-2728 50 26 2π‘›πœ‚(𝑋1 2π‘›πœ‚(𝑋1 NUM cana-2728 50 27 ) ) PUNCT cana-2728 50 28 , , PUNCT cana-2728 50 29 𝑄(πœ‰ 𝑄(πœ‰ NUM cana-2728 50 30 ) ) PUNCT cana-2728 50 31 = = SYM cana-2728 50 32 2𝑛 2𝑛 NUM cana-2728 50 33 , , PUNCT cana-2728 50 34 ( ( PUNCT cana-2728 50 35 2.11 2.11 NUM cana-2728 50 36 ) ) PUNCT cana-2728 50 37 communications communication NOUN cana-2728 50 38 on on ADP cana-2728 50 39 applied apply VERB cana-2728 50 40 nonlinear nonlinear ADJ cana-2728 50 41 analysis analysis NOUN cana-2728 50 42 issn issn NOUN cana-2728 50 43 : : PUNCT cana-2728 50 44 1074 1074 NUM cana-2728 50 45 - - PUNCT cana-2728 50 46 133x 133x NUM cana-2728 50 47 vol vol NOUN cana-2728 50 48 32 32 NUM cana-2728 51 1 no no NOUN cana-2728 51 2 . . PUNCT cana-2728 52 1 3s 3s NUM cana-2728 52 2 ( ( PUNCT cana-2728 52 3 2025 2025 NUM cana-2728 52 4 ) ) PUNCT cana-2728 52 5 701 701 NUM cana-2728 52 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 52 7 𝑆(πœ™π‘‹1 𝑆(πœ™π‘‹1 PROPN cana-2728 52 8 , , PUNCT cana-2728 52 9 πœ™π‘‹2 πœ™π‘‹2 PROPN cana-2728 52 10 ) ) PUNCT cana-2728 52 11 = = SYM cana-2728 53 1 𝑆(𝑋1 𝑆(𝑋1 PROPN cana-2728 53 2 , , PUNCT cana-2728 53 3 𝑋2 𝑋2 VERB cana-2728 53 4 ) ) PUNCT cana-2728 54 1 + + CCONJ cana-2728 54 2 2π‘›πœ‚(𝑋1)πœ‚(𝑋2 2π‘›πœ‚(𝑋1)πœ‚(𝑋2 NUM cana-2728 54 3 ) ) PUNCT cana-2728 54 4 , , PUNCT cana-2728 54 5 ( ( PUNCT cana-2728 54 6 2.12 2.12 NUM cana-2728 54 7 ) ) PUNCT cana-2728 54 8 for for ADP cana-2728 54 9 all all DET cana-2728 54 10 vector vector NOUN cana-2728 54 11 fields field NOUN cana-2728 54 12 𝑋1 𝑋1 PROPN cana-2728 54 13 and and CCONJ cana-2728 54 14 𝑋2 𝑋2 VERB cana-2728 54 15 on on ADP cana-2728 54 16 𝑀. 𝑀. PROPN cana-2728 54 17 let let NOUN cana-2728 54 18 ( ( PUNCT cana-2728 54 19 𝑔 𝑔 X cana-2728 54 20 , , PUNCT cana-2728 54 21 𝑉 𝑉 PROPN cana-2728 54 22 , , PUNCT cana-2728 54 23 πœ† πœ† NOUN cana-2728 54 24 ) ) PUNCT cana-2728 54 25 be be AUX cana-2728 54 26 a a DET cana-2728 54 27 ricci ricci NOUN cana-2728 54 28 soliton soliton NOUN cana-2728 54 29 in in ADP cana-2728 54 30 a a DET cana-2728 54 31 ( ( PUNCT cana-2728 54 32 2𝑛 2𝑛 NOUN cana-2728 54 33 + + CCONJ cana-2728 54 34 1)-dimensional 1)-dimensional NUM cana-2728 54 35 lorentzian lorentzian ADJ cana-2728 54 36 para para NOUN cana-2728 54 37 - - PUNCT cana-2728 54 38 kenmotsu kenmotsu NOUN cana-2728 54 39 manifold manifold PROPN cana-2728 54 40 𝑀. 𝑀. PROPN cana-2728 54 41 then then ADV cana-2728 54 42 , , PUNCT cana-2728 54 43 we we PRON cana-2728 54 44 have have VERB cana-2728 54 45 ( ( PUNCT cana-2728 54 46 ℒξ𝑔)(𝑋1 ℒξ𝑔)(𝑋1 PROPN cana-2728 54 47 , , PUNCT cana-2728 54 48 𝑋2 𝑋2 ADJ cana-2728 54 49 ) ) PUNCT cana-2728 55 1 = = SYM cana-2728 55 2 𝑔(βˆ‡πœ‰ 𝑔(βˆ‡πœ‰ NUM cana-2728 55 3 𝑋1 𝑋1 PROPN cana-2728 55 4 , , PUNCT cana-2728 55 5 𝑋2 𝑋2 VERB cana-2728 55 6 ) ) PUNCT cana-2728 56 1 + + CCONJ cana-2728 56 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 56 3 , , PUNCT cana-2728 56 4 βˆ‡πœ‰π‘‹2 βˆ‡πœ‰π‘‹2 NUM cana-2728 56 5 ) ) PUNCT cana-2728 56 6 . . PUNCT cana-2728 57 1 ( ( PUNCT cana-2728 57 2 2.13 2.13 NUM cana-2728 57 3 ) ) PUNCT cana-2728 57 4 using use VERB cana-2728 57 5 [ [ X cana-2728 57 6 2.6 2.6 NUM cana-2728 57 7 ] ] PUNCT cana-2728 57 8 in in ADP cana-2728 57 9 [ [ X cana-2728 57 10 2.13 2.13 NUM cana-2728 57 11 ] ] PUNCT cana-2728 57 12 , , PUNCT cana-2728 57 13 we we PRON cana-2728 57 14 have have VERB cana-2728 57 15 ( ( PUNCT cana-2728 57 16 ℒξ𝑔)(𝑋1 ℒξ𝑔)(𝑋1 PROPN cana-2728 57 17 , , PUNCT cana-2728 57 18 𝑋2 𝑋2 ADJ cana-2728 57 19 ) ) PUNCT cana-2728 57 20 = = SYM cana-2728 57 21 βˆ’2𝑔(𝑋1 βˆ’2𝑔(𝑋1 ADJ cana-2728 57 22 , , PUNCT cana-2728 57 23 𝑋2 𝑋2 VERB cana-2728 57 24 ) ) PUNCT cana-2728 58 1 βˆ’ βˆ’ PROPN cana-2728 58 2 2πœ‚(𝑋1)πœ‚(𝑋2 2πœ‚(𝑋1)πœ‚(𝑋2 NUM cana-2728 58 3 ) ) PUNCT cana-2728 58 4 . . PUNCT cana-2728 59 1 ( ( PUNCT cana-2728 59 2 2.14 2.14 NUM cana-2728 59 3 ) ) PUNCT cana-2728 59 4 from from ADP cana-2728 59 5 [ [ X cana-2728 59 6 1.1 1.1 NUM cana-2728 59 7 ] ] PUNCT cana-2728 59 8 and and CCONJ cana-2728 59 9 [ [ X cana-2728 59 10 2.14 2.14 NUM cana-2728 59 11 ] ] PUNCT cana-2728 59 12 , , PUNCT cana-2728 59 13 we we PRON cana-2728 59 14 get get VERB cana-2728 59 15 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 59 16 , , PUNCT cana-2728 59 17 𝑋2 𝑋2 ADJ cana-2728 59 18 ) ) PUNCT cana-2728 60 1 = = PUNCT cana-2728 60 2 ( ( PUNCT cana-2728 60 3 1 1 NUM cana-2728 60 4 βˆ’ βˆ’ NOUN cana-2728 60 5 πœ†)𝑔(𝑋1 πœ†)𝑔(𝑋1 NOUN cana-2728 60 6 , , PUNCT cana-2728 60 7 𝑋2 𝑋2 VERB cana-2728 60 8 ) ) PUNCT cana-2728 61 1 + + CCONJ cana-2728 61 2 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 NOUN cana-2728 61 3 ) ) PUNCT cana-2728 61 4 , , PUNCT cana-2728 61 5 ( ( PUNCT cana-2728 61 6 2.15 2.15 NUM cana-2728 61 7 ) ) PUNCT cana-2728 61 8 𝑄𝑋1 𝑄𝑋1 PROPN cana-2728 61 9 = = SYM cana-2728 61 10 ( ( PUNCT cana-2728 61 11 1 1 NUM cana-2728 61 12 βˆ’ βˆ’ NOUN cana-2728 61 13 πœ†)𝑋1 πœ†)𝑋1 PRON cana-2728 62 1 + + X cana-2728 62 2 πœ‚(𝑋1)πœ‰ πœ‚(𝑋1)πœ‰ PROPN cana-2728 62 3 , , PUNCT cana-2728 62 4 π‘Ÿ π‘Ÿ X cana-2728 62 5 = = SYM cana-2728 62 6 2𝑛 2𝑛 PROPN cana-2728 63 1 βˆ’ βˆ’ PROPN cana-2728 63 2 πœ†(2𝑛 πœ†(2𝑛 SYM cana-2728 63 3 + + NOUN cana-2728 63 4 1 1 NUM cana-2728 63 5 ) ) PUNCT cana-2728 63 6 . . PUNCT cana-2728 64 1 ( ( PUNCT cana-2728 64 2 2.16 2.16 NUM cana-2728 64 3 ) ) PUNCT cana-2728 64 4 in in ADP cana-2728 64 5 view view NOUN cana-2728 64 6 of of ADP cana-2728 64 7 [ [ X cana-2728 64 8 2.3 2.3 NUM cana-2728 64 9 ] ] PUNCT cana-2728 64 10 and and CCONJ cana-2728 64 11 [ [ X cana-2728 64 12 2.15 2.15 NUM cana-2728 64 13 ] ] PUNCT cana-2728 64 14 , , PUNCT cana-2728 64 15 we we PRON cana-2728 64 16 have have VERB cana-2728 64 17 𝑆(𝑋1 𝑆(𝑋1 NOUN cana-2728 64 18 , , PUNCT cana-2728 64 19 πœ‰ πœ‰ NOUN cana-2728 64 20 ) ) PUNCT cana-2728 64 21 = = SYM cana-2728 64 22 βˆ’πœ†πœ‚(𝑋1 βˆ’πœ†πœ‚(𝑋1 NUM cana-2728 64 23 ) ) PUNCT cana-2728 64 24 , , PUNCT cana-2728 64 25 ( ( PUNCT cana-2728 64 26 2.17 2.17 NUM cana-2728 64 27 ) ) PUNCT cana-2728 65 1 π‘„πœ‰ π‘„πœ‰ PROPN cana-2728 65 2 = = SYM cana-2728 65 3 βˆ’πœ†πœ‰. βˆ’πœ†πœ‰. X cana-2728 65 4 ( ( PUNCT cana-2728 65 5 2.18 2.18 NUM cana-2728 65 6 ) ) PUNCT cana-2728 65 7 definition definition NOUN cana-2728 65 8 2.1 2.1 NUM cana-2728 65 9 [ [ X cana-2728 65 10 7 7 NUM cana-2728 65 11 ] ] X cana-2728 65 12 : : PUNCT cana-2728 65 13 a a PRON cana-2728 65 14 ( ( PUNCT cana-2728 65 15 2𝑛 2𝑛 NOUN cana-2728 65 16 + + CCONJ cana-2728 65 17 1)-dimensional 1)-dimensional NUM cana-2728 65 18 lorentzian lorentzian ADJ cana-2728 65 19 para para NOUN cana-2728 65 20 - - PUNCT cana-2728 65 21 kenmotsu kenmotsu PROPN cana-2728 65 22 manifold manifold PROPN cana-2728 65 23 𝑀 𝑀 PROPN cana-2728 65 24 is be AUX cana-2728 65 25 called call VERB cana-2728 65 26 an an DET cana-2728 65 27 πœ‚einstein πœ‚einstein ADV cana-2728 65 28 manifold manifold ADJ cana-2728 65 29 if if SCONJ cana-2728 65 30 its its PRON cana-2728 65 31 ricci ricci PROPN cana-2728 65 32 tensor tensor NOUN cana-2728 65 33 𝑆 𝑆 PROPN cana-2728 65 34 satisfies satisfy VERB cana-2728 65 35 the the DET cana-2728 65 36 following follow VERB cana-2728 65 37 equation equation NOUN cana-2728 65 38 : : PUNCT cana-2728 65 39 𝑆(𝑋1 𝑆(𝑋1 VERB cana-2728 65 40 , , PUNCT cana-2728 65 41 𝑋2 𝑋2 VERB cana-2728 65 42 ) ) PUNCT cana-2728 65 43 = = SYM cana-2728 65 44 𝛼𝑔(𝑋1 𝛼𝑔(𝑋1 PROPN cana-2728 65 45 , , PUNCT cana-2728 65 46 𝑋2 𝑋2 VERB cana-2728 65 47 ) ) PUNCT cana-2728 66 1 + + NUM cana-2728 66 2 π›½πœ‚(𝑋1)πœ‚(𝑋2 π›½πœ‚(𝑋1)πœ‚(𝑋2 NUM cana-2728 66 3 ) ) PUNCT cana-2728 66 4 , , PUNCT cana-2728 66 5 ( ( PUNCT cana-2728 66 6 2.19 2.19 NUM cana-2728 66 7 ) ) PUNCT cana-2728 66 8 where where SCONJ cana-2728 66 9 𝛼 𝛼 X cana-2728 66 10 and and CCONJ cana-2728 66 11 𝛽 𝛽 PROPN cana-2728 66 12 are be AUX cana-2728 66 13 scalars scalar NOUN cana-2728 66 14 . . PUNCT cana-2728 67 1 2 2 X cana-2728 67 2 . . X cana-2728 67 3 parallel parallel ADJ cana-2728 67 4 symmetric symmetric ADJ cana-2728 67 5 second second ADJ cana-2728 67 6 order order NOUN cana-2728 67 7 tensor tensor NOUN cana-2728 67 8 and and CCONJ cana-2728 67 9 ricci ricci PROPN cana-2728 67 10 solitons soliton NOUN cana-2728 67 11 in in ADP cana-2728 67 12 lorentzian lorentzian ADJ cana-2728 67 13 parakenmotsu parakenmotsu NOUN cana-2728 67 14 manifolds manifold NOUN cana-2728 67 15 in in ADP cana-2728 67 16 this this DET cana-2728 67 17 section section NOUN cana-2728 67 18 , , PUNCT cana-2728 67 19 we we PRON cana-2728 67 20 study study VERB cana-2728 67 21 parallel parallel ADJ cana-2728 67 22 symmetric symmetric ADJ cana-2728 67 23 second second ADJ cana-2728 67 24 order order NOUN cana-2728 67 25 tensor tensor NOUN cana-2728 67 26 and and CCONJ cana-2728 67 27 ricci ricci PROPN cana-2728 67 28 solitons soliton NOUN cana-2728 67 29 in in ADP cana-2728 67 30 lorentzian lorentzian ADJ cana-2728 67 31 parakenmotsu parakenmotsu NOUN cana-2728 67 32 manifolds manifolds PROPN cana-2728 67 33 . . PUNCT cana-2728 68 1 theorem theorem VERB cana-2728 68 2 3.1 3.1 NUM cana-2728 68 3 : : PUNCT cana-2728 68 4 a a DET cana-2728 68 5 symmetric symmetric ADJ cana-2728 68 6 parallel parallel ADJ cana-2728 68 7 second second ADJ cana-2728 68 8 order order NOUN cana-2728 68 9 covariant covariant ADJ cana-2728 68 10 tensor tensor NOUN cana-2728 68 11 in in ADP cana-2728 68 12 a a DET cana-2728 68 13 lorentzian lorentzian ADJ cana-2728 68 14 para para NOUN cana-2728 68 15 - - PUNCT cana-2728 68 16 kenmotsu kenmotsu PROPN cana-2728 68 17 manifold manifold NOUN cana-2728 68 18 is be AUX cana-2728 68 19 a a DET cana-2728 68 20 constant constant ADJ cana-2728 68 21 multiple multiple NOUN cana-2728 68 22 of of ADP cana-2728 68 23 the the DET cana-2728 68 24 metric metric ADJ cana-2728 68 25 tensor tensor NOUN cana-2728 68 26 . . PUNCT cana-2728 69 1 proof proof NOUN cana-2728 69 2 : : PUNCT cana-2728 69 3 let let VERB cana-2728 69 4 β„Ž β„Ž PART cana-2728 69 5 be be AUX cana-2728 69 6 a a DET cana-2728 69 7 symmetric symmetric ADJ cana-2728 69 8 tensor tensor NOUN cana-2728 69 9 field field NOUN cana-2728 69 10 of of ADP cana-2728 69 11 ( ( PUNCT cana-2728 69 12 0,2)-type 0,2)-type NUM cana-2728 69 13 which which PRON cana-2728 69 14 is be AUX cana-2728 69 15 parallel parallel VERB cana-2728 69 16 with with ADP cana-2728 69 17 respect respect NOUN cana-2728 69 18 to to ADP cana-2728 69 19 βˆ‡ βˆ‡ PROPN cana-2728 69 20 that that PRON cana-2728 69 21 is be AUX cana-2728 69 22 βˆ‡β„Ž βˆ‡β„Ž PROPN cana-2728 69 23 = = SYM cana-2728 69 24 0 0 NUM cana-2728 69 25 . . PUNCT cana-2728 70 1 then then ADV cana-2728 70 2 by by ADP cana-2728 70 3 applying apply VERB cana-2728 70 4 the the DET cana-2728 70 5 ricci ricci NOUN cana-2728 70 6 identity identity NOUN cana-2728 70 7 [ [ X cana-2728 70 8 8 8 NUM cana-2728 70 9 ] ] PUNCT cana-2728 70 10 , , PUNCT cana-2728 70 11 we we PRON cana-2728 70 12 obtain obtain VERB cana-2728 70 13 𝛻2β„Ž(𝑋1 𝛻2β„Ž(𝑋1 PROPN cana-2728 70 14 , , PUNCT cana-2728 70 15 𝑋2 𝑋2 VERB cana-2728 70 16 ; ; PUNCT cana-2728 70 17 𝑋3 𝑋3 NOUN cana-2728 70 18 , , PUNCT cana-2728 70 19 𝑋4 𝑋4 VERB cana-2728 70 20 ) ) PUNCT cana-2728 70 21 βˆ’ βˆ’ PROPN cana-2728 70 22 𝛻2β„Ž(𝑋1 𝛻2β„Ž(𝑋1 PROPN cana-2728 70 23 , , PUNCT cana-2728 70 24 𝑋2 𝑋2 VERB cana-2728 70 25 ; ; PUNCT cana-2728 70 26 𝑋4 𝑋4 NOUN cana-2728 70 27 , , PUNCT cana-2728 70 28 𝑋3 𝑋3 NOUN cana-2728 70 29 ) ) PUNCT cana-2728 70 30 = = SYM cana-2728 70 31 0 0 NUM cana-2728 70 32 , , PUNCT cana-2728 70 33 ( ( PUNCT cana-2728 70 34 3.1 3.1 NUM cana-2728 70 35 ) ) PUNCT cana-2728 70 36 which which PRON cana-2728 70 37 implies imply VERB cana-2728 70 38 that that SCONJ cana-2728 70 39 β„Ž(𝑅(𝑋1 β„Ž(𝑅(𝑋1 PROPN cana-2728 70 40 , , PUNCT cana-2728 70 41 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 70 42 , , PUNCT cana-2728 70 43 𝑋4 𝑋4 VERB cana-2728 70 44 ) ) PUNCT cana-2728 71 1 + + CCONJ cana-2728 71 2 β„Ž(𝑋3 β„Ž(𝑋3 PROPN cana-2728 71 3 , , PUNCT cana-2728 71 4 𝑅(𝑋1 𝑅(𝑋1 ADJ cana-2728 71 5 , , PUNCT cana-2728 71 6 𝑋2)𝑋4 𝑋2)𝑋4 ADJ cana-2728 71 7 ) ) PUNCT cana-2728 71 8 = = SYM cana-2728 71 9 0 0 X cana-2728 71 10 . . PUNCT cana-2728 72 1 ( ( PUNCT cana-2728 72 2 3.2 3.2 NUM cana-2728 72 3 ) ) PUNCT cana-2728 72 4 replacing replace VERB cana-2728 72 5 𝑋3 𝑋3 NOUN cana-2728 72 6 = = SYM cana-2728 72 7 𝑋4 𝑋4 VERB cana-2728 72 8 = = PUNCT cana-2728 72 9 πœ‰ πœ‰ X cana-2728 72 10 in in ADP cana-2728 72 11 ( ( PUNCT cana-2728 72 12 3.2 3.2 NUM cana-2728 72 13 ) ) PUNCT cana-2728 72 14 and and CCONJ cana-2728 72 15 using use VERB cana-2728 72 16 ( ( PUNCT cana-2728 72 17 2.10 2.10 NUM cana-2728 72 18 ) ) PUNCT cana-2728 72 19 and and CCONJ cana-2728 72 20 the the DET cana-2728 72 21 symmetry symmetry NOUN cana-2728 72 22 of of ADP cana-2728 72 23 β„Ž β„Ž PROPN cana-2728 72 24 , , PUNCT cana-2728 72 25 we we PRON cana-2728 72 26 get get VERB cana-2728 72 27 2[πœ‚(𝑋2)β„Ž(𝑋1 2[πœ‚(𝑋2)β„Ž(𝑋1 NUM cana-2728 72 28 , , PUNCT cana-2728 72 29 πœ‰ πœ‰ NOUN cana-2728 72 30 ) ) PUNCT cana-2728 72 31 βˆ’ βˆ’ NOUN cana-2728 72 32 πœ‚(𝑋1)β„Ž(𝑋2 πœ‚(𝑋1)β„Ž(𝑋2 NOUN cana-2728 72 33 , , PUNCT cana-2728 72 34 πœ‰ πœ‰ NOUN cana-2728 72 35 ) ) PUNCT cana-2728 72 36 ] ] PUNCT cana-2728 73 1 = = PUNCT cana-2728 73 2 0 0 X cana-2728 73 3 . . PUNCT cana-2728 74 1 ( ( PUNCT cana-2728 74 2 3.3 3.3 NUM cana-2728 74 3 ) ) PUNCT cana-2728 74 4 putting put VERB cana-2728 74 5 𝑋1 𝑋1 NOUN cana-2728 74 6 = = PUNCT cana-2728 74 7 πœ‰ πœ‰ NOUN cana-2728 74 8 in in ADP cana-2728 74 9 ( ( PUNCT cana-2728 74 10 3.3 3.3 NUM cana-2728 74 11 ) ) PUNCT cana-2728 74 12 , , PUNCT cana-2728 74 13 we we PRON cana-2728 74 14 have have VERB cana-2728 74 15 communications communication NOUN cana-2728 74 16 on on ADP cana-2728 74 17 applied apply VERB cana-2728 74 18 nonlinear nonlinear ADJ cana-2728 74 19 analysis analysis NOUN cana-2728 74 20 issn issn NOUN cana-2728 74 21 : : PUNCT cana-2728 74 22 1074 1074 NUM cana-2728 74 23 - - PUNCT cana-2728 74 24 133x 133x NUM cana-2728 74 25 vol vol NOUN cana-2728 74 26 32 32 NUM cana-2728 75 1 no no NOUN cana-2728 75 2 . . PUNCT cana-2728 76 1 3s 3s NUM cana-2728 76 2 ( ( PUNCT cana-2728 76 3 2025 2025 NUM cana-2728 76 4 ) ) PUNCT cana-2728 76 5 702 702 NUM cana-2728 76 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 76 7 β„Ž(𝑋2 β„Ž(𝑋2 PROPN cana-2728 76 8 , , PUNCT cana-2728 76 9 πœ‰ πœ‰ NOUN cana-2728 76 10 ) ) PUNCT cana-2728 76 11 = = SYM cana-2728 76 12 βˆ’πœ‚(𝑋2)β„Ž(πœ‰ βˆ’πœ‚(𝑋2)β„Ž(πœ‰ NOUN cana-2728 76 13 , , PUNCT cana-2728 76 14 πœ‰ πœ‰ NOUN cana-2728 76 15 ) ) PUNCT cana-2728 76 16 . . PUNCT cana-2728 77 1 ( ( PUNCT cana-2728 77 2 3.4 3.4 NUM cana-2728 77 3 ) ) PUNCT cana-2728 77 4 now now ADV cana-2728 77 5 , , PUNCT cana-2728 77 6 differentiating differentiate VERB cana-2728 77 7 ( ( PUNCT cana-2728 77 8 3.4 3.4 NUM cana-2728 77 9 ) ) PUNCT cana-2728 77 10 covariantly covariantly ADV cana-2728 77 11 with with ADP cana-2728 77 12 respect respect NOUN cana-2728 77 13 to to ADP cana-2728 77 14 𝑋1 𝑋1 PROPN cana-2728 77 15 , , PUNCT cana-2728 77 16 we we PRON cana-2728 77 17 obtain obtain VERB cana-2728 77 18 ( ( PUNCT cana-2728 77 19 𝛻𝑋1 𝛻𝑋1 PROPN cana-2728 77 20 β„Ž)(𝑋2 β„Ž)(𝑋2 PROPN cana-2728 77 21 , , PUNCT cana-2728 77 22 πœ‰ πœ‰ X cana-2728 77 23 ) ) PUNCT cana-2728 77 24 + + CCONJ cana-2728 78 1 β„Ž(𝛻𝑋1 β„Ž(𝛻𝑋1 PROPN cana-2728 78 2 𝑋2 𝑋2 VERB cana-2728 78 3 , , PUNCT cana-2728 78 4 πœ‰ πœ‰ X cana-2728 78 5 ) ) PUNCT cana-2728 78 6 + + X cana-2728 79 1 β„Ž(𝑋2 β„Ž(𝑋2 PROPN cana-2728 79 2 , , PUNCT cana-2728 79 3 𝛻𝑋1 𝛻𝑋1 PROPN cana-2728 79 4 πœ‰ πœ‰ PROPN cana-2728 79 5 ) ) PUNCT cana-2728 79 6 = = SYM cana-2728 79 7 βˆ’[(𝛻𝑋1 βˆ’[(𝛻𝑋1 PROPN cana-2728 79 8 πœ‚)(𝑋2 πœ‚)(𝑋2 PROPN cana-2728 79 9 ) ) PUNCT cana-2728 80 1 + + CCONJ cana-2728 80 2 πœ‚(𝛻𝑋1 πœ‚(𝛻𝑋1 PUNCT cana-2728 80 3 𝑋2)]β„Ž(πœ‰ 𝑋2)]β„Ž(πœ‰ NOUN cana-2728 80 4 , , PUNCT cana-2728 80 5 πœ‰ πœ‰ NOUN cana-2728 80 6 ) ) PUNCT cana-2728 80 7 + + ADJ cana-2728 80 8 πœ‚(𝑋2)[(𝛻𝑋1 πœ‚(𝑋2)[(𝛻𝑋1 ADJ cana-2728 80 9 β„Ž)(πœ‰ β„Ž)(πœ‰ ADJ cana-2728 80 10 , , PUNCT cana-2728 80 11 πœ‰ πœ‰ NOUN cana-2728 80 12 ) ) PUNCT cana-2728 81 1 + + CCONJ cana-2728 81 2 2β„Ž(𝛻𝑋1 2β„Ž(𝛻𝑋1 NUM cana-2728 81 3 πœ‰ πœ‰ SYM cana-2728 81 4 , , PUNCT cana-2728 81 5 πœ‰ πœ‰ NOUN cana-2728 81 6 ) ) PUNCT cana-2728 81 7 ] ] PUNCT cana-2728 81 8 . . PUNCT cana-2728 82 1 ( ( PUNCT cana-2728 82 2 3.5 3.5 NUM cana-2728 82 3 ) ) PUNCT cana-2728 82 4 using use VERB cana-2728 82 5 ( ( PUNCT cana-2728 82 6 2.6 2.6 NUM cana-2728 82 7 ) ) PUNCT cana-2728 82 8 , , PUNCT cana-2728 82 9 ( ( PUNCT cana-2728 82 10 3.4 3.4 NUM cana-2728 82 11 ) ) PUNCT cana-2728 82 12 and and CCONJ cana-2728 82 13 the the DET cana-2728 82 14 parallel parallel ADJ cana-2728 82 15 condition condition NOUN cana-2728 82 16 βˆ‡β„Ž βˆ‡β„Ž PROPN cana-2728 82 17 = = SYM cana-2728 82 18 0 0 NUM cana-2728 82 19 in in ADP cana-2728 82 20 ( ( PUNCT cana-2728 82 21 3.5 3.5 NUM cana-2728 82 22 ) ) PUNCT cana-2728 82 23 , , PUNCT cana-2728 82 24 we we PRON cana-2728 82 25 get get VERB cana-2728 82 26 β„Ž(𝑋2 β„Ž(𝑋2 PROPN cana-2728 82 27 , , PUNCT cana-2728 82 28 𝛻𝑋1 𝛻𝑋1 PROPN cana-2728 82 29 πœ‰ πœ‰ PROPN cana-2728 82 30 ) ) PUNCT cana-2728 82 31 = = SYM cana-2728 82 32 βˆ’(𝛻𝑋1 βˆ’(𝛻𝑋1 SYM cana-2728 82 33 πœ‚)(𝑋2)β„Ž(πœ‰ πœ‚)(𝑋2)β„Ž(πœ‰ NOUN cana-2728 82 34 , , PUNCT cana-2728 82 35 πœ‰ πœ‰ NOUN cana-2728 82 36 ) ) PUNCT cana-2728 82 37 . . PUNCT cana-2728 83 1 ( ( PUNCT cana-2728 83 2 3.6 3.6 NUM cana-2728 83 3 ) ) PUNCT cana-2728 83 4 in in ADP cana-2728 83 5 consequences consequence NOUN cana-2728 83 6 of of ADP cana-2728 83 7 ( ( PUNCT cana-2728 83 8 2.6 2.6 NUM cana-2728 83 9 ) ) PUNCT cana-2728 83 10 , , PUNCT cana-2728 83 11 ( ( PUNCT cana-2728 83 12 2.7 2.7 NUM cana-2728 83 13 ) ) PUNCT cana-2728 83 14 , , PUNCT cana-2728 83 15 ( ( PUNCT cana-2728 83 16 3.4 3.4 NUM cana-2728 83 17 ) ) PUNCT cana-2728 83 18 and and CCONJ cana-2728 83 19 ( ( PUNCT cana-2728 83 20 3.6 3.6 NUM cana-2728 83 21 ) ) PUNCT cana-2728 83 22 , , PUNCT cana-2728 83 23 it it PRON cana-2728 83 24 yields yield VERB cana-2728 83 25 β„Ž(𝑋1 β„Ž(𝑋1 PROPN cana-2728 83 26 , , PUNCT cana-2728 83 27 𝑋2 𝑋2 VERB cana-2728 83 28 ) ) PUNCT cana-2728 83 29 = = SYM cana-2728 83 30 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 ADJ cana-2728 83 31 , , PUNCT cana-2728 83 32 𝑋2)β„Ž(πœ‰ 𝑋2)β„Ž(πœ‰ NOUN cana-2728 83 33 , , PUNCT cana-2728 83 34 πœ‰ πœ‰ NOUN cana-2728 83 35 ) ) PUNCT cana-2728 83 36 . . PUNCT cana-2728 84 1 ( ( PUNCT cana-2728 84 2 3.7 3.7 NUM cana-2728 84 3 ) ) PUNCT cana-2728 84 4 from from ADP cana-2728 84 5 the the DET cana-2728 84 6 above above ADJ cana-2728 84 7 ( ( PUNCT cana-2728 84 8 3.7 3.7 NUM cana-2728 84 9 ) ) PUNCT cana-2728 84 10 and and CCONJ cana-2728 84 11 ( ( PUNCT cana-2728 84 12 3.4 3.4 NUM cana-2728 84 13 ) ) PUNCT cana-2728 84 14 , , PUNCT cana-2728 84 15 we we PRON cana-2728 84 16 can can AUX cana-2728 84 17 conclude conclude VERB cana-2728 84 18 that that PRON cana-2728 84 19 β„Ž(πœ‰ β„Ž(πœ‰ NOUN cana-2728 84 20 , , PUNCT cana-2728 84 21 πœ‰ πœ‰ X cana-2728 84 22 ) ) PUNCT cana-2728 84 23 is be AUX cana-2728 84 24 a a DET cana-2728 84 25 constant constant ADJ cana-2728 84 26 . . PUNCT cana-2728 85 1 this this PRON cana-2728 85 2 completes complete VERB cana-2728 85 3 the the DET cana-2728 85 4 proof proof NOUN cana-2728 85 5 of of ADP cana-2728 85 6 the the DET cana-2728 85 7 theorem theorem PROPN cana-2728 85 8 . . PUNCT cana-2728 85 9 theorem theorem PROPN cana-2728 85 10 3.2 3.2 NUM cana-2728 85 11 . . PUNCT cana-2728 86 1 let let VERB cana-2728 86 2 𝑀 𝑀 PRON cana-2728 86 3 be be AUX cana-2728 86 4 a a DET cana-2728 86 5 lorentzian lorentzian ADJ cana-2728 86 6 para para NOUN cana-2728 86 7 - - PUNCT cana-2728 86 8 kenmotsu kenmotsu NOUN cana-2728 86 9 manifold manifold ADJ cana-2728 86 10 . . PUNCT cana-2728 87 1 assume assume VERB cana-2728 87 2 that that SCONJ cana-2728 87 3 a a DET cana-2728 87 4 symmetric symmetric ADJ cana-2728 87 5 metric metric ADJ cana-2728 87 6 tensor tensor NOUN cana-2728 87 7 field field NOUN cana-2728 87 8 β„Ž β„Ž PROPN cana-2728 87 9 = = SYM cana-2728 87 10 ℒ𝒱 ℒ𝒱 PROPN cana-2728 87 11 𝑔 𝑔 PROPN cana-2728 87 12 + + CCONJ cana-2728 87 13 2𝑆 2𝑆 PROPN cana-2728 87 14 is be AUX cana-2728 87 15 parallel parallel ADJ cana-2728 87 16 with with ADP cana-2728 87 17 respect respect NOUN cana-2728 87 18 to to ADP cana-2728 87 19 the the DET cana-2728 87 20 levi levi PROPN cana-2728 87 21 - - PUNCT cana-2728 87 22 civita civita PROPN cana-2728 87 23 connection connection NOUN cana-2728 87 24 associated associate VERB cana-2728 87 25 with with ADP cana-2728 87 26 𝑔. 𝑔. NOUN cana-2728 87 27 then then ADV cana-2728 87 28 ( ( PUNCT cana-2728 87 29 𝑔 𝑔 X cana-2728 87 30 , , PUNCT cana-2728 87 31 𝑉 𝑉 PROPN cana-2728 87 32 , , PUNCT cana-2728 87 33 Ξ» Ξ» NOUN cana-2728 87 34 ) ) PUNCT cana-2728 87 35 yields yield VERB cana-2728 87 36 a a DET cana-2728 87 37 ricci ricci NOUN cana-2728 87 38 - - PUNCT cana-2728 87 39 soliton soliton NOUN cana-2728 87 40 on on ADP cana-2728 87 41 𝑀. 𝑀. NOUN cana-2728 87 42 proof proof NOUN cana-2728 87 43 : : PUNCT cana-2728 87 44 let let VERB cana-2728 87 45 us we PRON cana-2728 87 46 assume assume VERB cana-2728 87 47 that that SCONJ cana-2728 87 48 a a DET cana-2728 87 49 symmetric symmetric ADJ cana-2728 87 50 tensor tensor NOUN cana-2728 87 51 field field NOUN cana-2728 87 52 β„Ž β„Ž NOUN cana-2728 87 53 = = SYM cana-2728 87 54 ℒ𝒱𝑔 ℒ𝒱𝑔 PROPN cana-2728 87 55 + + CCONJ cana-2728 87 56 2𝑆 2𝑆 PROPN cana-2728 87 57 is be AUX cana-2728 87 58 parallel parallel ADJ cana-2728 87 59 with with ADP cana-2728 87 60 respect respect NOUN cana-2728 87 61 to to ADP cana-2728 87 62 the the DET cana-2728 87 63 levicivita levicivita NOUN cana-2728 87 64 connection connection NOUN cana-2728 87 65 associated associate VERB cana-2728 87 66 with with ADP cana-2728 87 67 𝑔. 𝑔. NOUN cana-2728 87 68 then then ADV cana-2728 87 69 , , PUNCT cana-2728 87 70 β„Ž(πœ‰ β„Ž(πœ‰ NOUN cana-2728 87 71 , , PUNCT cana-2728 87 72 πœ‰ πœ‰ NOUN cana-2728 87 73 ) ) PUNCT cana-2728 88 1 = = SYM cana-2728 88 2 2πœ† 2πœ† NUM cana-2728 88 3 , , PUNCT cana-2728 88 4 this this PRON cana-2728 88 5 shows show VERB cana-2728 88 6 that that SCONJ cana-2728 88 7 πœ† πœ† X cana-2728 88 8 = = SYM cana-2728 88 9 1 1 NUM cana-2728 88 10 2 2 NUM cana-2728 88 11 β„Ž(πœ‰ β„Ž(πœ‰ NOUN cana-2728 88 12 , , PUNCT cana-2728 88 13 πœ‰ πœ‰ NOUN cana-2728 88 14 ) ) PUNCT cana-2728 88 15 . . PUNCT cana-2728 89 1 now now ADV cana-2728 89 2 , , PUNCT cana-2728 89 3 as as SCONJ cana-2728 89 4 β„Ž β„Ž PROPN cana-2728 89 5 is be AUX cana-2728 89 6 parallel parallel ADJ cana-2728 89 7 with with ADP cana-2728 89 8 respect respect NOUN cana-2728 89 9 to to ADP cana-2728 89 10 𝑔 𝑔 NOUN cana-2728 89 11 , , PUNCT cana-2728 89 12 then then ADV cana-2728 89 13 from from ADP cana-2728 89 14 ( ( PUNCT cana-2728 89 15 3.7 3.7 NUM cana-2728 89 16 ) ) PUNCT cana-2728 89 17 we we PRON cana-2728 89 18 get get VERB cana-2728 89 19 𝐻(𝑋1 𝐻(𝑋1 ADJ cana-2728 89 20 , , PUNCT cana-2728 89 21 𝑋2 𝑋2 VERB cana-2728 89 22 ) ) PUNCT cana-2728 90 1 = = SYM cana-2728 91 1 βˆ’2πœ†π‘”(𝑋1 βˆ’2πœ†π‘”(𝑋1 PROPN cana-2728 91 2 , , PUNCT cana-2728 91 3 𝑋2 𝑋2 VERB cana-2728 91 4 ) ) PUNCT cana-2728 91 5 , , PUNCT cana-2728 91 6 ( ( PUNCT cana-2728 91 7 3.8 3.8 NUM cana-2728 91 8 ) ) PUNCT cana-2728 91 9 for for ADP cana-2728 91 10 all all DET cana-2728 91 11 vector vector NOUN cana-2728 91 12 fields field NOUN cana-2728 91 13 𝑋1 𝑋1 PROPN cana-2728 91 14 and and CCONJ cana-2728 91 15 𝑋2 𝑋2 VERB cana-2728 91 16 on on ADP cana-2728 91 17 𝑀 𝑀 PROPN cana-2728 91 18 , , PUNCT cana-2728 91 19 which which PRON cana-2728 91 20 leads lead VERB cana-2728 91 21 to to ADP cana-2728 91 22 ℒ𝒱𝑔(𝑋1 ℒ𝒱𝑔(𝑋1 VERB cana-2728 91 23 , , PUNCT cana-2728 91 24 𝑋2 𝑋2 VERB cana-2728 91 25 ) ) PUNCT cana-2728 91 26 = = SYM cana-2728 92 1 βˆ’2πœ†π‘”(𝑋1 βˆ’2πœ†π‘”(𝑋1 PROPN cana-2728 92 2 , , PUNCT cana-2728 92 3 𝑋2 𝑋2 VERB cana-2728 92 4 ) ) PUNCT cana-2728 92 5 βˆ’ βˆ’ ADP cana-2728 92 6 2𝑆(𝑋1 2𝑆(𝑋1 NUM cana-2728 92 7 , , PUNCT cana-2728 92 8 𝑋2 𝑋2 VERB cana-2728 92 9 ) ) PUNCT cana-2728 92 10 . . PUNCT cana-2728 93 1 ( ( PUNCT cana-2728 93 2 3.9 3.9 NUM cana-2728 93 3 ) ) PUNCT cana-2728 93 4 hence hence ADV cana-2728 93 5 , , PUNCT cana-2728 93 6 we we PRON cana-2728 93 7 complete complete VERB cana-2728 93 8 the the DET cana-2728 93 9 proof proof NOUN cana-2728 93 10 of of ADP cana-2728 93 11 the the DET cana-2728 93 12 theorem theorem PROPN cana-2728 93 13 . . PUNCT cana-2728 93 14 theorem theorem VERB cana-2728 93 15 3.3 3.3 NUM cana-2728 93 16 . . PUNCT cana-2728 94 1 a a DET cana-2728 94 2 ricci ricci PROPN cana-2728 94 3 semi semi ADJ cana-2728 94 4 - - ADJ cana-2728 94 5 symmetric symmetric ADJ cana-2728 94 6 lorentzian lorentzian ADJ cana-2728 94 7 para para NOUN cana-2728 94 8 - - PUNCT cana-2728 94 9 kenmotsu kenmotsu PROPN cana-2728 94 10 manifold manifold NOUN cana-2728 94 11 is be AUX cana-2728 94 12 an an DET cana-2728 94 13 einstein einstein ADJ cana-2728 94 14 manifold manifold NOUN cana-2728 94 15 . . PUNCT cana-2728 95 1 proof proof NOUN cana-2728 95 2 : : PUNCT cana-2728 95 3 we we PRON cana-2728 95 4 consider consider VERB cana-2728 95 5 a a DET cana-2728 95 6 ricci ricci NOUN cana-2728 95 7 semi semi ADJ cana-2728 95 8 - - ADJ cana-2728 95 9 symmetric symmetric ADJ cana-2728 95 10 lorentzian lorentzian ADJ cana-2728 95 11 para para NOUN cana-2728 95 12 - - PUNCT cana-2728 95 13 kenmotsu kenmotsu NOUN cana-2728 95 14 manifold manifold ADJ cana-2728 95 15 , , PUNCT cana-2728 95 16 i.e. i.e. X cana-2728 95 17 , , PUNCT cana-2728 95 18 𝑅 𝑅 NOUN cana-2728 95 19 β€ˆ β€ˆ SPACE cana-2728 95 20 ∘ ∘ NOUN cana-2728 95 21 𝑆 𝑆 PROPN cana-2728 95 22 = = SYM cana-2728 95 23 0 0 X cana-2728 95 24 . . PUNCT cana-2728 96 1 we we PRON cana-2728 96 2 have have VERB cana-2728 96 3 [ [ X cana-2728 96 4 8 8 NUM cana-2728 96 5 ] ] PUNCT cana-2728 96 6 , , PUNCT cana-2728 96 7 ( ( PUNCT cana-2728 96 8 𝑅(𝑋1 𝑅(𝑋1 X cana-2728 96 9 , , PUNCT cana-2728 96 10 𝑋2 𝑋2 VERB cana-2728 96 11 ) ) PUNCT cana-2728 96 12 ∘ ∘ PROPN cana-2728 96 13 β€ˆ β€ˆ SPACE cana-2728 96 14 𝑆)(𝑋3 𝑆)(𝑋3 NUM cana-2728 96 15 , , PUNCT cana-2728 96 16 𝑋4 𝑋4 VERB cana-2728 96 17 ) ) PUNCT cana-2728 96 18 = = SYM cana-2728 97 1 βˆ’π‘†(𝑅(𝑋1 βˆ’π‘†(𝑅(𝑋1 X cana-2728 97 2 , , PUNCT cana-2728 97 3 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 97 4 , , PUNCT cana-2728 97 5 𝑋4 𝑋4 VERB cana-2728 97 6 ) ) PUNCT cana-2728 98 1 βˆ’ βˆ’ PROPN cana-2728 98 2 𝑆(𝑋3 𝑆(𝑋3 PROPN cana-2728 98 3 , , PUNCT cana-2728 98 4 𝑅(𝑋1 𝑅(𝑋1 NUM cana-2728 98 5 , , PUNCT cana-2728 98 6 𝑋2)𝑋4 𝑋2)𝑋4 ADJ cana-2728 98 7 ) ) PUNCT cana-2728 98 8 . . PUNCT cana-2728 99 1 ( ( PUNCT cana-2728 99 2 3.10 3.10 NUM cana-2728 99 3 ) ) PUNCT cana-2728 99 4 putting put VERB cana-2728 99 5 𝑋1 𝑋1 NOUN cana-2728 99 6 = = SYM cana-2728 99 7 πœ‰ πœ‰ NOUN cana-2728 99 8 and and CCONJ cana-2728 99 9 using use VERB cana-2728 99 10 𝑅 𝑅 PROPN cana-2728 99 11 ∘ ∘ NOUN cana-2728 99 12 𝑆 𝑆 PROPN cana-2728 99 13 = = SYM cana-2728 99 14 0 0 NUM cana-2728 99 15 in in ADP cana-2728 99 16 ( ( PUNCT cana-2728 99 17 3.10 3.10 NUM cana-2728 99 18 ) ) PUNCT cana-2728 99 19 , , PUNCT cana-2728 99 20 we we PRON cana-2728 99 21 have have VERB cana-2728 99 22 𝑆(𝑅(πœ‰ 𝑆(𝑅(πœ‰ PROPN cana-2728 99 23 , , PUNCT cana-2728 99 24 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 99 25 , , PUNCT cana-2728 99 26 𝑋4 𝑋4 VERB cana-2728 99 27 ) ) PUNCT cana-2728 100 1 + + CCONJ cana-2728 100 2 𝑆(𝑋3 𝑆(𝑋3 ADJ cana-2728 100 3 , , PUNCT cana-2728 100 4 𝑅(πœ‰ 𝑅(πœ‰ PRON cana-2728 100 5 , , PUNCT cana-2728 100 6 𝑋2 𝑋2 ADJ cana-2728 100 7 ) ) PUNCT cana-2728 100 8 , , PUNCT cana-2728 100 9 𝑋4 𝑋4 VERB cana-2728 100 10 ) ) PUNCT cana-2728 100 11 = = SYM cana-2728 101 1 0 0 X cana-2728 101 2 . . PUNCT cana-2728 102 1 ( ( PUNCT cana-2728 102 2 3.11 3.11 NUM cana-2728 102 3 ) ) PUNCT cana-2728 102 4 using use VERB cana-2728 102 5 ( ( PUNCT cana-2728 102 6 2.9 2.9 NUM cana-2728 102 7 ) ) PUNCT cana-2728 102 8 in in ADP cana-2728 102 9 ( ( PUNCT cana-2728 102 10 3.11 3.11 NUM cana-2728 102 11 ) ) PUNCT cana-2728 102 12 , , PUNCT cana-2728 102 13 we we PRON cana-2728 102 14 obtain obtain VERB cana-2728 102 15 βˆ’π‘”(𝑋2 βˆ’π‘”(𝑋2 PROPN cana-2728 102 16 , , PUNCT cana-2728 102 17 𝑋3)𝑆(πœ‰ 𝑋3)𝑆(πœ‰ PROPN cana-2728 102 18 , , PUNCT cana-2728 102 19 𝑋4 𝑋4 VERB cana-2728 102 20 ) ) PUNCT cana-2728 102 21 βˆ’ βˆ’ PROPN cana-2728 102 22 πœ‚(𝑋3)𝑆(𝑋2 πœ‚(𝑋3)𝑆(𝑋2 ADJ cana-2728 102 23 , , PUNCT cana-2728 102 24 𝑋4 𝑋4 VERB cana-2728 102 25 ) ) PUNCT cana-2728 102 26 + + CCONJ cana-2728 102 27 𝑆(𝑋3 𝑆(𝑋3 ADJ cana-2728 102 28 , , PUNCT cana-2728 102 29 πœ‰)𝑔(𝑋2 πœ‰)𝑔(𝑋2 NOUN cana-2728 102 30 , , PUNCT cana-2728 102 31 𝑋4 𝑋4 VERB cana-2728 102 32 ) ) PUNCT cana-2728 103 1 + + CCONJ cana-2728 103 2 πœ‚(𝑋4)𝑆(𝑋3 πœ‚(𝑋4)𝑆(𝑋3 VERB cana-2728 103 3 , , PUNCT cana-2728 103 4 𝑋2 𝑋2 VERB cana-2728 103 5 ) ) PUNCT cana-2728 104 1 = = SYM cana-2728 104 2 0 0 X cana-2728 104 3 . . PUNCT cana-2728 105 1 ( ( PUNCT cana-2728 105 2 3.12 3.12 NUM cana-2728 105 3 ) ) PUNCT cana-2728 105 4 communications communication NOUN cana-2728 105 5 on on ADP cana-2728 105 6 applied apply VERB cana-2728 105 7 nonlinear nonlinear ADJ cana-2728 105 8 analysis analysis NOUN cana-2728 105 9 issn issn NOUN cana-2728 105 10 : : PUNCT cana-2728 105 11 1074 1074 NUM cana-2728 105 12 - - PUNCT cana-2728 105 13 133x 133x NUM cana-2728 105 14 vol vol NOUN cana-2728 105 15 32 32 NUM cana-2728 105 16 no no NOUN cana-2728 105 17 . . PUNCT cana-2728 106 1 3s 3s NUM cana-2728 106 2 ( ( PUNCT cana-2728 106 3 2025 2025 NUM cana-2728 106 4 ) ) PUNCT cana-2728 106 5 703 703 NUM cana-2728 106 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 106 7 setting set VERB cana-2728 106 8 𝑋3 𝑋3 NOUN cana-2728 106 9 = = SYM cana-2728 106 10 πœ‰ πœ‰ X cana-2728 106 11 in in ADP cana-2728 106 12 ( ( PUNCT cana-2728 106 13 3.12 3.12 NUM cana-2728 106 14 ) ) PUNCT cana-2728 106 15 and and CCONJ cana-2728 106 16 by by ADP cana-2728 106 17 making make VERB cana-2728 106 18 used use VERB cana-2728 106 19 of of ADP cana-2728 106 20 ( ( PUNCT cana-2728 106 21 2.3 2.3 NUM cana-2728 106 22 ) ) PUNCT cana-2728 106 23 and and CCONJ cana-2728 106 24 ( ( PUNCT cana-2728 106 25 2.11 2.11 NUM cana-2728 106 26 ) ) PUNCT cana-2728 106 27 , , PUNCT cana-2728 106 28 we we PRON cana-2728 106 29 have have AUX cana-2728 106 30 𝑆(𝑋2 𝑆(𝑋2 PROPN cana-2728 106 31 , , PUNCT cana-2728 106 32 𝑋4 𝑋4 VERB cana-2728 106 33 ) ) PUNCT cana-2728 106 34 = = SYM cana-2728 107 1 2𝑛𝑔(𝑋2 2𝑛𝑔(𝑋2 NUM cana-2728 107 2 , , PUNCT cana-2728 107 3 𝑋4 𝑋4 VERB cana-2728 107 4 ) ) PUNCT cana-2728 107 5 . . PUNCT cana-2728 108 1 ( ( PUNCT cana-2728 108 2 3.13 3.13 NUM cana-2728 108 3 ) ) PUNCT cana-2728 108 4 thus thus ADV cana-2728 108 5 , , PUNCT cana-2728 108 6 we we PRON cana-2728 108 7 complete complete VERB cana-2728 108 8 the the DET cana-2728 108 9 proof proof NOUN cana-2728 108 10 . . PUNCT cana-2728 109 1 proposition proposition NOUN cana-2728 109 2 3.1 3.1 NUM cana-2728 109 3 . . PUNCT cana-2728 110 1 if if SCONJ cana-2728 110 2 a a PRON cana-2728 110 3 ( ( PUNCT cana-2728 110 4 2𝑛 2𝑛 NOUN cana-2728 110 5 + + CCONJ cana-2728 110 6 1)-dimensional 1)-dimensional NUM cana-2728 110 7 lorentzian lorentzian ADJ cana-2728 110 8 para para NOUN cana-2728 110 9 - - PUNCT cana-2728 110 10 kenmotsu kenmotsu PROPN cana-2728 110 11 manifold manifold NOUN cana-2728 110 12 is be AUX cana-2728 110 13 an an DET cana-2728 110 14 πœ‚-einstein πœ‚-einstein NOUN cana-2728 110 15 manifold manifold NOUN cana-2728 110 16 , , PUNCT cana-2728 110 17 then then ADV cana-2728 110 18 the the DET cana-2728 110 19 ricci ricci PROPN cana-2728 110 20 soliton soliton NOUN cana-2728 110 21 with with ADP cana-2728 110 22 constant constant ADJ cana-2728 110 23 scalar scalar ADJ cana-2728 110 24 curvature curvature NOUN cana-2728 110 25 is be AUX cana-2728 110 26 shrinking shrink VERB cana-2728 110 27 . . PUNCT cana-2728 111 1 proof proof NOUN cana-2728 111 2 : : PUNCT cana-2728 111 3 suppose suppose VERB cana-2728 111 4 that that SCONJ cana-2728 111 5 the the DET cana-2728 111 6 lorentzian lorentzian ADJ cana-2728 111 7 para para NOUN cana-2728 111 8 - - PUNCT cana-2728 111 9 kenmotsu kenmotsu PROPN cana-2728 111 10 manifold manifold NOUN cana-2728 111 11 is be AUX cana-2728 111 12 an an DET cana-2728 111 13 πœ‚-einstein πœ‚-einstein NOUN cana-2728 111 14 manifold manifold NOUN cana-2728 111 15 . . PUNCT cana-2728 112 1 then then ADV cana-2728 112 2 , , PUNCT cana-2728 112 3 we we PRON cana-2728 112 4 will will AUX cana-2728 112 5 find find VERB cana-2728 112 6 the the DET cana-2728 112 7 values value NOUN cana-2728 112 8 of of ADP cana-2728 112 9 𝛼 𝛼 PRON cana-2728 112 10 and and CCONJ cana-2728 112 11 𝛽. 𝛽. NOUN cana-2728 112 12 let let VERB cana-2728 112 13 { { PUNCT cana-2728 112 14 𝑒1 𝑒1 NOUN cana-2728 112 15 , , PUNCT cana-2728 112 16 𝑒2 𝑒2 PROPN cana-2728 112 17 , , PUNCT cana-2728 112 18 … … PUNCT cana-2728 112 19 , , PUNCT cana-2728 112 20 𝑒2𝑛+1 𝑒2𝑛+1 PROPN cana-2728 112 21 } } PUNCT cana-2728 112 22 be be AUX cana-2728 112 23 an an DET cana-2728 112 24 orthonormal orthonormal ADJ cana-2728 112 25 basis basis NOUN cana-2728 112 26 of of ADP cana-2728 112 27 the the DET cana-2728 112 28 tangent tangent NOUN cana-2728 112 29 at at ADP cana-2728 112 30 any any DET cana-2728 112 31 point point NOUN cana-2728 112 32 of of ADP cana-2728 112 33 the the DET cana-2728 112 34 manifold manifold NOUN cana-2728 112 35 . . PUNCT cana-2728 113 1 putting put VERB cana-2728 113 2 𝑋1 𝑋1 NOUN cana-2728 113 3 = = SYM cana-2728 113 4 𝑋2 𝑋2 VERB cana-2728 113 5 = = PUNCT cana-2728 113 6 𝑒𝑖 𝑒𝑖 X cana-2728 113 7 in in ADP cana-2728 113 8 ( ( PUNCT cana-2728 113 9 2.19 2.19 NUM cana-2728 113 10 ) ) PUNCT cana-2728 113 11 and and CCONJ cana-2728 113 12 taking take VERB cana-2728 113 13 summation summation NOUN cana-2728 113 14 over over ADP cana-2728 113 15 𝑖 𝑖 ADP cana-2728 113 16 , , PUNCT cana-2728 113 17 we we PRON cana-2728 113 18 get get VERB cana-2728 113 19 π‘Ÿ π‘Ÿ NOUN cana-2728 113 20 = = SYM cana-2728 113 21 ( ( PUNCT cana-2728 113 22 2𝑛 2𝑛 PROPN cana-2728 113 23 + + CCONJ cana-2728 113 24 1)𝛼 1)𝛼 NUM cana-2728 113 25 βˆ’ βˆ’ PROPN cana-2728 113 26 𝛽. 𝛽. NOUN cana-2728 113 27 ( ( PUNCT cana-2728 113 28 3.14 3.14 NUM cana-2728 113 29 ) ) PUNCT cana-2728 113 30 again again ADV cana-2728 113 31 , , PUNCT cana-2728 113 32 setting set VERB cana-2728 113 33 𝑋1 𝑋1 NOUN cana-2728 113 34 = = SYM cana-2728 113 35 𝑋2 𝑋2 VERB cana-2728 113 36 = = SYM cana-2728 113 37 πœ‰ πœ‰ NOUN cana-2728 113 38 in in ADP cana-2728 113 39 ( ( PUNCT cana-2728 113 40 2.19 2.19 NUM cana-2728 113 41 ) ) PUNCT cana-2728 113 42 , , PUNCT cana-2728 113 43 and and CCONJ cana-2728 113 44 using use VERB cana-2728 113 45 ( ( PUNCT cana-2728 113 46 2.11 2.11 NUM cana-2728 113 47 ) ) PUNCT cana-2728 113 48 , , PUNCT cana-2728 113 49 we we PRON cana-2728 113 50 have have VERB cana-2728 113 51 βˆ’2𝑛 βˆ’2𝑛 NOUN cana-2728 113 52 = = SYM cana-2728 113 53 βˆ’π›Ό βˆ’π›Ό PROPN cana-2728 113 54 + + CCONJ cana-2728 113 55 𝛽. 𝛽. NOUN cana-2728 113 56 ( ( PUNCT cana-2728 113 57 3.15 3.15 NUM cana-2728 113 58 ) ) PUNCT cana-2728 113 59 then then ADV cana-2728 113 60 from from ADP cana-2728 113 61 ( ( PUNCT cana-2728 113 62 3.14 3.14 NUM cana-2728 113 63 ) ) PUNCT cana-2728 113 64 and and CCONJ cana-2728 113 65 ( ( PUNCT cana-2728 113 66 3.15 3.15 NUM cana-2728 113 67 ) ) PUNCT cana-2728 113 68 , , PUNCT cana-2728 113 69 we we PRON cana-2728 113 70 get get VERB cana-2728 113 71 𝛼 𝛼 NOUN cana-2728 113 72 = = PUNCT cana-2728 113 73 [ [ PUNCT cana-2728 113 74 π‘Ÿ π‘Ÿ X cana-2728 113 75 2𝑛 2𝑛 PROPN cana-2728 113 76 βˆ’ βˆ’ PROPN cana-2728 113 77 1 1 NUM cana-2728 113 78 ] ] PUNCT cana-2728 113 79 , , PUNCT cana-2728 113 80 𝛽 𝛽 NOUN cana-2728 113 81 = = PUNCT cana-2728 114 1 [ [ X cana-2728 114 2 βˆ’2𝑛 βˆ’2𝑛 PROPN cana-2728 114 3 βˆ’ βˆ’ NUM cana-2728 114 4 1 1 NUM cana-2728 114 5 + + NUM cana-2728 114 6 π‘Ÿ π‘Ÿ NOUN cana-2728 114 7 2𝑛 2𝑛 PROPN cana-2728 114 8 ] ] PUNCT cana-2728 114 9 . . PUNCT cana-2728 115 1 ( ( PUNCT cana-2728 115 2 3.16 3.16 NUM cana-2728 115 3 ) ) PUNCT cana-2728 115 4 substituting substitute VERB cana-2728 115 5 the the DET cana-2728 115 6 value value NOUN cana-2728 115 7 of of ADP cana-2728 115 8 𝛼 𝛼 PRON cana-2728 115 9 and and CCONJ cana-2728 115 10 𝛽 𝛽 NOUN cana-2728 115 11 in in ADP cana-2728 115 12 ( ( PUNCT cana-2728 115 13 2.19 2.19 NUM cana-2728 115 14 ) ) PUNCT cana-2728 115 15 , , PUNCT cana-2728 115 16 we we PRON cana-2728 115 17 have have AUX cana-2728 115 18 𝑆(𝑋1 𝑆(𝑋1 VERB cana-2728 115 19 , , PUNCT cana-2728 115 20 𝑋2 𝑋2 VERB cana-2728 115 21 ) ) PUNCT cana-2728 116 1 = = PUNCT cana-2728 116 2 [ [ PUNCT cana-2728 116 3 π‘Ÿ π‘Ÿ X cana-2728 116 4 2𝑛 2𝑛 PROPN cana-2728 116 5 βˆ’ βˆ’ PROPN cana-2728 116 6 1 1 NUM cana-2728 116 7 ] ] X cana-2728 116 8 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 116 9 , , PUNCT cana-2728 116 10 𝑋2 𝑋2 VERB cana-2728 116 11 ) ) PUNCT cana-2728 117 1 + + CCONJ cana-2728 118 1 [ [ X cana-2728 118 2 βˆ’2𝑛 βˆ’2𝑛 X cana-2728 118 3 βˆ’ βˆ’ ADP cana-2728 118 4 1 1 NUM cana-2728 118 5 + + NUM cana-2728 118 6 π‘Ÿ π‘Ÿ NOUN cana-2728 118 7 2𝑛 2𝑛 NOUN cana-2728 118 8 ] ] PUNCT cana-2728 118 9 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 NOUN cana-2728 118 10 ) ) PUNCT cana-2728 118 11 . . PUNCT cana-2728 119 1 ( ( PUNCT cana-2728 119 2 3.17 3.17 NUM cana-2728 119 3 ) ) PUNCT cana-2728 119 4 for for ADP cana-2728 119 5 a a PRON cana-2728 119 6 ( ( PUNCT cana-2728 119 7 2𝑛 2𝑛 NOUN cana-2728 119 8 + + CCONJ cana-2728 119 9 1)-dimensional 1)-dimensional NUM cana-2728 119 10 lorentzian lorentzian ADJ cana-2728 119 11 para para NOUN cana-2728 119 12 - - PUNCT cana-2728 119 13 kenmotsu kenmotsu NOUN cana-2728 119 14 manifold manifold VERB cana-2728 119 15 the the DET cana-2728 119 16 symmetric symmetric ADJ cana-2728 119 17 parallel parallel ADJ cana-2728 119 18 covariant covariant PROPN cana-2728 119 19 tensor tensor NOUN cana-2728 119 20 β„Ž(𝑋1 β„Ž(𝑋1 PROPN cana-2728 119 21 , , PUNCT cana-2728 119 22 𝑋2 𝑋2 VERB cana-2728 119 23 ) ) PUNCT cana-2728 119 24 of of ADP cana-2728 119 25 type type NOUN cana-2728 119 26 ( ( PUNCT cana-2728 119 27 0,2 0,2 NUM cana-2728 119 28 ) ) PUNCT cana-2728 119 29 is be AUX cana-2728 119 30 given give VERB cana-2728 119 31 by by ADP cana-2728 119 32 β„Ž(𝑋1 β„Ž(𝑋1 PROPN cana-2728 119 33 , , PUNCT cana-2728 119 34 𝑋2 𝑋2 VERB cana-2728 119 35 ) ) PUNCT cana-2728 120 1 = = SYM cana-2728 120 2 ( ( PUNCT cana-2728 120 3 β„’ΞΎ β„’ΞΎ PROPN cana-2728 120 4 𝑔)(𝑋1 𝑔)(𝑋1 PROPN cana-2728 120 5 , , PUNCT cana-2728 120 6 𝑋2 𝑋2 VERB cana-2728 120 7 ) ) PUNCT cana-2728 121 1 + + CCONJ cana-2728 121 2 2𝑆(𝑋1 2𝑆(𝑋1 NUM cana-2728 121 3 , , PUNCT cana-2728 121 4 𝑋2 𝑋2 ADJ cana-2728 121 5 ) ) PUNCT cana-2728 121 6 . . PUNCT cana-2728 122 1 ( ( PUNCT cana-2728 122 2 3.18 3.18 NUM cana-2728 122 3 ) ) PUNCT cana-2728 122 4 using use VERB cana-2728 122 5 ( ( PUNCT cana-2728 122 6 2.13 2.13 NUM cana-2728 122 7 ) ) PUNCT cana-2728 122 8 and and CCONJ cana-2728 122 9 ( ( PUNCT cana-2728 122 10 3.17 3.17 NUM cana-2728 122 11 ) ) PUNCT cana-2728 122 12 in in ADP cana-2728 122 13 ( ( PUNCT cana-2728 122 14 3.18 3.18 NUM cana-2728 122 15 ) ) PUNCT cana-2728 122 16 , , PUNCT cana-2728 122 17 we we PRON cana-2728 122 18 get get VERB cana-2728 122 19 β„Ž(𝑋1 β„Ž(𝑋1 PROPN cana-2728 122 20 , , PUNCT cana-2728 122 21 𝑋2 𝑋2 VERB cana-2728 122 22 ) ) PUNCT cana-2728 123 1 = = PUNCT cana-2728 123 2 [ [ PUNCT cana-2728 123 3 2π‘Ÿ 2π‘Ÿ NUM cana-2728 123 4 2𝑛 2𝑛 NOUN cana-2728 123 5 βˆ’ βˆ’ ADP cana-2728 123 6 4 4 NUM cana-2728 123 7 ] ] X cana-2728 123 8 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 123 9 , , PUNCT cana-2728 123 10 𝑋2 𝑋2 VERB cana-2728 123 11 ) ) PUNCT cana-2728 124 1 + + CCONJ cana-2728 125 1 [ [ X cana-2728 125 2 βˆ’4𝑛 βˆ’4𝑛 PROPN cana-2728 125 3 + + NUM cana-2728 125 4 2π‘Ÿ 2π‘Ÿ NUM cana-2728 125 5 2𝑛 2𝑛 NOUN cana-2728 125 6 βˆ’ βˆ’ NOUN cana-2728 125 7 4 4 NUM cana-2728 125 8 ] ] PUNCT cana-2728 125 9 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 NOUN cana-2728 125 10 ) ) PUNCT cana-2728 125 11 . . PUNCT cana-2728 126 1 ( ( PUNCT cana-2728 126 2 3.19 3.19 NUM cana-2728 126 3 ) ) PUNCT cana-2728 126 4 taking take VERB cana-2728 126 5 covariant covariant ADJ cana-2728 126 6 derivative derivative NOUN cana-2728 126 7 of of ADP cana-2728 126 8 ( ( PUNCT cana-2728 126 9 3.19 3.19 NUM cana-2728 126 10 ) ) PUNCT cana-2728 126 11 with with ADP cana-2728 126 12 respect respect NOUN cana-2728 126 13 to to ADP cana-2728 126 14 𝑋3 𝑋3 NOUN cana-2728 126 15 , , PUNCT cana-2728 126 16 we we PRON cana-2728 126 17 have have VERB cana-2728 126 18 ( ( PUNCT cana-2728 126 19 𝛻𝑋3 𝛻𝑋3 PROPN cana-2728 126 20 β„Ž)(𝑋1 β„Ž)(𝑋1 PROPN cana-2728 126 21 , , PUNCT cana-2728 126 22 𝑋2 𝑋2 VERB cana-2728 126 23 ) ) PUNCT cana-2728 127 1 = = PUNCT cana-2728 127 2 [ [ PUNCT cana-2728 127 3 2(𝛻𝑋3 2(𝛻𝑋3 NUM cana-2728 127 4 π‘Ÿ π‘Ÿ NOUN cana-2728 127 5 ) ) PUNCT cana-2728 127 6 2𝑛 2𝑛 NOUN cana-2728 127 7 ] ] PUNCT cana-2728 128 1 [ [ X cana-2728 128 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 128 3 , , PUNCT cana-2728 128 4 𝑋2 𝑋2 VERB cana-2728 128 5 ) ) PUNCT cana-2728 128 6 + + CCONJ cana-2728 128 7 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 NOUN cana-2728 128 8 ) ) PUNCT cana-2728 128 9 ] ] PUNCT cana-2728 129 1 + + CCONJ cana-2728 129 2 [ [ PUNCT cana-2728 129 3 2π‘Ÿ 2π‘Ÿ NUM cana-2728 129 4 2𝑛 2𝑛 PROPN cana-2728 129 5 βˆ’ βˆ’ PROPN cana-2728 129 6 4𝑛 4𝑛 NOUN cana-2728 129 7 βˆ’ βˆ’ NOUN cana-2728 130 1 4 4 NUM cana-2728 130 2 ] ] X cana-2728 131 1 [ [ X cana-2728 131 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 131 3 , , PUNCT cana-2728 131 4 𝛻𝑋3 𝛻𝑋3 PROPN cana-2728 131 5 πœ‰)πœ‚(𝑋2 πœ‰)πœ‚(𝑋2 NOUN cana-2728 131 6 ) ) PUNCT cana-2728 131 7 + + CCONJ cana-2728 132 1 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 132 2 , , PUNCT cana-2728 132 3 𝛻𝑋3 𝛻𝑋3 PROPN cana-2728 132 4 πœ‰)πœ‚(𝑋1 πœ‰)πœ‚(𝑋1 ADJ cana-2728 132 5 ) ) PUNCT cana-2728 132 6 ] ] PUNCT cana-2728 132 7 . . PUNCT cana-2728 133 1 ( ( PUNCT cana-2728 133 2 3.20 3.20 NUM cana-2728 133 3 ) ) PUNCT cana-2728 133 4 by by ADP cana-2728 133 5 putting put VERB cana-2728 133 6 𝑋3 𝑋3 NOUN cana-2728 133 7 = = SYM cana-2728 133 8 πœ‰ πœ‰ NOUN cana-2728 133 9 and and CCONJ cana-2728 133 10 𝑋1 𝑋1 NOUN cana-2728 133 11 = = SYM cana-2728 133 12 𝑋2 𝑋2 VERB cana-2728 133 13 ∈ ∈ PROPN cana-2728 133 14 ( ( PUNCT cana-2728 133 15 π‘ π‘π‘Žπ‘›πœ‰)βŠ₯ π‘ π‘π‘Žπ‘›πœ‰)βŠ₯ PROPN cana-2728 133 16 in in ADP cana-2728 133 17 ( ( PUNCT cana-2728 133 18 3.20 3.20 NUM cana-2728 133 19 ) ) PUNCT cana-2728 133 20 and and CCONJ cana-2728 133 21 by by ADP cana-2728 133 22 using use VERB cana-2728 133 23 βˆ‡β„Ž βˆ‡β„Ž PROPN cana-2728 133 24 = = SYM cana-2728 133 25 0 0 NUM cana-2728 133 26 , , PUNCT cana-2728 133 27 we we PRON cana-2728 133 28 obtain obtain VERB cana-2728 133 29 2π›»πœ‰π‘Ÿ 2π›»πœ‰π‘Ÿ NUM cana-2728 133 30 = = SYM cana-2728 133 31 0 0 PROPN cana-2728 133 32 . . PUNCT cana-2728 134 1 ( ( PUNCT cana-2728 134 2 3.21 3.21 NUM cana-2728 134 3 ) ) PUNCT cana-2728 134 4 on on ADP cana-2728 134 5 integrating integrate VERB cana-2728 134 6 ( ( PUNCT cana-2728 134 7 3.21 3.21 NUM cana-2728 134 8 ) ) PUNCT cana-2728 134 9 , , PUNCT cana-2728 134 10 we we PRON cana-2728 134 11 get get VERB cana-2728 134 12 π‘Ÿ π‘Ÿ NOUN cana-2728 134 13 = = SYM cana-2728 134 14 𝑐 𝑐 NOUN cana-2728 134 15 , , PUNCT cana-2728 134 16 ( ( PUNCT cana-2728 134 17 3.22 3.22 NUM cana-2728 134 18 ) ) PUNCT cana-2728 134 19 communications communication NOUN cana-2728 134 20 on on ADP cana-2728 134 21 applied apply VERB cana-2728 134 22 nonlinear nonlinear ADJ cana-2728 134 23 analysis analysis NOUN cana-2728 134 24 issn issn NOUN cana-2728 134 25 : : PUNCT cana-2728 134 26 1074 1074 NUM cana-2728 134 27 - - PUNCT cana-2728 134 28 133x 133x NUM cana-2728 134 29 vol vol NOUN cana-2728 134 30 32 32 NUM cana-2728 134 31 no no NOUN cana-2728 134 32 . . PUNCT cana-2728 135 1 3s 3s NUM cana-2728 135 2 ( ( PUNCT cana-2728 135 3 2025 2025 NUM cana-2728 135 4 ) ) PUNCT cana-2728 135 5 704 704 NUM cana-2728 135 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 135 7 where where SCONJ cana-2728 135 8 𝑐 𝑐 PROPN cana-2728 135 9 is be AUX cana-2728 135 10 some some DET cana-2728 135 11 integral integral ADJ cana-2728 135 12 constant constant ADJ cana-2728 135 13 . . PUNCT cana-2728 136 1 thus thus ADV cana-2728 136 2 , , PUNCT cana-2728 136 3 from from ADP cana-2728 136 4 ( ( PUNCT cana-2728 136 5 3.22 3.22 NUM cana-2728 136 6 ) ) PUNCT cana-2728 136 7 we we PRON cana-2728 136 8 have have VERB cana-2728 136 9 π‘Ÿ π‘Ÿ NOUN cana-2728 136 10 is be AUX cana-2728 136 11 constant constant ADJ cana-2728 136 12 scalar scalar ADJ cana-2728 136 13 curvature curvature NOUN cana-2728 136 14 . . PUNCT cana-2728 137 1 finally finally ADV cana-2728 137 2 , , PUNCT cana-2728 137 3 we we PRON cana-2728 137 4 will will AUX cana-2728 137 5 check check VERB cana-2728 137 6 the the DET cana-2728 137 7 nature nature NOUN cana-2728 137 8 of of ADP cana-2728 137 9 the the DET cana-2728 137 10 ricci ricci NOUN cana-2728 137 11 - - PUNCT cana-2728 137 12 soliton soliton NOUN cana-2728 137 13 . . PUNCT cana-2728 138 1 from from ADP cana-2728 138 2 ( ( PUNCT cana-2728 138 3 3.18 3.18 NUM cana-2728 138 4 ) ) PUNCT cana-2728 138 5 , , PUNCT cana-2728 138 6 we we PRON cana-2728 138 7 have have VERB cana-2728 138 8 β„Ž(𝑋1 β„Ž(𝑋1 PROPN cana-2728 138 9 , , PUNCT cana-2728 138 10 𝑋2 𝑋2 VERB cana-2728 138 11 ) ) PUNCT cana-2728 139 1 = = SYM cana-2728 140 1 βˆ’2πœ†π‘”(𝑋1 βˆ’2πœ†π‘”(𝑋1 PROPN cana-2728 140 2 , , PUNCT cana-2728 140 3 𝑋2 𝑋2 ADJ cana-2728 140 4 ) ) PUNCT cana-2728 140 5 , , PUNCT cana-2728 140 6 then then ADV cana-2728 140 7 putting put VERB cana-2728 140 8 𝑋1 𝑋1 NOUN cana-2728 140 9 = = SYM cana-2728 140 10 𝑋2 𝑋2 VERB cana-2728 140 11 = = SYM cana-2728 140 12 πœ‰ πœ‰ NOUN cana-2728 140 13 , , PUNCT cana-2728 140 14 we we PRON cana-2728 140 15 have have VERB cana-2728 140 16 β„Ž(πœ‰ β„Ž(πœ‰ NOUN cana-2728 140 17 , , PUNCT cana-2728 140 18 πœ‰ πœ‰ NOUN cana-2728 140 19 ) ) PUNCT cana-2728 140 20 = = SYM cana-2728 140 21 2πœ†. 2πœ†. NUM cana-2728 140 22 ( ( PUNCT cana-2728 140 23 3.23 3.23 NUM cana-2728 140 24 ) ) PUNCT cana-2728 140 25 if if SCONJ cana-2728 140 26 we we PRON cana-2728 140 27 put put VERB cana-2728 140 28 𝑋1 𝑋1 NOUN cana-2728 140 29 = = SYM cana-2728 140 30 𝑋2 𝑋2 VERB cana-2728 140 31 = = SYM cana-2728 140 32 πœ‰ πœ‰ NOUN cana-2728 140 33 in in ADP cana-2728 140 34 ( ( PUNCT cana-2728 140 35 3.19 3.19 NUM cana-2728 140 36 ) ) PUNCT cana-2728 140 37 , , PUNCT cana-2728 140 38 that that PRON cana-2728 140 39 is be AUX cana-2728 140 40 β„Ž(πœ‰ β„Ž(πœ‰ NOUN cana-2728 140 41 , , PUNCT cana-2728 140 42 πœ‰ πœ‰ NOUN cana-2728 140 43 ) ) PUNCT cana-2728 140 44 = = SYM cana-2728 141 1 βˆ’ βˆ’ PROPN cana-2728 141 2 [ [ PUNCT cana-2728 141 3 2π‘Ÿ 2π‘Ÿ NUM cana-2728 141 4 2𝑛 2𝑛 NOUN cana-2728 141 5 βˆ’ βˆ’ ADP cana-2728 141 6 4 4 NUM cana-2728 141 7 ] ] PUNCT cana-2728 142 1 + + CCONJ cana-2728 142 2 [ [ X cana-2728 142 3 βˆ’4𝑛 βˆ’4𝑛 PROPN cana-2728 142 4 + + NUM cana-2728 142 5 2π‘Ÿ 2π‘Ÿ NUM cana-2728 142 6 2𝑛 2𝑛 NOUN cana-2728 142 7 βˆ’ βˆ’ PROPN cana-2728 142 8 4 4 NUM cana-2728 142 9 ] ] PUNCT cana-2728 142 10 . . PUNCT cana-2728 143 1 ( ( PUNCT cana-2728 143 2 3.24 3.24 NUM cana-2728 143 3 ) ) PUNCT cana-2728 143 4 the the DET cana-2728 143 5 above above ADJ cana-2728 143 6 equation equation NOUN cana-2728 143 7 is be AUX cana-2728 143 8 reduced reduce VERB cana-2728 143 9 as as ADP cana-2728 143 10 β„Ž(πœ‰ β„Ž(πœ‰ NOUN cana-2728 143 11 , , PUNCT cana-2728 143 12 πœ‰ πœ‰ NOUN cana-2728 143 13 ) ) PUNCT cana-2728 143 14 = = SYM cana-2728 144 1 βˆ’4𝑛. βˆ’4𝑛. X cana-2728 144 2 ( ( PUNCT cana-2728 144 3 3.25 3.25 NUM cana-2728 144 4 ) ) PUNCT cana-2728 144 5 equating equate VERB cana-2728 144 6 ( ( PUNCT cana-2728 144 7 3.23 3.23 NUM cana-2728 144 8 ) ) PUNCT cana-2728 144 9 and and CCONJ cana-2728 144 10 ( ( PUNCT cana-2728 144 11 3.25 3.25 NUM cana-2728 144 12 ) ) PUNCT cana-2728 144 13 , , PUNCT cana-2728 144 14 we we PRON cana-2728 144 15 obtain obtain VERB cana-2728 144 16 πœ† πœ† PRON cana-2728 144 17 = = SYM cana-2728 144 18 βˆ’2𝑛 βˆ’2𝑛 X cana-2728 144 19 < < X cana-2728 144 20 0 0 NUM cana-2728 144 21 , , PUNCT cana-2728 144 22 ( ( PUNCT cana-2728 144 23 3.26 3.26 NUM cana-2728 144 24 ) ) PUNCT cana-2728 144 25 that that PRON cana-2728 144 26 is be AUX cana-2728 144 27 the the DET cana-2728 144 28 ricci ricci PROPN cana-2728 144 29 soliton soliton NOUN cana-2728 144 30 in in ADP cana-2728 144 31 a a DET cana-2728 144 32 lorentzian lorentzian ADJ cana-2728 144 33 para para NOUN cana-2728 144 34 - - PUNCT cana-2728 144 35 kenmotsu kenmotsu PROPN cana-2728 144 36 manifold manifold NOUN cana-2728 144 37 is be AUX cana-2728 144 38 shrinking shrink VERB cana-2728 144 39 . . PUNCT cana-2728 145 1 hence hence ADV cana-2728 145 2 , , PUNCT cana-2728 145 3 the the DET cana-2728 145 4 theorem theorem NOUN cana-2728 145 5 is be AUX cana-2728 145 6 proved prove VERB cana-2728 145 7 . . PUNCT cana-2728 146 1 3 3 X cana-2728 146 2 . . X cana-2728 146 3 ricci ricci PROPN cana-2728 146 4 solitons soliton NOUN cana-2728 146 5 in in ADP cana-2728 146 6 a a DET cana-2728 146 7 π‘ΎπŸ-semisymmetric π‘ΎπŸ-semisymmetric ADJ cana-2728 146 8 lorentzian lorentzian ADJ cana-2728 146 9 para para NOUN cana-2728 146 10 - - PUNCT cana-2728 146 11 kenmotsu kenmotsu NOUN cana-2728 146 12 manifold manifold ADJ cana-2728 146 13 here here ADV cana-2728 146 14 , , PUNCT cana-2728 146 15 we we PRON cana-2728 146 16 study study VERB cana-2728 146 17 the the DET cana-2728 146 18 conditions condition NOUN cana-2728 146 19 of of ADP cana-2728 146 20 ricci ricci PROPN cana-2728 146 21 solitons soliton NOUN cana-2728 146 22 in in ADP cana-2728 146 23 a a DET cana-2728 146 24 π‘Š2semisymmetric π‘Š2semisymmetric ADJ cana-2728 146 25 lorentzian lorentzian ADJ cana-2728 146 26 para para NOUN cana-2728 146 27 - - PUNCT cana-2728 146 28 kenmotsu kenmotsu NOUN cana-2728 146 29 manifold manifold NOUN cana-2728 146 30 . . PUNCT cana-2728 147 1 definition definition NOUN cana-2728 147 2 4.1 4.1 NUM cana-2728 147 3 . . PUNCT cana-2728 148 1 [ [ X cana-2728 148 2 13 13 NUM cana-2728 148 3 ] ] PUNCT cana-2728 148 4 in in ADP cana-2728 148 5 a a DET cana-2728 148 6 ( ( PUNCT cana-2728 148 7 2𝑛 2𝑛 NOUN cana-2728 148 8 + + CCONJ cana-2728 148 9 1)-dimensional 1)-dimensional NUM cana-2728 148 10 lorentzian lorentzian ADJ cana-2728 148 11 para para NOUN cana-2728 148 12 - - PUNCT cana-2728 148 13 kenmotsu kenmotsu PROPN cana-2728 148 14 manifold manifold PROPN cana-2728 148 15 𝑀 𝑀 PROPN cana-2728 148 16 , , PUNCT cana-2728 148 17 the the DET cana-2728 148 18 π‘Š2curvature π‘Š2curvature NOUN cana-2728 148 19 tensor tensor NOUN cana-2728 148 20 is be AUX cana-2728 148 21 defined define VERB cana-2728 148 22 as as ADP cana-2728 148 23 π‘Š2(𝑋1 π‘Š2(𝑋1 PROPN cana-2728 148 24 , , PUNCT cana-2728 148 25 𝑋2)𝑋3 𝑋2)𝑋3 ADJ cana-2728 148 26 = = SYM cana-2728 148 27 𝑅(𝑋1 𝑅(𝑋1 PROPN cana-2728 148 28 , , PUNCT cana-2728 148 29 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 148 30 + + NOUN cana-2728 148 31 1 1 NUM cana-2728 148 32 2𝑛 2𝑛 NOUN cana-2728 149 1 [ [ X cana-2728 149 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 149 3 , , PUNCT cana-2728 149 4 𝑋3)𝑄𝑋2 𝑋3)𝑄𝑋2 X cana-2728 149 5 βˆ’ βˆ’ PROPN cana-2728 149 6 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 149 7 , , PUNCT cana-2728 149 8 𝑋3)𝑄𝑋1 𝑋3)𝑄𝑋1 PROPN cana-2728 149 9 ] ] PUNCT cana-2728 149 10 , , PUNCT cana-2728 149 11 ( ( PUNCT cana-2728 149 12 4.1 4.1 NUM cana-2728 149 13 ) ) PUNCT cana-2728 149 14 for for ADP cana-2728 149 15 all all DET cana-2728 149 16 𝑋1 𝑋1 PROPN cana-2728 149 17 , , PUNCT cana-2728 149 18 𝑋2 𝑋2 VERB cana-2728 149 19 and and CCONJ cana-2728 149 20 𝑋3 𝑋3 NOUN cana-2728 149 21 in in ADP cana-2728 149 22 𝑀. 𝑀. PROPN cana-2728 149 23 theorem theorem NOUN cana-2728 149 24 4.1 4.1 NUM cana-2728 149 25 . . PUNCT cana-2728 150 1 a a DET cana-2728 150 2 ricci ricci PROPN cana-2728 150 3 soliton soliton NOUN cana-2728 150 4 in in ADP cana-2728 150 5 a a DET cana-2728 150 6 π‘Š2 π‘Š2 ADJ cana-2728 150 7 - - PUNCT cana-2728 150 8 semi semi ADV cana-2728 150 9 symmetric symmetric ADJ cana-2728 150 10 lorentzian lorentzian ADJ cana-2728 150 11 para para NOUN cana-2728 150 12 - - PUNCT cana-2728 150 13 kenmotsu kenmotsu PROPN cana-2728 150 14 manifold manifold PROPN cana-2728 150 15 𝑀 𝑀 PROPN cana-2728 150 16 of of ADP cana-2728 150 17 dimension dimension NOUN cana-2728 150 18 ( ( PUNCT cana-2728 150 19 2𝑛 2𝑛 NOUN cana-2728 150 20 + + CCONJ cana-2728 150 21 1 1 X cana-2728 150 22 ) ) PUNCT cana-2728 150 23 is be AUX cana-2728 150 24 shrinking shrink VERB cana-2728 150 25 . . PUNCT cana-2728 151 1 proof proof NOUN cana-2728 151 2 : : PUNCT cana-2728 151 3 putting put VERB cana-2728 151 4 𝑋1 𝑋1 NOUN cana-2728 151 5 = = SYM cana-2728 151 6 πœ‰ πœ‰ NOUN cana-2728 151 7 in in ADP cana-2728 151 8 ( ( PUNCT cana-2728 151 9 4.1 4.1 NUM cana-2728 151 10 ) ) PUNCT cana-2728 151 11 and and CCONJ cana-2728 151 12 using use VERB cana-2728 151 13 ( ( PUNCT cana-2728 151 14 2.3 2.3 NUM cana-2728 151 15 ) ) PUNCT cana-2728 151 16 and and CCONJ cana-2728 151 17 ( ( PUNCT cana-2728 151 18 2.9 2.9 NUM cana-2728 151 19 ) ) PUNCT cana-2728 151 20 , , PUNCT cana-2728 151 21 we we PRON cana-2728 151 22 have have AUX cana-2728 151 23 π‘Š2(πœ‰ π‘Š2(πœ‰ NOUN cana-2728 151 24 , , PUNCT cana-2728 151 25 𝑋2)𝑋3 𝑋2)𝑋3 X cana-2728 151 26 = = SYM cana-2728 151 27 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 151 28 , , PUNCT cana-2728 151 29 𝑋3)πœ‰ 𝑋3)πœ‰ PROPN cana-2728 151 30 βˆ’ βˆ’ NOUN cana-2728 152 1 πœ‚(𝑋3)𝑋2 πœ‚(𝑋3)𝑋2 PROPN cana-2728 152 2 + + NOUN cana-2728 152 3 1 1 NUM cana-2728 152 4 2𝑛 2𝑛 NOUN cana-2728 152 5 [ [ X cana-2728 152 6 πœ‚(𝑋3)𝑄𝑋2 πœ‚(𝑋3)𝑄𝑋2 PUNCT cana-2728 152 7 βˆ’ βˆ’ PROPN cana-2728 152 8 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 152 9 , , PUNCT cana-2728 152 10 𝑋3)π‘„πœ‰ 𝑋3)π‘„πœ‰ PROPN cana-2728 152 11 ] ] PUNCT cana-2728 152 12 . . PUNCT cana-2728 153 1 ( ( PUNCT cana-2728 153 2 4.2 4.2 NUM cana-2728 153 3 ) ) PUNCT cana-2728 153 4 taking take VERB cana-2728 153 5 inner inner ADJ cana-2728 153 6 product product NOUN cana-2728 153 7 on on ADP cana-2728 153 8 both both DET cana-2728 153 9 sides side NOUN cana-2728 153 10 of of ADP cana-2728 153 11 ( ( PUNCT cana-2728 153 12 4.1 4.1 NUM cana-2728 153 13 ) ) PUNCT cana-2728 153 14 with with ADP cana-2728 153 15 respect respect NOUN cana-2728 153 16 to to ADP cana-2728 153 17 πœ‰ πœ‰ ADP cana-2728 153 18 , , PUNCT cana-2728 153 19 we we PRON cana-2728 153 20 get get VERB cana-2728 153 21 πœ‚(π‘Š2(𝑋1 πœ‚(π‘Š2(𝑋1 NOUN cana-2728 153 22 , , PUNCT cana-2728 153 23 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 153 24 ) ) PUNCT cana-2728 154 1 = = SYM cana-2728 154 2 πœ‚(𝑅(𝑋1 πœ‚(𝑅(𝑋1 NOUN cana-2728 154 3 , , PUNCT cana-2728 154 4 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 154 5 ) ) PUNCT cana-2728 155 1 + + CCONJ cana-2728 155 2 1 1 NUM cana-2728 155 3 2𝑛 2𝑛 NOUN cana-2728 156 1 [ [ X cana-2728 156 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 156 3 , , PUNCT cana-2728 156 4 𝑋3)𝑔(𝑄𝑋2 𝑋3)𝑔(𝑄𝑋2 PROPN cana-2728 156 5 , , PUNCT cana-2728 156 6 πœ‰ πœ‰ NOUN cana-2728 156 7 ) ) PUNCT cana-2728 156 8 βˆ’ βˆ’ PROPN cana-2728 157 1 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 157 2 , , PUNCT cana-2728 157 3 𝑋3)𝑔(𝑄𝑋1 𝑋3)𝑔(𝑄𝑋1 PROPN cana-2728 157 4 , , PUNCT cana-2728 157 5 πœ‰ πœ‰ NOUN cana-2728 157 6 ) ) PUNCT cana-2728 157 7 ] ] PUNCT cana-2728 157 8 . . PUNCT cana-2728 158 1 ( ( PUNCT cana-2728 158 2 4.3 4.3 NUM cana-2728 158 3 ) ) PUNCT cana-2728 158 4 using use VERB cana-2728 158 5 ( ( PUNCT cana-2728 158 6 2.8 2.8 NUM cana-2728 158 7 ) ) PUNCT cana-2728 158 8 and and CCONJ cana-2728 158 9 ( ( PUNCT cana-2728 158 10 2.17 2.17 NUM cana-2728 158 11 ) ) PUNCT cana-2728 158 12 in in ADP cana-2728 158 13 ( ( PUNCT cana-2728 158 14 4.3 4.3 NUM cana-2728 158 15 ) ) PUNCT cana-2728 158 16 , , PUNCT cana-2728 158 17 we we PRON cana-2728 158 18 obtain obtain VERB cana-2728 158 19 πœ‚(π‘Š2(𝑋1 πœ‚(π‘Š2(𝑋1 NOUN cana-2728 158 20 , , PUNCT cana-2728 158 21 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 158 22 ) ) PUNCT cana-2728 158 23 = = PUNCT cana-2728 159 1 ( ( PUNCT cana-2728 159 2 1 1 NUM cana-2728 159 3 + + CCONJ cana-2728 159 4 πœ† πœ† PROPN cana-2728 159 5 2𝑛 2𝑛 PROPN cana-2728 159 6 ) ) PUNCT cana-2728 160 1 [ [ X cana-2728 160 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 160 3 , , PUNCT cana-2728 160 4 𝑋3)πœ‚(𝑋1 𝑋3)πœ‚(𝑋1 PROPN cana-2728 160 5 ) ) PUNCT cana-2728 160 6 βˆ’ βˆ’ PROPN cana-2728 160 7 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 160 8 , , PUNCT cana-2728 160 9 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 160 10 ) ) PUNCT cana-2728 160 11 ] ] PUNCT cana-2728 160 12 . . PUNCT cana-2728 161 1 ( ( PUNCT cana-2728 161 2 4.4 4.4 NUM cana-2728 161 3 ) ) PUNCT cana-2728 161 4 suppose suppose VERB cana-2728 161 5 that that SCONJ cana-2728 161 6 the the DET cana-2728 161 7 condition condition NOUN cana-2728 161 8 , , PUNCT cana-2728 161 9 𝑅(πœ‰ 𝑅(πœ‰ PROPN cana-2728 161 10 , , PUNCT cana-2728 161 11 𝑋1 𝑋1 PROPN cana-2728 161 12 ) ) PUNCT cana-2728 161 13 ∘ ∘ PROPN cana-2728 161 14 π‘Š2(𝑋2 π‘Š2(𝑋2 PROPN cana-2728 161 15 , , PUNCT cana-2728 161 16 𝑋3)𝑋4 𝑋3)𝑋4 NOUN cana-2728 161 17 = = SYM cana-2728 161 18 0 0 NUM cana-2728 161 19 holds hold VERB cana-2728 161 20 in in ADP cana-2728 161 21 𝑀. 𝑀. PROPN cana-2728 161 22 then then ADV cana-2728 161 23 by by ADP cana-2728 161 24 definition definition NOUN cana-2728 161 25 , , PUNCT cana-2728 161 26 we we PRON cana-2728 161 27 have have VERB cana-2728 161 28 𝑅(πœ‰ 𝑅(πœ‰ NUM cana-2728 161 29 , , PUNCT cana-2728 161 30 𝑋1)π‘Š2(𝑋2 𝑋1)π‘Š2(𝑋2 NOUN cana-2728 161 31 , , PUNCT cana-2728 161 32 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 161 33 βˆ’ βˆ’ PROPN cana-2728 161 34 π‘Š2(𝑅(πœ‰ π‘Š2(𝑅(πœ‰ PROPN cana-2728 161 35 , , PUNCT cana-2728 161 36 𝑋1)𝑋2 𝑋1)𝑋2 PROPN cana-2728 161 37 , , PUNCT cana-2728 161 38 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 161 39 βˆ’ βˆ’ PROPN cana-2728 161 40 π‘Š2(𝑋2 π‘Š2(𝑋2 PROPN cana-2728 161 41 , , PUNCT cana-2728 161 42 𝑅(πœ‰ 𝑅(πœ‰ PRON cana-2728 161 43 , , PUNCT cana-2728 161 44 𝑋1)𝑋3)𝑋4 𝑋1)𝑋3)𝑋4 NOUN cana-2728 161 45 communications communication NOUN cana-2728 161 46 on on ADP cana-2728 161 47 applied apply VERB cana-2728 161 48 nonlinear nonlinear ADJ cana-2728 161 49 analysis analysis NOUN cana-2728 161 50 issn issn NOUN cana-2728 161 51 : : PUNCT cana-2728 161 52 1074 1074 NUM cana-2728 161 53 - - PUNCT cana-2728 161 54 133x 133x NUM cana-2728 161 55 vol vol NOUN cana-2728 161 56 32 32 NUM cana-2728 161 57 no no NOUN cana-2728 161 58 . . PUNCT cana-2728 162 1 3s 3s NUM cana-2728 162 2 ( ( PUNCT cana-2728 162 3 2025 2025 NUM cana-2728 162 4 ) ) PUNCT cana-2728 162 5 705 705 NUM cana-2728 162 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 162 7 βˆ’π‘Š2(𝑋2 βˆ’π‘Š2(𝑋2 NOUN cana-2728 162 8 , , PUNCT cana-2728 162 9 𝑋3)𝑅(πœ‰ 𝑋3)𝑅(πœ‰ VERB cana-2728 162 10 , , PUNCT cana-2728 162 11 𝑋1)𝑋4 𝑋1)𝑋4 NOUN cana-2728 162 12 = = SYM cana-2728 162 13 0 0 NUM cana-2728 162 14 , , PUNCT cana-2728 162 15 ( ( PUNCT cana-2728 162 16 4.5 4.5 NUM cana-2728 162 17 ) ) PUNCT cana-2728 162 18 for for ADP cana-2728 162 19 all all DET cana-2728 162 20 vector vector NOUN cana-2728 162 21 fields field NOUN cana-2728 162 22 𝑋1 𝑋1 PROPN cana-2728 162 23 , , PUNCT cana-2728 162 24 𝑋2 𝑋2 VERB cana-2728 162 25 , , PUNCT cana-2728 162 26 𝑋3 𝑋3 NOUN cana-2728 162 27 and and CCONJ cana-2728 162 28 𝑋4 𝑋4 VERB cana-2728 162 29 on on ADP cana-2728 162 30 𝑀. 𝑀. PROPN cana-2728 162 31 in in ADP cana-2728 162 32 view view NOUN cana-2728 162 33 of of ADP cana-2728 162 34 ( ( PUNCT cana-2728 162 35 2.9 2.9 NUM cana-2728 162 36 ) ) PUNCT cana-2728 162 37 and and CCONJ cana-2728 162 38 ( ( PUNCT cana-2728 162 39 4.5 4.5 NUM cana-2728 162 40 ) ) PUNCT cana-2728 162 41 , , PUNCT cana-2728 162 42 we we PRON cana-2728 162 43 get get VERB cana-2728 162 44 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 162 45 , , PUNCT cana-2728 162 46 π‘Š2(𝑋2 π‘Š2(𝑋2 PROPN cana-2728 162 47 , , PUNCT cana-2728 162 48 𝑋3)𝑋4)πœ‰ 𝑋3)𝑋4)πœ‰ PROPN cana-2728 162 49 βˆ’ βˆ’ PROPN cana-2728 163 1 πœ‚(π‘Š2(𝑋2 πœ‚(π‘Š2(𝑋2 NOUN cana-2728 163 2 , , PUNCT cana-2728 163 3 𝑋3)𝑋4)𝑋1 𝑋3)𝑋4)𝑋1 NOUN cana-2728 163 4 βˆ’ βˆ’ PROPN cana-2728 163 5 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 163 6 , , PUNCT cana-2728 163 7 𝑋2)π‘Š2(πœ‰ 𝑋2)π‘Š2(πœ‰ NOUN cana-2728 163 8 , , PUNCT cana-2728 163 9 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 163 10 + + PROPN cana-2728 163 11 πœ‚(𝑋2)π‘Š2(𝑋1 πœ‚(𝑋2)π‘Š2(𝑋1 ADJ cana-2728 163 12 , , PUNCT cana-2728 163 13 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 163 14 βˆ’ βˆ’ PROPN cana-2728 163 15 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 163 16 , , PUNCT cana-2728 163 17 𝑋3)π‘Š2(𝑋2 𝑋3)π‘Š2(𝑋2 NOUN cana-2728 163 18 , , PUNCT cana-2728 163 19 πœ‰)𝑋4 πœ‰)𝑋4 PROPN cana-2728 163 20 + + CCONJ cana-2728 163 21 πœ‚(𝑋3)π‘Š2(𝑋2 πœ‚(𝑋3)π‘Š2(𝑋2 PROPN cana-2728 163 22 , , PUNCT cana-2728 163 23 𝑋1)𝑋4 𝑋1)𝑋4 NOUN cana-2728 163 24 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 ADJ cana-2728 163 25 , , PUNCT cana-2728 163 26 𝑋4)π‘Š2(𝑋2 𝑋4)π‘Š2(𝑋2 PROPN cana-2728 163 27 , , PUNCT cana-2728 163 28 𝑋3)πœ‰ 𝑋3)πœ‰ X cana-2728 163 29 + + NUM cana-2728 163 30 πœ‚(𝑋4)π‘Š2(𝑋2 πœ‚(𝑋4)π‘Š2(𝑋2 NOUN cana-2728 163 31 , , PUNCT cana-2728 163 32 𝑋3)𝑋1 𝑋3)𝑋1 X cana-2728 163 33 = = SYM cana-2728 163 34 0 0 X cana-2728 163 35 . . PUNCT cana-2728 164 1 ( ( PUNCT cana-2728 164 2 4.6 4.6 NUM cana-2728 164 3 ) ) PUNCT cana-2728 164 4 again again ADV cana-2728 164 5 , , PUNCT cana-2728 164 6 taking take VERB cana-2728 164 7 inner inner ADJ cana-2728 164 8 product product NOUN cana-2728 164 9 on on ADP cana-2728 164 10 both both DET cana-2728 164 11 sides side NOUN cana-2728 164 12 of of ADP cana-2728 164 13 ( ( PUNCT cana-2728 164 14 4.6 4.6 NUM cana-2728 164 15 ) ) PUNCT cana-2728 164 16 with with ADP cana-2728 164 17 πœ‰ πœ‰ NOUN cana-2728 164 18 and and CCONJ cana-2728 164 19 using use VERB cana-2728 164 20 ( ( PUNCT cana-2728 164 21 2.3 2.3 NUM cana-2728 164 22 ) ) PUNCT cana-2728 164 23 , , PUNCT cana-2728 164 24 we we PRON cana-2728 164 25 have have VERB cana-2728 164 26 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 ADJ cana-2728 164 27 , , PUNCT cana-2728 164 28 π‘Š2(𝑋2 π‘Š2(𝑋2 PROPN cana-2728 164 29 , , PUNCT cana-2728 164 30 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 164 31 ) ) PUNCT cana-2728 164 32 βˆ’ βˆ’ PROPN cana-2728 165 1 πœ‚(π‘Š2(𝑋2 πœ‚(π‘Š2(𝑋2 PROPN cana-2728 165 2 , , PUNCT cana-2728 165 3 𝑋3)𝑋4)πœ‚(𝑋1 𝑋3)𝑋4)πœ‚(𝑋1 PROPN cana-2728 165 4 ) ) PUNCT cana-2728 165 5 βˆ’ βˆ’ PROPN cana-2728 165 6 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 165 7 , , PUNCT cana-2728 165 8 𝑋2)πœ‚(π‘Š2(πœ‰ 𝑋2)πœ‚(π‘Š2(πœ‰ NOUN cana-2728 165 9 , , PUNCT cana-2728 165 10 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 165 11 ) ) PUNCT cana-2728 165 12 + + NOUN cana-2728 165 13 πœ‚(𝑋2)πœ‚(π‘Š2(𝑋1 πœ‚(𝑋2)πœ‚(π‘Š2(𝑋1 NOUN cana-2728 165 14 , , PUNCT cana-2728 165 15 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 165 16 ) ) PUNCT cana-2728 165 17 βˆ’ βˆ’ PROPN cana-2728 165 18 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 165 19 , , PUNCT cana-2728 165 20 𝑋3)πœ‚(π‘Š2(𝑋2 𝑋3)πœ‚(π‘Š2(𝑋2 PROPN cana-2728 165 21 , , PUNCT cana-2728 165 22 πœ‰)𝑋4 πœ‰)𝑋4 NOUN cana-2728 165 23 ) ) PUNCT cana-2728 165 24 + + CCONJ cana-2728 165 25 πœ‚(𝑋3)πœ‚(π‘Š2(𝑋2 πœ‚(𝑋3)πœ‚(π‘Š2(𝑋2 VERB cana-2728 165 26 , , PUNCT cana-2728 165 27 𝑋1)𝑋4 𝑋1)𝑋4 NOUN cana-2728 165 28 ) ) PUNCT cana-2728 165 29 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 ADJ cana-2728 165 30 , , PUNCT cana-2728 165 31 𝑋4)πœ‚(π‘Š2(𝑋2 𝑋4)πœ‚(π‘Š2(𝑋2 PROPN cana-2728 165 32 , , PUNCT cana-2728 165 33 𝑋3)πœ‰ 𝑋3)πœ‰ NOUN cana-2728 165 34 ) ) PUNCT cana-2728 165 35 + + CCONJ cana-2728 165 36 πœ‚(𝑋4)πœ‚(π‘Š2(𝑋2 πœ‚(𝑋4)πœ‚(π‘Š2(𝑋2 ADJ cana-2728 165 37 , , PUNCT cana-2728 165 38 𝑋3)𝑋1 𝑋3)𝑋1 PROPN cana-2728 165 39 ) ) PUNCT cana-2728 165 40 = = SYM cana-2728 165 41 0 0 X cana-2728 165 42 . . PUNCT cana-2728 165 43 ( ( PUNCT cana-2728 165 44 4.7 4.7 NUM cana-2728 165 45 ) ) PUNCT cana-2728 165 46 in in ADP cana-2728 165 47 consequence consequence NOUN cana-2728 165 48 of of ADP cana-2728 165 49 ( ( PUNCT cana-2728 165 50 4.2 4.2 NUM cana-2728 165 51 ) ) PUNCT cana-2728 165 52 , , PUNCT cana-2728 165 53 ( ( PUNCT cana-2728 165 54 4.4 4.4 NUM cana-2728 165 55 ) ) PUNCT cana-2728 165 56 and and CCONJ cana-2728 165 57 ( ( PUNCT cana-2728 165 58 4.7 4.7 NUM cana-2728 165 59 ) ) PUNCT cana-2728 165 60 , , PUNCT cana-2728 165 61 it it PRON cana-2728 165 62 yields yield VERB cana-2728 165 63 𝑔(𝑅(𝑋2 𝑔(𝑅(𝑋2 PROPN cana-2728 165 64 , , PUNCT cana-2728 165 65 𝑋3)𝑋4 𝑋3)𝑋4 PROPN cana-2728 165 66 , , PUNCT cana-2728 165 67 𝑋1 𝑋1 PROPN cana-2728 165 68 ) ) PUNCT cana-2728 166 1 + + CCONJ cana-2728 166 2 1 1 NUM cana-2728 166 3 2𝑛 2𝑛 NOUN cana-2728 166 4 [ [ X cana-2728 166 5 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 166 6 , , PUNCT cana-2728 166 7 𝑋4)𝑆(𝑋1 𝑋4)𝑆(𝑋1 NOUN cana-2728 166 8 , , PUNCT cana-2728 166 9 𝑋3 𝑋3 NOUN cana-2728 166 10 ) ) PUNCT cana-2728 167 1 βˆ’ βˆ’ PROPN cana-2728 167 2 𝑔(𝑋3 𝑔(𝑋3 PROPN cana-2728 167 3 , , PUNCT cana-2728 167 4 𝑋4)𝑆(𝑋1 𝑋4)𝑆(𝑋1 PROPN cana-2728 167 5 , , PUNCT cana-2728 167 6 𝑋2 𝑋2 VERB cana-2728 167 7 ) ) PUNCT cana-2728 167 8 ] ] PUNCT cana-2728 168 1 βˆ’ βˆ’ PROPN cana-2728 168 2 ( ( PUNCT cana-2728 168 3 1 1 NUM cana-2728 168 4 + + CCONJ cana-2728 168 5 πœ† πœ† PROPN cana-2728 168 6 2𝑛 2𝑛 PROPN cana-2728 168 7 ) ) PUNCT cana-2728 169 1 [ [ X cana-2728 169 2 πœ‚(𝑋1){𝑔(𝑋4 πœ‚(𝑋1){𝑔(𝑋4 ADP cana-2728 169 3 , , PUNCT cana-2728 169 4 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 NUM cana-2728 169 5 ) ) PUNCT cana-2728 169 6 βˆ’ βˆ’ PROPN cana-2728 170 1 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 170 2 , , PUNCT cana-2728 170 3 𝑋4)πœ‚(𝑋3 𝑋4)πœ‚(𝑋3 NUM cana-2728 170 4 ) ) PUNCT cana-2728 170 5 } } PUNCT cana-2728 171 1 + + PROPN cana-2728 171 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 171 3 , , PUNCT cana-2728 171 4 𝑋2){πœ‚(𝑋4)πœ‚(𝑋3 𝑋2){πœ‚(𝑋4)πœ‚(𝑋3 NOUN cana-2728 171 5 ) ) PUNCT cana-2728 171 6 + + CCONJ cana-2728 172 1 𝑔(𝑋3 𝑔(𝑋3 ADJ cana-2728 172 2 , , PUNCT cana-2728 172 3 𝑋4 𝑋4 NOUN cana-2728 172 4 ) ) PUNCT cana-2728 172 5 } } PUNCT cana-2728 173 1 + + PROPN cana-2728 173 2 πœ‚(𝑋2){𝑔(𝑋3 πœ‚(𝑋2){𝑔(𝑋3 PROPN cana-2728 173 3 , , PUNCT cana-2728 173 4 𝑋4)πœ‚(𝑋1 𝑋4)πœ‚(𝑋1 NUM cana-2728 173 5 ) ) PUNCT cana-2728 173 6 βˆ’ βˆ’ PROPN cana-2728 173 7 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 173 8 , , PUNCT cana-2728 173 9 𝑋4)πœ‚(𝑋3 𝑋4)πœ‚(𝑋3 NUM cana-2728 173 10 ) ) PUNCT cana-2728 173 11 } } PUNCT cana-2728 174 1 βˆ’ βˆ’ PROPN cana-2728 174 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 174 3 , , PUNCT cana-2728 174 4 𝑋3){πœ‚(𝑋4)πœ‚(𝑋2 𝑋3){πœ‚(𝑋4)πœ‚(𝑋2 PROPN cana-2728 174 5 ) ) PUNCT cana-2728 175 1 + + CCONJ cana-2728 176 1 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 176 2 , , PUNCT cana-2728 176 3 𝑋4 𝑋4 NOUN cana-2728 176 4 ) ) PUNCT cana-2728 176 5 } } PUNCT cana-2728 177 1 + + ADV cana-2728 177 2 πœ‚(𝑋3){𝑔(𝑋1 πœ‚(𝑋3){𝑔(𝑋1 NUM cana-2728 177 3 , , PUNCT cana-2728 177 4 𝑋4)πœ‚(𝑋2 𝑋4)πœ‚(𝑋2 NOUN cana-2728 177 5 ) ) PUNCT cana-2728 178 1 βˆ’ βˆ’ PROPN cana-2728 179 1 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 179 2 , , PUNCT cana-2728 179 3 𝑋4)πœ‚(𝑋1 𝑋4)πœ‚(𝑋1 NOUN cana-2728 179 4 ) ) PUNCT cana-2728 179 5 } } PUNCT cana-2728 180 1 + + NOUN cana-2728 180 2 πœ‚(𝑋4){𝑔(𝑋3 πœ‚(𝑋4){𝑔(𝑋3 ADJ cana-2728 180 3 , , PUNCT cana-2728 180 4 𝑋1)πœ‚(𝑋2 𝑋1)πœ‚(𝑋2 PROPN cana-2728 180 5 ) ) PUNCT cana-2728 180 6 βˆ’ βˆ’ PROPN cana-2728 181 1 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 181 2 , , PUNCT cana-2728 181 3 𝑋1)πœ‚(𝑋3 𝑋1)πœ‚(𝑋3 NUM cana-2728 181 4 ) ) PUNCT cana-2728 181 5 } } PUNCT cana-2728 181 6 ] ] PUNCT cana-2728 182 1 = = PUNCT cana-2728 182 2 0 0 X cana-2728 182 3 . . PUNCT cana-2728 182 4 ( ( PUNCT cana-2728 182 5 4.8 4.8 NUM cana-2728 182 6 ) ) PUNCT cana-2728 182 7 let let VERB cana-2728 182 8 { { PUNCT cana-2728 182 9 𝑒1 𝑒1 NOUN cana-2728 182 10 , , PUNCT cana-2728 182 11 𝑒2 𝑒2 PROPN cana-2728 182 12 , , PUNCT cana-2728 182 13 … … PUNCT cana-2728 182 14 , , PUNCT cana-2728 182 15 β€ˆ β€ˆ SPACE cana-2728 182 16 𝑒2𝑛+1 𝑒2𝑛+1 VERB cana-2728 182 17 } } PUNCT cana-2728 182 18 be be AUX cana-2728 182 19 an an DET cana-2728 182 20 orthonormal orthonormal ADJ cana-2728 182 21 basis basis NOUN cana-2728 182 22 . . PUNCT cana-2728 183 1 putting put VERB cana-2728 183 2 𝑋1 𝑋1 NOUN cana-2728 183 3 = = SYM cana-2728 183 4 𝑋2 𝑋2 VERB cana-2728 183 5 = = PUNCT cana-2728 183 6 𝑒𝑖 𝑒𝑖 X cana-2728 183 7 in in ADP cana-2728 183 8 ( ( PUNCT cana-2728 183 9 4.8 4.8 NUM cana-2728 183 10 ) ) PUNCT cana-2728 183 11 and and CCONJ cana-2728 183 12 taking take VERB cana-2728 183 13 summation summation NOUN cana-2728 183 14 over over ADP cana-2728 183 15 𝑖 𝑖 ADP cana-2728 183 16 , , PUNCT cana-2728 183 17 where where SCONJ cana-2728 183 18 1 1 NUM cana-2728 183 19 ≀ ≀ NOUN cana-2728 183 20 𝑖 𝑖 SYM cana-2728 183 21 ≀ ≀ NOUN cana-2728 183 22 ( ( PUNCT cana-2728 183 23 2𝑛 2𝑛 NOUN cana-2728 183 24 + + PROPN cana-2728 183 25 1 1 NUM cana-2728 183 26 ) ) PUNCT cana-2728 183 27 , , PUNCT cana-2728 183 28 we we PRON cana-2728 183 29 have have VERB cana-2728 183 30 [ [ PUNCT cana-2728 183 31 2𝑛+1 2𝑛+1 NUM cana-2728 183 32 2𝑛 2𝑛 PROPN cana-2728 183 33 ] ] PUNCT cana-2728 184 1 𝑆(𝑋3 𝑆(𝑋3 INTJ cana-2728 184 2 , , PUNCT cana-2728 184 3 𝑋4 𝑋4 VERB cana-2728 184 4 ) ) PUNCT cana-2728 185 1 = = SYM cana-2728 186 1 π‘Ÿ π‘Ÿ PRON cana-2728 186 2 2𝑛 2𝑛 PROPN cana-2728 186 3 𝑔(𝑋3 𝑔(𝑋3 PROPN cana-2728 186 4 , , PUNCT cana-2728 186 5 𝑋4 𝑋4 VERB cana-2728 186 6 ) ) PUNCT cana-2728 187 1 βˆ’ βˆ’ PROPN cana-2728 187 2 2 2 NUM cana-2728 187 3 ( ( PUNCT cana-2728 187 4 1 1 NUM cana-2728 187 5 + + CCONJ cana-2728 187 6 πœ† πœ† PROPN cana-2728 187 7 2𝑛 2𝑛 PROPN cana-2728 187 8 ) ) PUNCT cana-2728 188 1 [ [ X cana-2728 188 2 𝑛𝑔(𝑋3 𝑛𝑔(𝑋3 ADV cana-2728 188 3 , , PUNCT cana-2728 188 4 𝑋4 𝑋4 VERB cana-2728 188 5 ) ) PUNCT cana-2728 188 6 βˆ’(2𝑛 βˆ’(2𝑛 PUNCT cana-2728 189 1 + + NUM cana-2728 189 2 1)πœ‚(𝑋3)πœ‚(𝑋4 1)πœ‚(𝑋3)πœ‚(𝑋4 NUM cana-2728 189 3 ) ) PUNCT cana-2728 189 4 ] ] PUNCT cana-2728 189 5 . . PUNCT cana-2728 190 1 ( ( PUNCT cana-2728 190 2 4.9 4.9 NUM cana-2728 190 3 ) ) PUNCT cana-2728 190 4 again again ADV cana-2728 190 5 , , PUNCT cana-2728 190 6 taking take VERB cana-2728 190 7 orthonormal orthonormal ADJ cana-2728 190 8 frame frame NOUN cana-2728 190 9 field field NOUN cana-2728 190 10 over over ADP cana-2728 190 11 𝑋3 𝑋3 NOUN cana-2728 190 12 and and CCONJ cana-2728 190 13 𝑋4 𝑋4 VERB cana-2728 190 14 , , PUNCT cana-2728 190 15 we we PRON cana-2728 190 16 get get VERB cana-2728 190 17 πœ† πœ† PRON cana-2728 190 18 = = SYM cana-2728 190 19 βˆ’2𝑛 βˆ’2𝑛 X cana-2728 190 20 < < X cana-2728 190 21 0 0 NUM cana-2728 190 22 , , PUNCT cana-2728 190 23 which which PRON cana-2728 190 24 implies imply VERB cana-2728 190 25 that that SCONJ cana-2728 190 26 the the DET cana-2728 190 27 soliton soliton NOUN cana-2728 190 28 is be AUX cana-2728 190 29 shrinking shrink VERB cana-2728 190 30 . . PUNCT cana-2728 191 1 hence hence ADV cana-2728 191 2 , , PUNCT cana-2728 191 3 the the DET cana-2728 191 4 proof proof NOUN cana-2728 191 5 is be AUX cana-2728 191 6 completed complete VERB cana-2728 191 7 . . PUNCT cana-2728 192 1 theorem theorem VERB cana-2728 192 2 4.2 4.2 NUM cana-2728 192 3 . . PUNCT cana-2728 193 1 let let VERB cana-2728 193 2 𝑀 𝑀 PROPN cana-2728 193 3 be be AUX cana-2728 193 4 a a DET cana-2728 193 5 ( ( PUNCT cana-2728 193 6 2𝑛 2𝑛 NOUN cana-2728 193 7 + + CCONJ cana-2728 193 8 1)-dimensional 1)-dimensional NUM cana-2728 193 9 lorentzian lorentzian ADJ cana-2728 193 10 para para NOUN cana-2728 193 11 - - PUNCT cana-2728 193 12 kenmotsu kenmotsu NOUN cana-2728 193 13 manifold manifold ADJ cana-2728 193 14 and and CCONJ cana-2728 193 15 ( ( PUNCT cana-2728 193 16 𝑔 𝑔 ADJ cana-2728 193 17 , , PUNCT cana-2728 193 18 𝑉 𝑉 PROPN cana-2728 193 19 , , PUNCT cana-2728 193 20 Ξ» Ξ» NOUN cana-2728 193 21 ) ) PUNCT cana-2728 193 22 be be VERB cana-2728 193 23 a a DET cana-2728 193 24 ricci ricci NOUN cana-2728 193 25 soliton soliton NOUN cana-2728 193 26 satisfying satisfy VERB cana-2728 193 27 the the DET cana-2728 193 28 condition condition NOUN cana-2728 193 29 π‘Š2(πœ‰ π‘Š2(πœ‰ NOUN cana-2728 193 30 , , PUNCT cana-2728 193 31 𝑋1 𝑋1 PROPN cana-2728 193 32 ) ) PUNCT cana-2728 194 1 ∘ ∘ NOUN cana-2728 194 2 𝑆 𝑆 PROPN cana-2728 194 3 = = SYM cana-2728 194 4 0 0 PROPN cana-2728 194 5 in in ADP cana-2728 194 6 𝑀 𝑀 PROPN cana-2728 194 7 , , PUNCT cana-2728 194 8 then then ADV cana-2728 194 9 the the DET cana-2728 194 10 ricci ricci PROPN cana-2728 194 11 soliton soliton NOUN cana-2728 194 12 is be AUX cana-2728 194 13 steady steady ADJ cana-2728 194 14 . . PUNCT cana-2728 195 1 proof proof NOUN cana-2728 195 2 : : PUNCT cana-2728 195 3 let let VERB cana-2728 195 4 𝑀 𝑀 PRON cana-2728 195 5 be be AUX cana-2728 195 6 a a DET cana-2728 195 7 ( ( PUNCT cana-2728 195 8 2𝑛 2𝑛 NOUN cana-2728 195 9 + + CCONJ cana-2728 195 10 1)-dimensional 1)-dimensional NUM cana-2728 195 11 lorentzian lorentzian ADJ cana-2728 195 12 para para NOUN cana-2728 195 13 - - PUNCT cana-2728 195 14 kenmotsu kenmotsu NOUN cana-2728 195 15 manifold manifold ADJ cana-2728 195 16 and and CCONJ cana-2728 195 17 ( ( PUNCT cana-2728 195 18 𝑔 𝑔 ADJ cana-2728 195 19 , , PUNCT cana-2728 195 20 𝑉 𝑉 PROPN cana-2728 195 21 , , PUNCT cana-2728 195 22 Ξ» Ξ» NOUN cana-2728 195 23 ) ) PUNCT cana-2728 195 24 be be VERB cana-2728 195 25 a a DET cana-2728 195 26 ricci ricci NOUN cana-2728 195 27 soliton soliton NOUN cana-2728 195 28 in in ADP cana-2728 195 29 𝑀. 𝑀. NOUN cana-2728 195 30 communications communication NOUN cana-2728 195 31 on on ADP cana-2728 195 32 applied apply VERB cana-2728 195 33 nonlinear nonlinear ADJ cana-2728 195 34 analysis analysis NOUN cana-2728 195 35 issn issn NOUN cana-2728 195 36 : : PUNCT cana-2728 195 37 1074 1074 NUM cana-2728 195 38 - - PUNCT cana-2728 195 39 133x 133x NUM cana-2728 195 40 vol vol NOUN cana-2728 195 41 32 32 NUM cana-2728 195 42 no no NOUN cana-2728 195 43 . . PUNCT cana-2728 196 1 3s 3s NUM cana-2728 196 2 ( ( PUNCT cana-2728 196 3 2025 2025 NUM cana-2728 196 4 ) ) PUNCT cana-2728 196 5 706 706 NUM cana-2728 196 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 197 1 we we PRON cana-2728 197 2 assume assume VERB cana-2728 197 3 that that SCONJ cana-2728 197 4 the the DET cana-2728 197 5 condition condition NOUN cana-2728 197 6 π‘Š2(πœ‰ π‘Š2(πœ‰ NOUN cana-2728 197 7 , , PUNCT cana-2728 197 8 𝑋1 𝑋1 PROPN cana-2728 197 9 ) ) PUNCT cana-2728 197 10 ∘ ∘ NOUN cana-2728 197 11 𝑆 𝑆 PROPN cana-2728 197 12 = = SYM cana-2728 197 13 0 0 NUM cana-2728 197 14 holds hold NOUN cana-2728 197 15 in in ADP cana-2728 197 16 𝑀 𝑀 PROPN cana-2728 197 17 , , PUNCT cana-2728 197 18 then then ADV cana-2728 197 19 we we PRON cana-2728 197 20 have have VERB cana-2728 197 21 𝑆(π‘Š2(πœ‰ 𝑆(π‘Š2(πœ‰ ADJ cana-2728 197 22 , , PUNCT cana-2728 197 23 𝑋1)𝑋2 𝑋1)𝑋2 ADJ cana-2728 197 24 , , PUNCT cana-2728 197 25 𝑋3 𝑋3 NOUN cana-2728 197 26 ) ) PUNCT cana-2728 197 27 + + CCONJ cana-2728 198 1 𝑆(𝑋2 𝑆(𝑋2 PROPN cana-2728 198 2 , , PUNCT cana-2728 198 3 π‘Š2(πœ‰ π‘Š2(πœ‰ NOUN cana-2728 198 4 , , PUNCT cana-2728 198 5 𝑋1)𝑋3 𝑋1)𝑋3 ADJ cana-2728 198 6 ) ) PUNCT cana-2728 198 7 = = SYM cana-2728 198 8 0 0 X cana-2728 198 9 . . PUNCT cana-2728 199 1 ( ( PUNCT cana-2728 199 2 4.10 4.10 NUM cana-2728 199 3 ) ) PUNCT cana-2728 199 4 using use VERB cana-2728 199 5 ( ( PUNCT cana-2728 199 6 2.17 2.17 NUM cana-2728 199 7 ) ) PUNCT cana-2728 199 8 , , PUNCT cana-2728 199 9 ( ( PUNCT cana-2728 199 10 4.2 4.2 NUM cana-2728 199 11 ) ) PUNCT cana-2728 199 12 and and CCONJ cana-2728 199 13 ( ( PUNCT cana-2728 199 14 4.10 4.10 NUM cana-2728 199 15 ) ) PUNCT cana-2728 199 16 , , PUNCT cana-2728 199 17 we we PRON cana-2728 199 18 obtain obtain VERB cana-2728 199 19 βˆ’πœ†π‘”(𝑋1 βˆ’πœ†π‘”(𝑋1 NUM cana-2728 199 20 , , PUNCT cana-2728 199 21 𝑋2)πœ‚(𝑋3 𝑋2)πœ‚(𝑋3 NUM cana-2728 199 22 ) ) PUNCT cana-2728 199 23 βˆ’ βˆ’ NOUN cana-2728 199 24 πœ†π‘”(𝑋1 πœ†π‘”(𝑋1 PUNCT cana-2728 199 25 , , PUNCT cana-2728 199 26 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 199 27 ) ) PUNCT cana-2728 200 1 + + CCONJ cana-2728 200 2 1 1 NUM cana-2728 200 3 2𝑛 2𝑛 NOUN cana-2728 200 4 [ [ X cana-2728 200 5 𝑆(𝑄𝑋1 𝑆(𝑄𝑋1 PROPN cana-2728 200 6 , , PUNCT cana-2728 200 7 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 200 8 ) ) PUNCT cana-2728 200 9 + + X cana-2728 200 10 𝑆(𝑄𝑋1 𝑆(𝑄𝑋1 PROPN cana-2728 200 11 , , PUNCT cana-2728 200 12 𝑋2)πœ‚(𝑋3 𝑋2)πœ‚(𝑋3 NUM cana-2728 200 13 ) ) PUNCT cana-2728 200 14 ] ] PUNCT cana-2728 201 1 βˆ’π‘†(𝑋1 βˆ’π‘†(𝑋1 X cana-2728 201 2 , , PUNCT cana-2728 201 3 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 201 4 ) ) PUNCT cana-2728 201 5 βˆ’ βˆ’ PUNCT cana-2728 202 1 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 202 2 , , PUNCT cana-2728 202 3 𝑋2)πœ‚(𝑋3 𝑋2)πœ‚(𝑋3 NUM cana-2728 202 4 ) ) PUNCT cana-2728 202 5 βˆ’ βˆ’ PROPN cana-2728 203 1 1 1 NUM cana-2728 203 2 2𝑛 2𝑛 PROPN cana-2728 204 1 [ [ X cana-2728 204 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 204 3 , , PUNCT cana-2728 204 4 𝑋2)𝑆(π‘„πœ‰ 𝑋2)𝑆(π‘„πœ‰ ADJ cana-2728 204 5 , , PUNCT cana-2728 204 6 𝑋3 𝑋3 NOUN cana-2728 204 7 ) ) PUNCT cana-2728 204 8 + + ADP cana-2728 204 9 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 204 10 , , PUNCT cana-2728 204 11 𝑋3)𝑆(π‘„πœ‰ 𝑋3)𝑆(π‘„πœ‰ ADP cana-2728 204 12 , , PUNCT cana-2728 204 13 𝑋2 𝑋2 ADJ cana-2728 204 14 ) ) PUNCT cana-2728 204 15 ] ] PUNCT cana-2728 205 1 = = PUNCT cana-2728 205 2 0 0 X cana-2728 205 3 . . PUNCT cana-2728 206 1 ( ( PUNCT cana-2728 206 2 4.11 4.11 NUM cana-2728 206 3 ) ) PUNCT cana-2728 206 4 setting set VERB cana-2728 206 5 𝑋3 𝑋3 NOUN cana-2728 206 6 = = SYM cana-2728 206 7 πœ‰ πœ‰ X cana-2728 206 8 in in ADP cana-2728 206 9 ( ( PUNCT cana-2728 206 10 4.11 4.11 NUM cana-2728 206 11 ) ) PUNCT cana-2728 206 12 and and CCONJ cana-2728 206 13 using use VERB cana-2728 206 14 ( ( PUNCT cana-2728 206 15 2.1 2.1 NUM cana-2728 206 16 ) ) PUNCT cana-2728 206 17 , , PUNCT cana-2728 206 18 ( ( PUNCT cana-2728 206 19 2.3 2.3 NUM cana-2728 206 20 ) ) PUNCT cana-2728 206 21 and and CCONJ cana-2728 206 22 ( ( PUNCT cana-2728 206 23 2.17 2.17 NUM cana-2728 206 24 ) ) PUNCT cana-2728 206 25 , , PUNCT cana-2728 206 26 we we PRON cana-2728 206 27 get get VERB cana-2728 206 28 ( ( PUNCT cana-2728 206 29 πœ† πœ† PROPN cana-2728 206 30 + + SYM cana-2728 206 31 1 1 NUM cana-2728 206 32 2𝑛 2𝑛 PROPN cana-2728 206 33 πœ†2 πœ†2 PROPN cana-2728 206 34 ) ) PUNCT cana-2728 207 1 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 207 2 , , PUNCT cana-2728 207 3 𝑋2 𝑋2 VERB cana-2728 207 4 ) ) PUNCT cana-2728 208 1 + + CCONJ cana-2728 208 2 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 208 3 , , PUNCT cana-2728 208 4 𝑋2 𝑋2 VERB cana-2728 208 5 ) ) PUNCT cana-2728 209 1 + + CCONJ cana-2728 209 2 1 1 NUM cana-2728 209 3 2𝑛 2𝑛 NOUN cana-2728 210 1 [ [ X cana-2728 210 2 πœ†2πœ‚(𝑋1)πœ‚(𝑋2 πœ†2πœ‚(𝑋1)πœ‚(𝑋2 X cana-2728 210 3 ) ) PUNCT cana-2728 210 4 βˆ’ βˆ’ PROPN cana-2728 210 5 𝑆(𝑄𝑋1 𝑆(𝑄𝑋1 PROPN cana-2728 210 6 , , PUNCT cana-2728 210 7 𝑋2 𝑋2 VERB cana-2728 210 8 ) ) PUNCT cana-2728 210 9 ] ] PUNCT cana-2728 211 1 βˆ’ βˆ’ PROPN cana-2728 211 2 1 1 NUM cana-2728 211 3 2𝑛 2𝑛 PROPN cana-2728 211 4 πœ‚(𝑋1)𝑆(π‘„πœ‰ πœ‚(𝑋1)𝑆(π‘„πœ‰ PROPN cana-2728 211 5 , , PUNCT cana-2728 211 6 𝑋2 𝑋2 VERB cana-2728 211 7 ) ) PUNCT cana-2728 212 1 = = SYM cana-2728 212 2 0 0 X cana-2728 212 3 . . PUNCT cana-2728 212 4 ( ( PUNCT cana-2728 212 5 4.12 4.12 NUM cana-2728 212 6 ) ) PUNCT cana-2728 212 7 again again ADV cana-2728 212 8 , , PUNCT cana-2728 212 9 putting put VERB cana-2728 212 10 𝑋2 𝑋2 VERB cana-2728 212 11 = = X cana-2728 212 12 πœ‰ πœ‰ NOUN cana-2728 212 13 in in ADP cana-2728 212 14 ( ( PUNCT cana-2728 212 15 4.12 4.12 NUM cana-2728 212 16 ) ) PUNCT cana-2728 212 17 and and CCONJ cana-2728 212 18 using use VERB cana-2728 212 19 ( ( PUNCT cana-2728 212 20 2.1 2.1 NUM cana-2728 212 21 ) ) PUNCT cana-2728 212 22 and and CCONJ cana-2728 212 23 ( ( PUNCT cana-2728 212 24 2.17 2.17 NUM cana-2728 212 25 ) ) PUNCT cana-2728 212 26 , , PUNCT cana-2728 212 27 we we PRON cana-2728 212 28 have have VERB cana-2728 212 29 πœ† πœ† PRON cana-2728 212 30 = = SYM cana-2728 212 31 0 0 PROPN cana-2728 212 32 . . PUNCT cana-2728 213 1 the the DET cana-2728 213 2 above above ADJ cana-2728 213 3 equation equation NOUN cana-2728 213 4 implies imply VERB cana-2728 213 5 that that SCONJ cana-2728 213 6 the the DET cana-2728 213 7 ricci ricci PROPN cana-2728 213 8 soliton soliton NOUN cana-2728 213 9 is be AUX cana-2728 213 10 steady steady ADJ cana-2728 213 11 . . PUNCT cana-2728 214 1 hence hence ADV cana-2728 214 2 , , PUNCT cana-2728 214 3 we we PRON cana-2728 214 4 complete complete VERB cana-2728 214 5 the the DET cana-2728 214 6 proof proof NOUN cana-2728 214 7 . . PUNCT cana-2728 215 1 4 4 X cana-2728 215 2 . . X cana-2728 215 3 ricci ricci PROPN cana-2728 215 4 tensor tensor NOUN cana-2728 215 5 of of ADP cana-2728 215 6 a a DET cana-2728 215 7 lorentzian lorentzian ADJ cana-2728 215 8 para para NOUN cana-2728 215 9 - - PUNCT cana-2728 215 10 kenmotsu kenmotsu NOUN cana-2728 215 11 manifold manifold ADJ cana-2728 215 12 admitting admit VERB cana-2728 215 13 a a DET cana-2728 215 14 ricci ricci PROPN cana-2728 215 15 soliton soliton NOUN cana-2728 215 16 in in ADP cana-2728 215 17 this this DET cana-2728 215 18 section section NOUN cana-2728 215 19 , , PUNCT cana-2728 215 20 we we PRON cana-2728 215 21 study study VERB cana-2728 215 22 ricci ricci PROPN cana-2728 215 23 tensor tensor NOUN cana-2728 215 24 of of ADP cana-2728 215 25 a a DET cana-2728 215 26 lorentzian lorentzian ADJ cana-2728 215 27 para para NOUN cana-2728 215 28 - - PUNCT cana-2728 215 29 kenmotsu kenmotsu NOUN cana-2728 215 30 manifold manifold ADJ cana-2728 215 31 admitting admit VERB cana-2728 215 32 a a DET cana-2728 215 33 ricci ricci PROPN cana-2728 215 34 soliton soliton NOUN cana-2728 215 35 . . PUNCT cana-2728 216 1 theorem theorem VERB cana-2728 216 2 5.1 5.1 NUM cana-2728 216 3 . . PUNCT cana-2728 217 1 let let VERB cana-2728 217 2 𝑀 𝑀 PRON cana-2728 217 3 be be AUX cana-2728 217 4 a a DET cana-2728 217 5 lorentzian lorentzian ADJ cana-2728 217 6 para para NOUN cana-2728 217 7 - - PUNCT cana-2728 217 8 kenmotsu kenmotsu NOUN cana-2728 217 9 manifold manifold ADJ cana-2728 217 10 admitting admit VERB cana-2728 217 11 a a DET cana-2728 217 12 ricci ricci PROPN cana-2728 217 13 soliton soliton NOUN cana-2728 217 14 ( ( PUNCT cana-2728 217 15 𝑔 𝑔 PROPN cana-2728 217 16 , , PUNCT cana-2728 217 17 𝑉 𝑉 PROPN cana-2728 217 18 , , PUNCT cana-2728 217 19 Ξ» Ξ» NOUN cana-2728 217 20 ) ) PUNCT cana-2728 217 21 . . PUNCT cana-2728 218 1 if if SCONJ cana-2728 218 2 the the DET cana-2728 218 3 ricci ricci PROPN cana-2728 218 4 tensor tensor NOUN cana-2728 218 5 𝑆 𝑆 PROPN cana-2728 218 6 of of ADP cana-2728 218 7 the the DET cana-2728 218 8 manifold manifold NOUN cana-2728 218 9 is be AUX cana-2728 218 10 πœ‚-recurrent πœ‚-recurrent PROPN cana-2728 218 11 , , PUNCT cana-2728 218 12 then then ADV cana-2728 218 13 the the DET cana-2728 218 14 ricci ricci PROPN cana-2728 218 15 soliton soliton NOUN cana-2728 218 16 is be AUX cana-2728 218 17 steady steady ADJ cana-2728 218 18 . . PUNCT cana-2728 219 1 proof proof NOUN cana-2728 219 2 : : PUNCT cana-2728 219 3 suppose suppose VERB cana-2728 219 4 that that SCONJ cana-2728 219 5 the the DET cana-2728 219 6 ricci ricci PROPN cana-2728 219 7 tensor tensor NOUN cana-2728 219 8 of of ADP cana-2728 219 9 the the DET cana-2728 219 10 lorentzian lorentzian ADJ cana-2728 219 11 para para NOUN cana-2728 219 12 - - PUNCT cana-2728 219 13 kenmotsu kenmotsu PROPN cana-2728 219 14 manifold manifold PROPN cana-2728 219 15 is be AUX cana-2728 219 16 πœ‚-recurrent πœ‚-recurrent PROPN cana-2728 219 17 , , PUNCT cana-2728 219 18 i.e. i.e. X cana-2728 219 19 , , PUNCT cana-2728 219 20 ( ( PUNCT cana-2728 219 21 𝛻𝑋1 𝛻𝑋1 PROPN cana-2728 219 22 𝑆)(𝑋2 𝑆)(𝑋2 NUM cana-2728 219 23 , , PUNCT cana-2728 219 24 𝑋3 𝑋3 NOUN cana-2728 219 25 ) ) PUNCT cana-2728 220 1 = = SYM cana-2728 220 2 πœ‚(𝑋1)𝑆(𝑋2 πœ‚(𝑋1)𝑆(𝑋2 NOUN cana-2728 220 3 , , PUNCT cana-2728 220 4 𝑋3 𝑋3 NOUN cana-2728 220 5 ) ) PUNCT cana-2728 220 6 , , PUNCT cana-2728 220 7 ( ( PUNCT cana-2728 220 8 5.1 5.1 NUM cana-2728 220 9 ) ) PUNCT cana-2728 220 10 for for ADP cana-2728 220 11 all all DET cana-2728 220 12 vector vector NOUN cana-2728 220 13 fields field NOUN cana-2728 220 14 𝑋1 𝑋1 PROPN cana-2728 220 15 , , PUNCT cana-2728 220 16 𝑋2 𝑋2 VERB cana-2728 220 17 , , PUNCT cana-2728 220 18 𝑋3 𝑋3 NOUN cana-2728 220 19 on on ADP cana-2728 220 20 𝑀. 𝑀. PROPN cana-2728 220 21 then then ADV cana-2728 220 22 , , PUNCT cana-2728 220 23 by by ADP cana-2728 220 24 using use VERB cana-2728 220 25 ( ( PUNCT cana-2728 220 26 2.15 2.15 NUM cana-2728 220 27 ) ) PUNCT cana-2728 220 28 , , PUNCT cana-2728 220 29 we we PRON cana-2728 220 30 have have VERB cana-2728 220 31 ( ( PUNCT cana-2728 220 32 𝛻𝑋1 𝛻𝑋1 PROPN cana-2728 220 33 𝑆)(𝑋2 𝑆)(𝑋2 NUM cana-2728 220 34 , , PUNCT cana-2728 220 35 𝑋3 𝑋3 NOUN cana-2728 220 36 ) ) PUNCT cana-2728 220 37 = = SYM cana-2728 221 1 βˆ’2πœ‚(𝑋1)πœ‚(𝑋2)πœ‚(𝑋3 βˆ’2πœ‚(𝑋1)πœ‚(𝑋2)πœ‚(𝑋3 ADJ cana-2728 221 2 ) ) PUNCT cana-2728 222 1 βˆ’ βˆ’ PROPN cana-2728 222 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 222 3 , , PUNCT cana-2728 222 4 𝑋2)πœ‚(𝑋3 𝑋2)πœ‚(𝑋3 NUM cana-2728 222 5 ) ) PUNCT cana-2728 222 6 βˆ’ βˆ’ PROPN cana-2728 222 7 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 222 8 , , PUNCT cana-2728 222 9 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 222 10 ) ) PUNCT cana-2728 222 11 . . PUNCT cana-2728 223 1 ( ( PUNCT cana-2728 223 2 5.2 5.2 X cana-2728 223 3 ) ) PUNCT cana-2728 223 4 using use VERB cana-2728 223 5 ( ( PUNCT cana-2728 223 6 2.15 2.15 NUM cana-2728 223 7 ) ) PUNCT cana-2728 223 8 in in ADP cana-2728 223 9 ( ( PUNCT cana-2728 223 10 5.1 5.1 NUM cana-2728 223 11 ) ) PUNCT cana-2728 223 12 and and CCONJ cana-2728 223 13 comparing compare VERB cana-2728 223 14 with with ADP cana-2728 223 15 ( ( PUNCT cana-2728 223 16 5.2 5.2 NUM cana-2728 223 17 ) ) PUNCT cana-2728 223 18 , , PUNCT cana-2728 223 19 we we PRON cana-2728 223 20 get get VERB cana-2728 223 21 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 ADJ cana-2728 223 22 , , PUNCT cana-2728 223 23 𝑋2)πœ‚(𝑋3 𝑋2)πœ‚(𝑋3 NUM cana-2728 223 24 ) ) PUNCT cana-2728 223 25 βˆ’ βˆ’ PROPN cana-2728 223 26 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 223 27 , , PUNCT cana-2728 223 28 𝑋3)πœ‚(𝑋2 𝑋3)πœ‚(𝑋2 PROPN cana-2728 223 29 ) ) PUNCT cana-2728 223 30 βˆ’ βˆ’ PROPN cana-2728 223 31 ( ( PUNCT cana-2728 223 32 1 1 NUM cana-2728 223 33 βˆ’ βˆ’ PROPN cana-2728 223 34 πœ†)𝑔(𝑋2 πœ†)𝑔(𝑋2 PROPN cana-2728 223 35 , , PUNCT cana-2728 223 36 𝑋3)πœ‚(𝑋1 𝑋3)πœ‚(𝑋1 X cana-2728 223 37 ) ) PUNCT cana-2728 223 38 = = SYM cana-2728 223 39 3πœ‚(𝑋1)πœ‚(𝑋2)πœ‚(𝑋3 3πœ‚(𝑋1)πœ‚(𝑋2)πœ‚(𝑋3 NUM cana-2728 223 40 ) ) PUNCT cana-2728 223 41 . . PUNCT cana-2728 224 1 ( ( PUNCT cana-2728 224 2 5.3 5.3 NUM cana-2728 224 3 ) ) PUNCT cana-2728 224 4 setting set VERB cana-2728 224 5 𝑋2 𝑋2 ADJ cana-2728 224 6 = = NOUN cana-2728 224 7 𝑋3 𝑋3 NOUN cana-2728 224 8 = = SYM cana-2728 224 9 πœ‰ πœ‰ X cana-2728 224 10 in in ADP cana-2728 224 11 ( ( PUNCT cana-2728 224 12 5.3 5.3 NUM cana-2728 224 13 ) ) PUNCT cana-2728 224 14 , , PUNCT cana-2728 224 15 we we PRON cana-2728 224 16 obtain obtain VERB cana-2728 224 17 βˆ’πœ†πœ‚(𝑋1 βˆ’πœ†πœ‚(𝑋1 NOUN cana-2728 224 18 ) ) PUNCT cana-2728 224 19 = = SYM cana-2728 225 1 0 0 X cana-2728 225 2 . . PUNCT cana-2728 226 1 ( ( PUNCT cana-2728 226 2 5.4 5.4 NUM cana-2728 226 3 ) ) PUNCT cana-2728 226 4 since since SCONJ cana-2728 226 5 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 226 6 ) ) PUNCT cana-2728 226 7 β‰  β‰  PROPN cana-2728 226 8 0 0 NUM cana-2728 226 9 , , PUNCT cana-2728 226 10 we we PRON cana-2728 226 11 have have VERB cana-2728 226 12 πœ† πœ† NOUN cana-2728 226 13 = = SYM cana-2728 226 14 0 0 NUM cana-2728 226 15 . . PUNCT cana-2728 227 1 therefore therefore ADV cana-2728 227 2 , , PUNCT cana-2728 227 3 the the DET cana-2728 227 4 ricci ricci PROPN cana-2728 227 5 soliton soliton NOUN cana-2728 227 6 is be AUX cana-2728 227 7 steady steady ADJ cana-2728 227 8 . . PUNCT cana-2728 228 1 communications communication NOUN cana-2728 228 2 on on ADP cana-2728 228 3 applied apply VERB cana-2728 228 4 nonlinear nonlinear ADJ cana-2728 228 5 analysis analysis NOUN cana-2728 228 6 issn issn NOUN cana-2728 228 7 : : PUNCT cana-2728 228 8 1074 1074 NUM cana-2728 228 9 - - PUNCT cana-2728 228 10 133x 133x NUM cana-2728 228 11 vol vol NOUN cana-2728 228 12 32 32 NUM cana-2728 228 13 no no NOUN cana-2728 228 14 . . PUNCT cana-2728 229 1 3s 3s NUM cana-2728 229 2 ( ( PUNCT cana-2728 229 3 2025 2025 NUM cana-2728 229 4 ) ) PUNCT cana-2728 229 5 707 707 NUM cana-2728 229 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 229 7 theorem theorem VERB cana-2728 229 8 5.2 5.2 NUM cana-2728 229 9 . . PUNCT cana-2728 230 1 let let VERB cana-2728 230 2 𝑀 𝑀 PROPN cana-2728 230 3 be be AUX cana-2728 230 4 an an DET cana-2728 230 5 lorentzian lorentzian ADJ cana-2728 230 6 para para NOUN cana-2728 230 7 - - PUNCT cana-2728 230 8 kenmotsu kenmotsu NOUN cana-2728 230 9 manifold manifold NOUN cana-2728 230 10 , , PUNCT cana-2728 230 11 admitting admit VERB cana-2728 230 12 a a DET cana-2728 230 13 ricci ricci PROPN cana-2728 230 14 soliton soliton NOUN cana-2728 230 15 ( ( PUNCT cana-2728 230 16 𝑔 𝑔 PROPN cana-2728 230 17 , , PUNCT cana-2728 230 18 𝑉 𝑉 PROPN cana-2728 230 19 , , PUNCT cana-2728 230 20 Ξ» Ξ» NOUN cana-2728 230 21 ) ) PUNCT cana-2728 230 22 . . PUNCT cana-2728 231 1 then then ADV cana-2728 231 2 𝑄 𝑄 PROPN cana-2728 231 3 and and CCONJ cana-2728 231 4 𝑆 𝑆 PROPN cana-2728 231 5 are be AUX cana-2728 231 6 parallel parallel ADJ cana-2728 231 7 along along ADP cana-2728 231 8 πœ‰ πœ‰ PROPN cana-2728 231 9 , , PUNCT cana-2728 231 10 where where SCONJ cana-2728 231 11 𝑄 𝑄 PRON cana-2728 231 12 is be AUX cana-2728 231 13 the the DET cana-2728 231 14 ricci ricci PROPN cana-2728 231 15 operator operator NOUN cana-2728 231 16 , , PUNCT cana-2728 231 17 defined define VERB cana-2728 231 18 by by ADP cana-2728 231 19 𝑆(𝑋1 𝑆(𝑋1 PROPN cana-2728 231 20 , , PUNCT cana-2728 231 21 𝑋2 𝑋2 VERB cana-2728 231 22 ) ) PUNCT cana-2728 232 1 = = SYM cana-2728 232 2 𝑔(𝑄𝑋1 𝑔(𝑄𝑋1 ADJ cana-2728 232 3 , , PUNCT cana-2728 232 4 𝑋2 𝑋2 VERB cana-2728 232 5 ) ) PUNCT cana-2728 232 6 and and CCONJ cana-2728 232 7 𝑆 𝑆 PROPN cana-2728 232 8 is be AUX cana-2728 232 9 the the DET cana-2728 232 10 ricci ricci PROPN cana-2728 232 11 tensor tensor NOUN cana-2728 232 12 of of ADP cana-2728 232 13 𝑀. 𝑀. PROPN cana-2728 232 14 proof proof NOUN cana-2728 232 15 : : PUNCT cana-2728 232 16 we we PRON cana-2728 232 17 can can AUX cana-2728 232 18 express express VERB cana-2728 232 19 the the DET cana-2728 232 20 equations equation NOUN cana-2728 232 21 for for ADP cana-2728 232 22 the the DET cana-2728 232 23 ricci ricci PROPN cana-2728 232 24 operator operator NOUN cana-2728 232 25 and and CCONJ cana-2728 232 26 ricci ricci PROPN cana-2728 232 27 tensor tensor NOUN cana-2728 232 28 along along ADP cana-2728 232 29 πœ‰ πœ‰ X cana-2728 232 30 as as SCONJ cana-2728 232 31 follows follow VERB cana-2728 232 32 , , PUNCT cana-2728 232 33 ( ( PUNCT cana-2728 232 34 βˆ‡πœ‰π‘„)𝑋1 βˆ‡πœ‰π‘„)𝑋1 NOUN cana-2728 232 35 = = SYM cana-2728 232 36 βˆ‡πœ‰π‘„(𝑋1 βˆ‡πœ‰π‘„(𝑋1 X cana-2728 232 37 ) ) PUNCT cana-2728 232 38 βˆ’ βˆ’ NOUN cana-2728 232 39 𝑄(βˆ‡πœ‰π‘‹1 𝑄(βˆ‡πœ‰π‘‹1 NOUN cana-2728 232 40 ) ) PUNCT cana-2728 232 41 ( ( PUNCT cana-2728 232 42 5.5 5.5 NUM cana-2728 232 43 ) ) PUNCT cana-2728 232 44 and and CCONJ cana-2728 232 45 ( ( PUNCT cana-2728 232 46 βˆ‡πœ‰ βˆ‡πœ‰ PROPN cana-2728 232 47 𝑆)(𝑋1 𝑆)(𝑋1 NUM cana-2728 232 48 , , PUNCT cana-2728 232 49 𝑋2 𝑋2 VERB cana-2728 232 50 ) ) PUNCT cana-2728 232 51 = = SYM cana-2728 232 52 βˆ‡πœ‰π‘†(𝑋1 βˆ‡πœ‰π‘†(𝑋1 X cana-2728 232 53 , , PUNCT cana-2728 232 54 𝑋2 𝑋2 VERB cana-2728 232 55 ) ) PUNCT cana-2728 233 1 βˆ’ βˆ’ PROPN cana-2728 233 2 𝑆(βˆ‡πœ‰π‘‹1 𝑆(βˆ‡πœ‰π‘‹1 PROPN cana-2728 233 3 , , PUNCT cana-2728 233 4 𝑋2 𝑋2 VERB cana-2728 233 5 ) ) PUNCT cana-2728 234 1 βˆ’ βˆ’ PROPN cana-2728 234 2 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 234 3 , , PUNCT cana-2728 234 4 βˆ‡πœ‰π‘‹2 βˆ‡πœ‰π‘‹2 NUM cana-2728 234 5 ) ) PUNCT cana-2728 234 6 . . PUNCT cana-2728 235 1 ( ( PUNCT cana-2728 235 2 5.6 5.6 NUM cana-2728 235 3 ) ) PUNCT cana-2728 235 4 using use VERB cana-2728 235 5 ( ( PUNCT cana-2728 235 6 2.16 2.16 NUM cana-2728 235 7 ) ) PUNCT cana-2728 235 8 in in ADP cana-2728 235 9 ( ( PUNCT cana-2728 235 10 5.5 5.5 NUM cana-2728 235 11 ) ) PUNCT cana-2728 235 12 , , PUNCT cana-2728 235 13 we we PRON cana-2728 235 14 can can AUX cana-2728 235 15 simplify simplify VERB cana-2728 235 16 to to PART cana-2728 235 17 obtain obtain VERB cana-2728 235 18 ( ( PUNCT cana-2728 235 19 βˆ‡πœ‰π‘„)𝑋1 βˆ‡πœ‰π‘„)𝑋1 NOUN cana-2728 235 20 = = SYM cana-2728 235 21 0 0 NUM cana-2728 235 22 . . PUNCT cana-2728 236 1 ( ( PUNCT cana-2728 236 2 5.7 5.7 NUM cana-2728 236 3 ) ) PUNCT cana-2728 236 4 similarly similarly ADV cana-2728 236 5 , , PUNCT cana-2728 236 6 applying apply VERB cana-2728 236 7 ( ( PUNCT cana-2728 236 8 2.15 2.15 NUM cana-2728 236 9 ) ) PUNCT cana-2728 236 10 to to ADP cana-2728 236 11 ( ( PUNCT cana-2728 236 12 5.6 5.6 NUM cana-2728 236 13 ) ) PUNCT cana-2728 236 14 , , PUNCT cana-2728 236 15 we we PRON cana-2728 236 16 get get VERB cana-2728 236 17 ( ( PUNCT cana-2728 236 18 βˆ‡πœ‰ βˆ‡πœ‰ PROPN cana-2728 236 19 𝑆)(𝑋1 𝑆)(𝑋1 NUM cana-2728 236 20 , , PUNCT cana-2728 236 21 𝑋2 𝑋2 VERB cana-2728 236 22 ) ) PUNCT cana-2728 237 1 = = SYM cana-2728 237 2 0 0 X cana-2728 237 3 . . PUNCT cana-2728 237 4 ( ( PUNCT cana-2728 237 5 5.8 5.8 NUM cana-2728 237 6 ) ) PUNCT cana-2728 237 7 therefore therefore ADV cana-2728 237 8 , , PUNCT cana-2728 237 9 from from ADP cana-2728 237 10 ( ( PUNCT cana-2728 237 11 5.7 5.7 NUM cana-2728 237 12 ) ) PUNCT cana-2728 237 13 and and CCONJ cana-2728 237 14 ( ( PUNCT cana-2728 237 15 5.8 5.8 NUM cana-2728 237 16 ) ) PUNCT cana-2728 237 17 , , PUNCT cana-2728 237 18 we we PRON cana-2728 237 19 can can AUX cana-2728 237 20 conclude conclude VERB cana-2728 237 21 that that SCONJ cana-2728 237 22 𝑄 𝑄 PROPN cana-2728 237 23 and and CCONJ cana-2728 237 24 𝑆 𝑆 PROPN cana-2728 237 25 are be AUX cana-2728 237 26 parallel parallel ADJ cana-2728 237 27 along along ADP cana-2728 237 28 πœ‰ πœ‰ PROPN cana-2728 237 29 , , PUNCT cana-2728 237 30 which which PRON cana-2728 237 31 completes complete VERB cana-2728 237 32 the the DET cana-2728 237 33 proof proof NOUN cana-2728 237 34 . . PUNCT cana-2728 238 1 5 5 X cana-2728 238 2 . . X cana-2728 238 3 ricci ricci PROPN cana-2728 238 4 soliton soliton NOUN cana-2728 238 5 in in ADP cana-2728 238 6 a a DET cana-2728 238 7 lorentzian lorentzian ADJ cana-2728 238 8 para para NOUN cana-2728 238 9 - - PUNCT cana-2728 238 10 kenmotsu kenmotsu NOUN cana-2728 238 11 manifold manifold ADJ cana-2728 238 12 and and CCONJ cana-2728 238 13 its its PRON cana-2728 238 14 curvature curvature NOUN cana-2728 238 15 properties property NOUN cana-2728 238 16 in in ADP cana-2728 238 17 this this DET cana-2728 238 18 section section NOUN cana-2728 238 19 , , PUNCT cana-2728 238 20 we we PRON cana-2728 238 21 explore explore VERB cana-2728 238 22 some some DET cana-2728 238 23 curvature curvature NOUN cana-2728 238 24 properties property NOUN cana-2728 238 25 of of ADP cana-2728 238 26 a a DET cana-2728 238 27 lorentzian lorentzian ADJ cana-2728 238 28 para para NOUN cana-2728 238 29 - - PUNCT cana-2728 238 30 kenmotsu kenmotsu NOUN cana-2728 238 31 manifold manifold ADJ cana-2728 238 32 admitting admit VERB cana-2728 238 33 a a DET cana-2728 238 34 ricci ricci PROPN cana-2728 238 35 soliton soliton NOUN cana-2728 238 36 . . PUNCT cana-2728 239 1 definition definition NOUN cana-2728 239 2 6.1 6.1 NUM cana-2728 239 3 . . PUNCT cana-2728 240 1 [ [ X cana-2728 240 2 19 19 NUM cana-2728 240 3 ] ] PUNCT cana-2728 240 4 in in ADP cana-2728 240 5 a a DET cana-2728 240 6 lorentzian lorentzian ADJ cana-2728 240 7 para para NOUN cana-2728 240 8 - - PUNCT cana-2728 240 9 kenmotsu kenmotsu PROPN cana-2728 240 10 manifold manifold PROPN cana-2728 240 11 𝑀 𝑀 PROPN cana-2728 240 12 , , PUNCT cana-2728 240 13 the the DET cana-2728 240 14 projective projective ADJ cana-2728 240 15 curvature curvature NOUN cana-2728 240 16 𝑃 𝑃 NOUN cana-2728 240 17 of of ADP cana-2728 240 18 the the DET cana-2728 240 19 manifold manifold NOUN cana-2728 240 20 is be AUX cana-2728 240 21 defined define VERB cana-2728 240 22 as as ADP cana-2728 240 23 𝑃(𝑋1 𝑃(𝑋1 NUM cana-2728 240 24 , , PUNCT cana-2728 240 25 𝑋2)𝑋3 𝑋2)𝑋3 X cana-2728 240 26 = = SYM cana-2728 240 27 𝑅(𝑋1 𝑅(𝑋1 PROPN cana-2728 240 28 , , PUNCT cana-2728 240 29 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 240 30 βˆ’ βˆ’ ADP cana-2728 240 31 1 1 NUM cana-2728 240 32 2𝑛 2𝑛 PROPN cana-2728 240 33 [ [ X cana-2728 240 34 𝑆(𝑋2 𝑆(𝑋2 PROPN cana-2728 240 35 , , PUNCT cana-2728 240 36 𝑋3 𝑋3 NOUN cana-2728 240 37 ) ) PUNCT cana-2728 240 38 𝑋1 𝑋1 PROPN cana-2728 241 1 βˆ’ βˆ’ PROPN cana-2728 242 1 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 242 2 , , PUNCT cana-2728 242 3 𝑋3 𝑋3 NOUN cana-2728 242 4 ) ) PUNCT cana-2728 242 5 𝑋2 𝑋2 VERB cana-2728 242 6 ] ] X cana-2728 242 7 , , PUNCT cana-2728 242 8 ( ( PUNCT cana-2728 243 1 6.1 6.1 NUM cana-2728 243 2 ) ) PUNCT cana-2728 243 3 for for ADP cana-2728 243 4 all all DET cana-2728 243 5 vector vector NOUN cana-2728 243 6 fields field NOUN cana-2728 243 7 𝑋1 𝑋1 PROPN cana-2728 243 8 , , PUNCT cana-2728 243 9 𝑋2 𝑋2 VERB cana-2728 243 10 and and CCONJ cana-2728 243 11 𝑋3 𝑋3 VERB cana-2728 243 12 on on ADP cana-2728 243 13 𝑀. 𝑀. NOUN cana-2728 243 14 proposition proposition NOUN cana-2728 243 15 6.1 6.1 NUM cana-2728 243 16 . . PUNCT cana-2728 244 1 a a DET cana-2728 244 2 lorentzian lorentzian ADJ cana-2728 244 3 para para NOUN cana-2728 244 4 - - PUNCT cana-2728 244 5 kenmotsu kenmotsu PROPN cana-2728 244 6 manifold manifold PROPN cana-2728 244 7 𝑀 𝑀 PROPN cana-2728 244 8 , , PUNCT cana-2728 244 9 admitting admit VERB cana-2728 244 10 a a DET cana-2728 244 11 ricci ricci PROPN cana-2728 244 12 soliton soliton NOUN cana-2728 244 13 ( ( PUNCT cana-2728 244 14 𝑔 𝑔 PROPN cana-2728 244 15 , , PUNCT cana-2728 244 16 𝑉 𝑉 PROPN cana-2728 244 17 , , PUNCT cana-2728 244 18 Ξ» Ξ» NOUN cana-2728 244 19 ) ) PUNCT cana-2728 244 20 is be AUX cana-2728 244 21 πœ‰projectively πœ‰projectively ADV cana-2728 244 22 flat flat ADJ cana-2728 244 23 iff iff NOUN cana-2728 244 24 the the DET cana-2728 244 25 soliton soliton NOUN cana-2728 244 26 is be AUX cana-2728 244 27 shrinking shrink VERB cana-2728 244 28 . . PUNCT cana-2728 245 1 proof proof NOUN cana-2728 245 2 : : PUNCT cana-2728 245 3 putting put VERB cana-2728 245 4 𝑋3 𝑋3 NOUN cana-2728 245 5 = = SYM cana-2728 245 6 πœ‰ πœ‰ X cana-2728 245 7 in in ADP cana-2728 245 8 ( ( PUNCT cana-2728 245 9 6.1 6.1 NUM cana-2728 245 10 ) ) PUNCT cana-2728 245 11 and and CCONJ cana-2728 245 12 by by ADP cana-2728 245 13 using use VERB cana-2728 245 14 ( ( PUNCT cana-2728 245 15 2.10 2.10 NUM cana-2728 245 16 ) ) PUNCT cana-2728 245 17 and and CCONJ cana-2728 245 18 ( ( PUNCT cana-2728 245 19 2.11 2.11 NUM cana-2728 245 20 ) ) PUNCT cana-2728 245 21 , , PUNCT cana-2728 245 22 we we PRON cana-2728 245 23 get get VERB cana-2728 245 24 𝑃(𝑋1 𝑃(𝑋1 NUM cana-2728 245 25 , , PUNCT cana-2728 245 26 𝑋2)πœ‰ 𝑋2)πœ‰ X cana-2728 245 27 = = PUNCT cana-2728 245 28 [ [ PUNCT cana-2728 245 29 2𝑛+πœ† 2𝑛+πœ† NUM cana-2728 245 30 2𝑛 2𝑛 NOUN cana-2728 245 31 ] ] PUNCT cana-2728 246 1 [ [ X cana-2728 246 2 πœ‚(𝑋2 πœ‚(𝑋2 X cana-2728 246 3 ) ) PUNCT cana-2728 246 4 𝑋1 𝑋1 PROPN cana-2728 246 5 βˆ’ βˆ’ PROPN cana-2728 246 6 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 246 7 ) ) PUNCT cana-2728 246 8 𝑋2 𝑋2 VERB cana-2728 246 9 ] ] PUNCT cana-2728 246 10 . . PUNCT cana-2728 247 1 ( ( PUNCT cana-2728 247 2 6.2 6.2 NUM cana-2728 247 3 ) ) PUNCT cana-2728 247 4 this this PRON cana-2728 247 5 implies imply VERB cana-2728 247 6 that that SCONJ cana-2728 247 7 𝑃(𝑋1 𝑃(𝑋1 PROPN cana-2728 247 8 , , PUNCT cana-2728 247 9 𝑋2)πœ‰ 𝑋2)πœ‰ X cana-2728 247 10 = = SYM cana-2728 247 11 0 0 PUNCT cana-2728 247 12 if if SCONJ cana-2728 247 13 and and CCONJ cana-2728 247 14 only only ADV cana-2728 247 15 if if SCONJ cana-2728 247 16 πœ† πœ† PRON cana-2728 247 17 = = SYM cana-2728 247 18 βˆ’2𝑛 βˆ’2𝑛 PROPN cana-2728 247 19 , , PUNCT cana-2728 247 20 which which PRON cana-2728 247 21 proves prove VERB cana-2728 247 22 the the DET cana-2728 247 23 proposition proposition NOUN cana-2728 247 24 . . PUNCT cana-2728 248 1 definition definition NOUN cana-2728 248 2 6.2 6.2 NUM cana-2728 248 3 . . PUNCT cana-2728 249 1 [ [ X cana-2728 249 2 13 13 NUM cana-2728 249 3 ] ] PUNCT cana-2728 249 4 in in ADP cana-2728 249 5 a a DET cana-2728 249 6 lorentzian lorentzian ADJ cana-2728 249 7 para para NOUN cana-2728 249 8 - - PUNCT cana-2728 249 9 kenmotsu kenmotsu PROPN cana-2728 249 10 manifold manifold PROPN cana-2728 249 11 𝑀 𝑀 PROPN cana-2728 249 12 , , PUNCT cana-2728 249 13 the the DET cana-2728 249 14 concircular concircular ADJ cana-2728 249 15 curvature curvature NOUN cana-2728 249 16 𝐢 𝐢 PROPN cana-2728 249 17 of of ADP cana-2728 249 18 the the DET cana-2728 249 19 manifold manifold NOUN cana-2728 249 20 is be AUX cana-2728 249 21 defined define VERB cana-2728 249 22 as as ADP cana-2728 249 23 𝐢(𝑋1 𝐢(𝑋1 PROPN cana-2728 249 24 , , PUNCT cana-2728 249 25 𝑋2)𝑋3 𝑋2)𝑋3 X cana-2728 249 26 = = SYM cana-2728 249 27 𝑅(𝑋1 𝑅(𝑋1 PROPN cana-2728 249 28 , , PUNCT cana-2728 249 29 𝑋2)𝑋3 𝑋2)𝑋3 PRON cana-2728 249 30 βˆ’ βˆ’ ADP cana-2728 249 31 π‘Ÿ π‘Ÿ SYM cana-2728 249 32 2𝑛(2𝑛+1 2𝑛(2𝑛+1 NUM cana-2728 249 33 ) ) PUNCT cana-2728 250 1 [ [ X cana-2728 250 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 250 3 , , PUNCT cana-2728 250 4 𝑋3)𝑋1 𝑋3)𝑋1 PROPN cana-2728 250 5 βˆ’ βˆ’ PROPN cana-2728 250 6 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 250 7 , , PUNCT cana-2728 250 8 𝑋3)𝑋2 𝑋3)𝑋2 PROPN cana-2728 250 9 ] ] X cana-2728 250 10 . . PUNCT cana-2728 251 1 ( ( PUNCT cana-2728 251 2 6.3 6.3 NUM cana-2728 251 3 ) ) PUNCT cana-2728 251 4 proposition proposition NOUN cana-2728 251 5 6.2 6.2 NUM cana-2728 251 6 . . PUNCT cana-2728 252 1 a a DET cana-2728 252 2 lorentzian lorentzian ADJ cana-2728 252 3 para para NOUN cana-2728 252 4 - - PUNCT cana-2728 252 5 kenmotsu kenmotsu PROPN cana-2728 252 6 manifold manifold PROPN cana-2728 252 7 𝑀 𝑀 PROPN cana-2728 252 8 , , PUNCT cana-2728 252 9 admitting admit VERB cana-2728 252 10 a a DET cana-2728 252 11 ricci ricci PROPN cana-2728 252 12 soliton soliton NOUN cana-2728 252 13 ( ( PUNCT cana-2728 252 14 𝑔 𝑔 PROPN cana-2728 252 15 , , PUNCT cana-2728 252 16 𝑉 𝑉 PROPN cana-2728 252 17 , , PUNCT cana-2728 252 18 Ξ» Ξ» NOUN cana-2728 252 19 ) ) PUNCT cana-2728 252 20 is be AUX cana-2728 252 21 πœ‰concircularly πœ‰concircularly ADV cana-2728 252 22 flat flat ADJ cana-2728 252 23 if if SCONJ cana-2728 252 24 the the DET cana-2728 252 25 soliton soliton NOUN cana-2728 252 26 is be AUX cana-2728 252 27 shrinking shrink VERB cana-2728 252 28 . . PUNCT cana-2728 253 1 proof proof NOUN cana-2728 253 2 : : PUNCT cana-2728 253 3 setting set VERB cana-2728 253 4 𝑋3 𝑋3 NOUN cana-2728 253 5 = = SYM cana-2728 253 6 πœ‰ πœ‰ X cana-2728 253 7 in in ADP cana-2728 253 8 ( ( PUNCT cana-2728 253 9 6.3 6.3 NUM cana-2728 253 10 ) ) PUNCT cana-2728 253 11 and and CCONJ cana-2728 253 12 using use VERB cana-2728 253 13 the the DET cana-2728 253 14 equations equation NOUN cana-2728 253 15 ( ( PUNCT cana-2728 253 16 2.3 2.3 NUM cana-2728 253 17 ) ) PUNCT cana-2728 253 18 and and CCONJ cana-2728 253 19 ( ( PUNCT cana-2728 253 20 2.16 2.16 NUM cana-2728 253 21 ) ) PUNCT cana-2728 253 22 , , PUNCT cana-2728 253 23 we we PRON cana-2728 253 24 get get VERB cana-2728 253 25 communications communication NOUN cana-2728 253 26 on on ADP cana-2728 253 27 applied apply VERB cana-2728 253 28 nonlinear nonlinear ADJ cana-2728 253 29 analysis analysis NOUN cana-2728 253 30 issn issn NOUN cana-2728 253 31 : : PUNCT cana-2728 253 32 1074 1074 NUM cana-2728 253 33 - - PUNCT cana-2728 253 34 133x 133x NUM cana-2728 253 35 vol vol NOUN cana-2728 253 36 32 32 NUM cana-2728 253 37 no no NOUN cana-2728 253 38 . . PUNCT cana-2728 254 1 3s 3s NUM cana-2728 254 2 ( ( PUNCT cana-2728 254 3 2025 2025 NUM cana-2728 254 4 ) ) PUNCT cana-2728 254 5 708 708 NUM cana-2728 254 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 254 7 𝐢(𝑋1 𝐢(𝑋1 PROPN cana-2728 254 8 , , PUNCT cana-2728 254 9 𝑋2)𝑋3 𝑋2)𝑋3 X cana-2728 254 10 = = PUNCT cana-2728 254 11 [ [ PUNCT cana-2728 254 12 4𝑛2+πœ†(2𝑛+1 4𝑛2+πœ†(2𝑛+1 NUM cana-2728 254 13 ) ) PUNCT cana-2728 254 14 2𝑛(2𝑛+1 2𝑛(2𝑛+1 NUM cana-2728 254 15 ) ) PUNCT cana-2728 254 16 ] ] PUNCT cana-2728 255 1 [ [ X cana-2728 255 2 πœ‚(𝑋2 πœ‚(𝑋2 X cana-2728 255 3 ) ) PUNCT cana-2728 255 4 𝑋1 𝑋1 PROPN cana-2728 255 5 βˆ’ βˆ’ NOUN cana-2728 255 6 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 255 7 ) ) PUNCT cana-2728 255 8 𝑋2 𝑋2 VERB cana-2728 255 9 ] ] PUNCT cana-2728 255 10 . . PUNCT cana-2728 256 1 ( ( PUNCT cana-2728 256 2 6.4 6.4 NUM cana-2728 256 3 ) ) PUNCT cana-2728 256 4 this this PRON cana-2728 256 5 shows show VERB cana-2728 256 6 that that SCONJ cana-2728 256 7 𝐢(𝑋1 𝐢(𝑋1 ADV cana-2728 256 8 , , PUNCT cana-2728 256 9 𝑋2)πœ‰ 𝑋2)πœ‰ PROPN cana-2728 256 10 = = SYM cana-2728 256 11 0 0 PUNCT cana-2728 256 12 if if SCONJ cana-2728 256 13 and and CCONJ cana-2728 256 14 only only ADV cana-2728 256 15 if if SCONJ cana-2728 256 16 πœ† πœ† PRON cana-2728 256 17 = = SYM cana-2728 256 18 βˆ’ βˆ’ PROPN cana-2728 257 1 4𝑛2 4𝑛2 NUM cana-2728 258 1 2𝑛+1 2𝑛+1 NOUN cana-2728 258 2 . . PUNCT cana-2728 259 1 hence hence ADV cana-2728 259 2 , , PUNCT cana-2728 259 3 we we PRON cana-2728 259 4 prove prove VERB cana-2728 259 5 the the DET cana-2728 259 6 proposition proposition NOUN cana-2728 259 7 . . PUNCT cana-2728 260 1 definition definition NOUN cana-2728 260 2 6.3 6.3 NUM cana-2728 260 3 . . PUNCT cana-2728 261 1 [ [ X cana-2728 261 2 9 9 NUM cana-2728 261 3 ] ] PUNCT cana-2728 261 4 in in ADP cana-2728 261 5 a a DET cana-2728 261 6 lorentzian lorentzian ADJ cana-2728 261 7 para para NOUN cana-2728 261 8 - - PUNCT cana-2728 261 9 kenmotsu kenmotsu PROPN cana-2728 261 10 manifold manifold PROPN cana-2728 261 11 𝑀 𝑀 PROPN cana-2728 261 12 , , PUNCT cana-2728 261 13 the the DET cana-2728 261 14 conharmonic conharmonic ADJ cana-2728 261 15 curvature curvature NOUN cana-2728 261 16 tensor tensor NOUN cana-2728 261 17 𝐻 𝐻 PROPN cana-2728 261 18 is be AUX cana-2728 261 19 defined define VERB cana-2728 261 20 as as ADP cana-2728 261 21 𝐻(𝑋1 𝐻(𝑋1 ADJ cana-2728 261 22 , , PUNCT cana-2728 261 23 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 261 24 = = SYM cana-2728 261 25 𝑅 𝑅 PROPN cana-2728 261 26 ( ( PUNCT cana-2728 261 27 𝑋1 𝑋1 PROPN cana-2728 261 28 , , PUNCT cana-2728 261 29 𝑋2)𝑋3 𝑋2)𝑋3 PROPN cana-2728 261 30 βˆ’ βˆ’ PROPN cana-2728 261 31 1 1 NUM cana-2728 261 32 2π‘›βˆ’1 2π‘›βˆ’1 NUM cana-2728 262 1 [ [ X cana-2728 262 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 262 3 , , PUNCT cana-2728 262 4 𝑋3 𝑋3 NOUN cana-2728 262 5 ) ) PUNCT cana-2728 263 1 𝑄𝑋1 𝑄𝑋1 PROPN cana-2728 263 2 βˆ’ βˆ’ PROPN cana-2728 263 3 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 263 4 , , PUNCT cana-2728 263 5 𝑋3 𝑋3 NOUN cana-2728 263 6 ) ) PUNCT cana-2728 263 7 𝑄𝑋2 𝑄𝑋2 PROPN cana-2728 263 8 ] ] PUNCT cana-2728 264 1 + + X cana-2728 264 2 𝑆(𝑋2 𝑆(𝑋2 ADJ cana-2728 264 3 , , PUNCT cana-2728 264 4 𝑋3 𝑋3 NOUN cana-2728 264 5 ) ) PUNCT cana-2728 264 6 𝑋1 𝑋1 PROPN cana-2728 264 7 βˆ’ βˆ’ PROPN cana-2728 264 8 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 264 9 , , PUNCT cana-2728 264 10 𝑋3 𝑋3 NOUN cana-2728 264 11 ) ) PUNCT cana-2728 264 12 𝑋2 𝑋2 VERB cana-2728 264 13 ] ] PUNCT cana-2728 264 14 . . PUNCT cana-2728 265 1 ( ( PUNCT cana-2728 265 2 6.5 6.5 NUM cana-2728 265 3 ) ) PUNCT cana-2728 265 4 proposition proposition NOUN cana-2728 265 5 6.3 6.3 NUM cana-2728 265 6 . . PUNCT cana-2728 266 1 a a DET cana-2728 266 2 lorentzian lorentzian ADJ cana-2728 266 3 para para NOUN cana-2728 266 4 - - PUNCT cana-2728 266 5 kenmotsu kenmotsu PROPN cana-2728 266 6 manifold manifold PROPN cana-2728 266 7 𝑀 𝑀 PROPN cana-2728 266 8 , , PUNCT cana-2728 266 9 admitting admit VERB cana-2728 266 10 a a DET cana-2728 266 11 ricci ricci PROPN cana-2728 266 12 soliton soliton NOUN cana-2728 266 13 ( ( PUNCT cana-2728 266 14 𝑔 𝑔 PROPN cana-2728 266 15 , , PUNCT cana-2728 266 16 𝑉 𝑉 PROPN cana-2728 266 17 , , PUNCT cana-2728 266 18 Ξ» Ξ» NOUN cana-2728 266 19 ) ) PUNCT cana-2728 266 20 is be AUX cana-2728 266 21 πœ‰conharmonically πœ‰conharmonically ADV cana-2728 266 22 flat flat ADJ cana-2728 266 23 if if SCONJ cana-2728 266 24 the the DET cana-2728 266 25 soliton soliton NOUN cana-2728 266 26 is be AUX cana-2728 266 27 shrinking shrink VERB cana-2728 266 28 . . PUNCT cana-2728 267 1 proof proof NOUN cana-2728 267 2 : : PUNCT cana-2728 267 3 putting put VERB cana-2728 267 4 𝑋3 𝑋3 NOUN cana-2728 267 5 = = SYM cana-2728 267 6 πœ‰ πœ‰ X cana-2728 267 7 in in ADP cana-2728 267 8 ( ( PUNCT cana-2728 267 9 6.5 6.5 NUM cana-2728 267 10 ) ) PUNCT cana-2728 267 11 , , PUNCT cana-2728 267 12 we we PRON cana-2728 267 13 obtain obtain VERB cana-2728 267 14 𝐻(𝑋1 𝐻(𝑋1 PROPN cana-2728 267 15 , , PUNCT cana-2728 267 16 𝑋2)πœ‰ 𝑋2)πœ‰ X cana-2728 267 17 = = SYM cana-2728 267 18 𝑅(𝑋1 𝑅(𝑋1 PROPN cana-2728 267 19 , , PUNCT cana-2728 267 20 𝑋2)πœ‰ 𝑋2)πœ‰ PROPN cana-2728 267 21 βˆ’ βˆ’ PROPN cana-2728 267 22 1 1 NUM cana-2728 267 23 2π‘›βˆ’1 2π‘›βˆ’1 NUM cana-2728 268 1 [ [ X cana-2728 268 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 268 3 , , PUNCT cana-2728 268 4 πœ‰)𝑄𝑋1 πœ‰)𝑄𝑋1 PUNCT cana-2728 269 1 βˆ’ βˆ’ PROPN cana-2728 269 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 269 3 , , PUNCT cana-2728 269 4 πœ‰)𝑄𝑋2 πœ‰)𝑄𝑋2 VERB cana-2728 269 5 + + ADV cana-2728 269 6 𝑆(𝑋2 𝑆(𝑋2 PROPN cana-2728 269 7 , , PUNCT cana-2728 269 8 πœ‰)𝑋1 πœ‰)𝑋1 VERB cana-2728 269 9 βˆ’ βˆ’ PROPN cana-2728 269 10 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 269 11 , , PUNCT cana-2728 269 12 πœ‰)𝑋2 πœ‰)𝑋2 VERB cana-2728 269 13 ] ] PUNCT cana-2728 269 14 . . PUNCT cana-2728 270 1 ( ( PUNCT cana-2728 270 2 6.6 6.6 NUM cana-2728 270 3 ) ) PUNCT cana-2728 270 4 using use VERB cana-2728 270 5 ( ( PUNCT cana-2728 270 6 2.3 2.3 NUM cana-2728 270 7 ) ) PUNCT cana-2728 270 8 , , PUNCT cana-2728 270 9 ( ( PUNCT cana-2728 270 10 2.16 2.16 NUM cana-2728 270 11 ) ) PUNCT cana-2728 270 12 and and CCONJ cana-2728 270 13 ( ( PUNCT cana-2728 270 14 2.17 2.17 NUM cana-2728 270 15 ) ) PUNCT cana-2728 270 16 in in ADP cana-2728 270 17 ( ( PUNCT cana-2728 270 18 6.6 6.6 NUM cana-2728 270 19 ) ) PUNCT cana-2728 270 20 , , PUNCT cana-2728 270 21 we we PRON cana-2728 270 22 have have VERB cana-2728 270 23 𝐻(𝑋1 𝐻(𝑋1 NOUN cana-2728 270 24 , , PUNCT cana-2728 270 25 𝑋2)πœ‰ 𝑋2)πœ‰ PROPN cana-2728 270 26 = = SYM cana-2728 270 27 2π‘›βˆ’2 2π‘›βˆ’2 NUM cana-2728 270 28 + + NOUN cana-2728 270 29 2πœ† 2πœ† NUM cana-2728 270 30 2π‘›βˆ’1 2π‘›βˆ’1 NUM cana-2728 271 1 [ [ X cana-2728 271 2 πœ‚(𝑋2)𝑋1 πœ‚(𝑋2)𝑋1 NOUN cana-2728 271 3 βˆ’ βˆ’ ADP cana-2728 271 4 πœ‚(𝑋1)𝑋2 πœ‚(𝑋1)𝑋2 NOUN cana-2728 271 5 ] ] PUNCT cana-2728 271 6 . . PUNCT cana-2728 272 1 ( ( PUNCT cana-2728 272 2 6.7 6.7 NUM cana-2728 272 3 ) ) PUNCT cana-2728 272 4 thus thus ADV cana-2728 272 5 , , PUNCT cana-2728 272 6 𝐻(𝑋1 𝐻(𝑋1 PROPN cana-2728 272 7 , , PUNCT cana-2728 272 8 𝑋2)πœ‰ 𝑋2)πœ‰ X cana-2728 272 9 = = SYM cana-2728 272 10 0 0 PUNCT cana-2728 272 11 if if SCONJ cana-2728 272 12 and and CCONJ cana-2728 272 13 only only ADV cana-2728 272 14 if if SCONJ cana-2728 272 15 Ξ»= Ξ»= NOUN cana-2728 272 16 βˆ’(𝑛 βˆ’(𝑛 VERB cana-2728 272 17 βˆ’ βˆ’ NOUN cana-2728 272 18 1 1 NUM cana-2728 272 19 ) ) PUNCT cana-2728 272 20 . . PUNCT cana-2728 273 1 hence hence ADV cana-2728 273 2 , , PUNCT cana-2728 273 3 the the DET cana-2728 273 4 proof proof NOUN cana-2728 273 5 is be AUX cana-2728 273 6 completed complete VERB cana-2728 273 7 . . PUNCT cana-2728 274 1 definition definition NOUN cana-2728 274 2 6.4 6.4 NUM cana-2728 274 3 . . PUNCT cana-2728 275 1 [ [ X cana-2728 275 2 3 3 X cana-2728 275 3 ] ] PUNCT cana-2728 275 4 in in ADP cana-2728 275 5 a a DET cana-2728 275 6 lorentzian lorentzian ADJ cana-2728 275 7 para para NOUN cana-2728 275 8 - - PUNCT cana-2728 275 9 kenmotsu kenmotsu PROPN cana-2728 275 10 manifold manifold PROPN cana-2728 275 11 𝑀 𝑀 PROPN cana-2728 275 12 , , PUNCT cana-2728 275 13 the the DET cana-2728 275 14 weyl weyl VERB cana-2728 275 15 conformal conformal NOUN cana-2728 275 16 curvature curvature NOUN cana-2728 275 17 tensor tensor NOUN cana-2728 275 18 π‘Š π‘Š PROPN cana-2728 275 19 is be AUX cana-2728 275 20 defined define VERB cana-2728 275 21 as as ADP cana-2728 275 22 π‘Š(𝑋1 π‘Š(𝑋1 NOUN cana-2728 275 23 , , PUNCT cana-2728 275 24 𝑋2)𝑋3 𝑋2)𝑋3 X cana-2728 275 25 = = SYM cana-2728 275 26 𝑅(𝑋1 𝑅(𝑋1 PROPN cana-2728 275 27 , , PUNCT cana-2728 275 28 𝑋2)𝑋3 𝑋2)𝑋3 PRON cana-2728 275 29 βˆ’ βˆ’ NOUN cana-2728 276 1 1 1 NUM cana-2728 276 2 2nβˆ’1 2nβˆ’1 NUM cana-2728 276 3 [ [ X cana-2728 276 4 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 276 5 , , PUNCT cana-2728 276 6 𝑋3)𝑄𝑋1 𝑋3)𝑄𝑋1 PROPN cana-2728 276 7 βˆ’ βˆ’ PROPN cana-2728 276 8 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 276 9 , , PUNCT cana-2728 276 10 𝑋3)𝑄𝑋2 𝑋3)𝑄𝑋2 X cana-2728 276 11 + + NOUN cana-2728 276 12 𝑆(𝑋2 𝑆(𝑋2 PROPN cana-2728 276 13 , , PUNCT cana-2728 276 14 𝑋3)𝑋1 𝑋3)𝑋1 X cana-2728 276 15 βˆ’ βˆ’ ADP cana-2728 276 16 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 276 17 , , PUNCT cana-2728 276 18 𝑋3)𝑋2 𝑋3)𝑋2 NOUN cana-2728 276 19 ] ] X cana-2728 276 20 + + NUM cana-2728 276 21 π‘Ÿ π‘Ÿ NOUN cana-2728 276 22 2𝑛(2π‘›βˆ’1 2𝑛(2π‘›βˆ’1 NUM cana-2728 276 23 ) ) PUNCT cana-2728 277 1 [ [ X cana-2728 277 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 277 3 , , PUNCT cana-2728 277 4 𝑋3)𝑋1 𝑋3)𝑋1 PROPN cana-2728 277 5 βˆ’ βˆ’ PROPN cana-2728 277 6 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 277 7 , , PUNCT cana-2728 277 8 𝑋3)𝑋2 𝑋3)𝑋2 PROPN cana-2728 277 9 ] ] X cana-2728 277 10 . . PUNCT cana-2728 278 1 ( ( PUNCT cana-2728 278 2 6.8 6.8 NUM cana-2728 278 3 ) ) PUNCT cana-2728 278 4 proposition proposition NOUN cana-2728 278 5 6.4 6.4 NUM cana-2728 278 6 . . PUNCT cana-2728 279 1 a a DET cana-2728 279 2 lorentzian lorentzian ADJ cana-2728 279 3 para para NOUN cana-2728 279 4 - - PUNCT cana-2728 279 5 kenmotsu kenmotsu PROPN cana-2728 279 6 manifold manifold PROPN cana-2728 279 7 𝑀 𝑀 PROPN cana-2728 279 8 , , PUNCT cana-2728 279 9 admitting admit VERB cana-2728 279 10 a a DET cana-2728 279 11 ricci ricci PROPN cana-2728 279 12 soliton soliton NOUN cana-2728 279 13 ( ( PUNCT cana-2728 279 14 𝑔 𝑔 PROPN cana-2728 279 15 , , PUNCT cana-2728 279 16 𝑉 𝑉 PROPN cana-2728 279 17 , , PUNCT cana-2728 279 18 Ξ» Ξ» NOUN cana-2728 279 19 ) ) PUNCT cana-2728 279 20 is be AUX cana-2728 279 21 πœ‰conformally πœ‰conformally ADV cana-2728 279 22 flat flat ADJ cana-2728 279 23 if if SCONJ cana-2728 279 24 the the DET cana-2728 279 25 soliton soliton NOUN cana-2728 279 26 is be AUX cana-2728 279 27 shrinking shrink VERB cana-2728 279 28 . . PUNCT cana-2728 280 1 proof proof NOUN cana-2728 280 2 : : PUNCT cana-2728 280 3 putting put VERB cana-2728 280 4 𝑋3 𝑋3 NOUN cana-2728 280 5 = = SYM cana-2728 280 6 πœ‰ πœ‰ X cana-2728 280 7 in in ADP cana-2728 280 8 ( ( PUNCT cana-2728 280 9 6.8 6.8 NUM cana-2728 280 10 ) ) PUNCT cana-2728 280 11 , , PUNCT cana-2728 280 12 we we PRON cana-2728 280 13 obtain obtain VERB cana-2728 280 14 π‘Š(𝑋1 π‘Š(𝑋1 NOUN cana-2728 280 15 , , PUNCT cana-2728 280 16 𝑋2)πœ‰ 𝑋2)πœ‰ X cana-2728 280 17 = = SYM cana-2728 280 18 𝑅(𝑋1 𝑅(𝑋1 PROPN cana-2728 280 19 , , PUNCT cana-2728 280 20 𝑋2)πœ‰ 𝑋2)πœ‰ PROPN cana-2728 280 21 βˆ’ βˆ’ NOUN cana-2728 280 22 π‘Ÿ π‘Ÿ SYM cana-2728 280 23 2π‘›βˆ’1 2π‘›βˆ’1 NUM cana-2728 281 1 [ [ X cana-2728 281 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 281 3 , , PUNCT cana-2728 281 4 πœ‰)𝑄𝑋1 πœ‰)𝑄𝑋1 PUNCT cana-2728 281 5 βˆ’π‘”(𝑋1 βˆ’π‘”(𝑋1 ADJ cana-2728 281 6 , , PUNCT cana-2728 281 7 πœ‰)𝑄𝑋2 πœ‰)𝑄𝑋2 NOUN cana-2728 281 8 + + CCONJ cana-2728 281 9 𝑆(𝑋2 𝑆(𝑋2 PROPN cana-2728 281 10 , , PUNCT cana-2728 281 11 πœ‰)𝑋1 πœ‰)𝑋1 VERB cana-2728 281 12 βˆ’ βˆ’ PROPN cana-2728 281 13 𝑆(𝑋1 𝑆(𝑋1 ADJ cana-2728 281 14 , , PUNCT cana-2728 281 15 πœ‰)𝑋2 πœ‰)𝑋2 NOUN cana-2728 281 16 ] ] PUNCT cana-2728 282 1 + + CCONJ cana-2728 282 2 π‘Ÿ π‘Ÿ NOUN cana-2728 282 3 2𝑛(2π‘›βˆ’1 2𝑛(2π‘›βˆ’1 NUM cana-2728 282 4 ) ) PUNCT cana-2728 283 1 [ [ X cana-2728 283 2 𝑔(𝑋2 𝑔(𝑋2 PROPN cana-2728 283 3 , , PUNCT cana-2728 283 4 πœ‰ πœ‰ NOUN cana-2728 283 5 ) ) PUNCT cana-2728 283 6 𝑋1 𝑋1 PROPN cana-2728 283 7 βˆ’ βˆ’ PROPN cana-2728 283 8 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 283 9 , , PUNCT cana-2728 283 10 πœ‰ πœ‰ NOUN cana-2728 283 11 ) ) PUNCT cana-2728 283 12 𝑋2 𝑋2 VERB cana-2728 283 13 ] ] PUNCT cana-2728 283 14 . . PUNCT cana-2728 284 1 ( ( PUNCT cana-2728 284 2 6.9 6.9 NUM cana-2728 284 3 ) ) PUNCT cana-2728 284 4 using use VERB cana-2728 284 5 ( ( PUNCT cana-2728 284 6 2.3 2.3 NUM cana-2728 284 7 ) ) PUNCT cana-2728 284 8 , , PUNCT cana-2728 284 9 ( ( PUNCT cana-2728 284 10 2.16 2.16 NUM cana-2728 284 11 ) ) PUNCT cana-2728 284 12 and and CCONJ cana-2728 284 13 ( ( PUNCT cana-2728 284 14 2.17 2.17 NUM cana-2728 284 15 ) ) PUNCT cana-2728 284 16 in in ADP cana-2728 284 17 ( ( PUNCT cana-2728 284 18 6.9 6.9 NUM cana-2728 284 19 ) ) PUNCT cana-2728 284 20 , , PUNCT cana-2728 284 21 we we PRON cana-2728 284 22 have have VERB cana-2728 284 23 communications communication NOUN cana-2728 284 24 on on ADP cana-2728 284 25 applied apply VERB cana-2728 284 26 nonlinear nonlinear ADJ cana-2728 284 27 analysis analysis NOUN cana-2728 284 28 issn issn NOUN cana-2728 284 29 : : PUNCT cana-2728 284 30 1074 1074 NUM cana-2728 284 31 - - PUNCT cana-2728 284 32 133x 133x NUM cana-2728 284 33 vol vol NOUN cana-2728 284 34 32 32 NUM cana-2728 285 1 no no NOUN cana-2728 285 2 . . PUNCT cana-2728 286 1 3s 3s NUM cana-2728 286 2 ( ( PUNCT cana-2728 286 3 2025 2025 NUM cana-2728 286 4 ) ) PUNCT cana-2728 287 1 709 709 NUM cana-2728 287 2 https://internationalpubls.com https://internationalpubls.com X cana-2728 287 3 π‘Š(𝑋1 π‘Š(𝑋1 NOUN cana-2728 287 4 , , PUNCT cana-2728 287 5 𝑋2)πœ‰ 𝑋2)πœ‰ PROPN cana-2728 287 6 = = SYM cana-2728 287 7 2𝑛+Ξ» 2𝑛+Ξ» NUM cana-2728 287 8 2𝑛 2𝑛 NUM cana-2728 288 1 [ [ X cana-2728 288 2 πœ‚(𝑋2)𝑋1 πœ‚(𝑋2)𝑋1 NOUN cana-2728 288 3 βˆ’ βˆ’ ADP cana-2728 288 4 πœ‚(𝑋1)𝑋2 πœ‚(𝑋1)𝑋2 NOUN cana-2728 288 5 ] ] PUNCT cana-2728 288 6 . . PUNCT cana-2728 289 1 ( ( PUNCT cana-2728 289 2 6.10 6.10 NUM cana-2728 289 3 ) ) PUNCT cana-2728 289 4 this this PRON cana-2728 289 5 implies imply VERB cana-2728 289 6 that that SCONJ cana-2728 289 7 π‘Š(𝑋1 π‘Š(𝑋1 NOUN cana-2728 289 8 , , PUNCT cana-2728 289 9 𝑋2)πœ‰ 𝑋2)πœ‰ X cana-2728 289 10 = = SYM cana-2728 289 11 0 0 PUNCT cana-2728 289 12 if if SCONJ cana-2728 289 13 and and CCONJ cana-2728 289 14 only only ADV cana-2728 289 15 if if SCONJ cana-2728 289 16 Ξ» Ξ» NOUN cana-2728 289 17 = = VERB cana-2728 289 18 βˆ’2𝑛. βˆ’2𝑛. NOUN cana-2728 289 19 this this PRON cana-2728 289 20 completes complete VERB cana-2728 289 21 the the DET cana-2728 289 22 proof proof NOUN cana-2728 289 23 of of ADP cana-2728 289 24 the the DET cana-2728 289 25 proposition proposition NOUN cana-2728 289 26 . . PUNCT cana-2728 290 1 6 6 NUM cana-2728 290 2 . . NOUN cana-2728 290 3 example example NOUN cana-2728 290 4 of of ADP cana-2728 290 5 a a DET cana-2728 290 6 lorentzian lorentzian ADJ cana-2728 290 7 para para NOUN cana-2728 290 8 - - PUNCT cana-2728 290 9 kenmotsu kenmotsu NOUN cana-2728 290 10 manifold manifold ADJ cana-2728 290 11 in in ADP cana-2728 290 12 this this DET cana-2728 290 13 section section NOUN cana-2728 290 14 we we PRON cana-2728 290 15 establish establish VERB cana-2728 290 16 an an DET cana-2728 290 17 example example NOUN cana-2728 290 18 of of ADP cana-2728 290 19 a a DET cana-2728 290 20 lorentzian lorentzian ADJ cana-2728 290 21 para para NOUN cana-2728 290 22 - - PUNCT cana-2728 290 23 kenmotsu kenmotsu NOUN cana-2728 290 24 manifold manifold ADJ cana-2728 290 25 . . PUNCT cana-2728 291 1 we we PRON cana-2728 291 2 consider consider VERB cana-2728 291 3 the the DET cana-2728 291 4 3dimensional 3dimensional ADJ cana-2728 291 5 manifold manifold ADJ cana-2728 291 6 𝑀 𝑀 NOUN cana-2728 291 7 = = SYM cana-2728 291 8 { { PUNCT cana-2728 291 9 ( ( PUNCT cana-2728 291 10 π‘₯1 π‘₯1 NOUN cana-2728 291 11 , , PUNCT cana-2728 291 12 π‘₯2 π‘₯2 NOUN cana-2728 291 13 , , PUNCT cana-2728 291 14 π‘₯3 π‘₯3 NOUN cana-2728 291 15 ) ) PUNCT cana-2728 291 16 ∈ ∈ PROPN cana-2728 291 17 𝑅3 𝑅3 PROPN cana-2728 291 18 : : PUNCT cana-2728 291 19 π‘₯3 π‘₯3 NOUN cana-2728 291 20 β‰  β‰  PROPN cana-2728 291 21 0 0 NUM cana-2728 291 22 } } PUNCT cana-2728 291 23 , , PUNCT cana-2728 291 24 where where SCONJ cana-2728 291 25 ( ( PUNCT cana-2728 291 26 π‘₯1 π‘₯1 NOUN cana-2728 291 27 , , PUNCT cana-2728 291 28 π‘₯2 π‘₯2 NOUN cana-2728 291 29 , , PUNCT cana-2728 291 30 π‘₯3 π‘₯3 NOUN cana-2728 291 31 ) ) PUNCT cana-2728 291 32 are be AUX cana-2728 291 33 the the DET cana-2728 291 34 standard standard ADJ cana-2728 291 35 coordinates coordinate NOUN cana-2728 291 36 in in ADP cana-2728 291 37 𝑅3 𝑅3 PROPN cana-2728 291 38 . . PUNCT cana-2728 292 1 let let VERB cana-2728 292 2 𝐸1 𝐸1 NOUN cana-2728 292 3 , , PUNCT cana-2728 292 4 𝐸2 𝐸2 ADJ cana-2728 292 5 and and CCONJ cana-2728 292 6 𝐸3 𝐸3 NOUN cana-2728 292 7 be be AUX cana-2728 292 8 a a DET cana-2728 292 9 linearly linearly ADV cana-2728 292 10 independent independent ADJ cana-2728 292 11 vector vector NOUN cana-2728 292 12 fields field NOUN cana-2728 292 13 in in ADP cana-2728 292 14 𝑀 𝑀 PROPN cana-2728 292 15 which which PRON cana-2728 292 16 satisfy satisfy VERB cana-2728 293 1 [ [ X cana-2728 293 2 𝐸1 𝐸1 NOUN cana-2728 293 3 , , PUNCT cana-2728 293 4 𝐸2 𝐸2 ADJ cana-2728 293 5 ] ] X cana-2728 293 6 = = PUNCT cana-2728 293 7 𝐸2 𝐸2 ADJ cana-2728 293 8 , , PUNCT cana-2728 293 9 [ [ X cana-2728 293 10 𝐸2 𝐸2 ADJ cana-2728 293 11 , , PUNCT cana-2728 293 12 𝐸3 𝐸3 NOUN cana-2728 293 13 ] ] X cana-2728 293 14 = = SYM cana-2728 293 15 0 0 NUM cana-2728 293 16 , , PUNCT cana-2728 293 17 [ [ X cana-2728 293 18 𝐸1 𝐸1 NOUN cana-2728 293 19 , , PUNCT cana-2728 293 20 𝐸3 𝐸3 NOUN cana-2728 293 21 ] ] X cana-2728 293 22 = = PUNCT cana-2728 293 23 𝐸3 𝐸3 NOUN cana-2728 293 24 . . PUNCT cana-2728 294 1 let let VERB cana-2728 294 2 𝑔 𝑔 PRON cana-2728 294 3 be be AUX cana-2728 294 4 the the DET cana-2728 294 5 lorentzian lorentzian ADJ cana-2728 294 6 metric metric NOUN cana-2728 294 7 defined define VERB cana-2728 294 8 by by ADP cana-2728 294 9 𝑔(𝐸1 𝑔(𝐸1 NOUN cana-2728 294 10 , , PUNCT cana-2728 294 11 𝐸1 𝐸1 NOUN cana-2728 294 12 ) ) PUNCT cana-2728 294 13 = = SYM cana-2728 294 14 βˆ’1 βˆ’1 NOUN cana-2728 294 15 , , PUNCT cana-2728 294 16 𝑔(𝐸2 𝑔(𝐸2 PROPN cana-2728 294 17 , , PUNCT cana-2728 294 18 𝐸2 𝐸2 ADJ cana-2728 294 19 ) ) PUNCT cana-2728 294 20 = = SYM cana-2728 295 1 𝑔(𝐸3 𝑔(𝐸3 PROPN cana-2728 295 2 , , PUNCT cana-2728 295 3 𝐸3 𝐸3 NOUN cana-2728 295 4 ) ) PUNCT cana-2728 295 5 = = SYM cana-2728 295 6 1 1 NUM cana-2728 295 7 , , PUNCT cana-2728 295 8 𝑔(𝐸1 𝑔(𝐸1 PROPN cana-2728 295 9 , , PUNCT cana-2728 295 10 𝐸2 𝐸2 ADJ cana-2728 295 11 ) ) PUNCT cana-2728 295 12 = = SYM cana-2728 295 13 𝑔(𝐸2 𝑔(𝐸2 PROPN cana-2728 295 14 , , PUNCT cana-2728 295 15 𝐸3 𝐸3 NOUN cana-2728 295 16 ) ) PUNCT cana-2728 295 17 = = SYM cana-2728 296 1 𝑔(𝐸1 𝑔(𝐸1 PROPN cana-2728 296 2 , , PUNCT cana-2728 296 3 𝐸3 𝐸3 NOUN cana-2728 296 4 ) ) PUNCT cana-2728 296 5 = = SYM cana-2728 297 1 0 0 X cana-2728 297 2 . . PUNCT cana-2728 298 1 let let VERB cana-2728 298 2 πœ‚ πœ‚ NOUN cana-2728 298 3 be be AUX cana-2728 298 4 the the DET cana-2728 298 5 1 1 NUM cana-2728 298 6 - - PUNCT cana-2728 298 7 form form NOUN cana-2728 298 8 defined define VERB cana-2728 298 9 by by ADP cana-2728 298 10 πœ‚(𝑋1 πœ‚(𝑋1 PROPN cana-2728 298 11 ) ) PUNCT cana-2728 298 12 = = SYM cana-2728 298 13 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 298 14 , , PUNCT cana-2728 298 15 𝐸3 𝐸3 NOUN cana-2728 298 16 ) ) PUNCT cana-2728 298 17 , , PUNCT cana-2728 298 18 for for ADP cana-2728 298 19 any any DET cana-2728 298 20 vector vector NOUN cana-2728 298 21 field field NOUN cana-2728 298 22 𝑋1 𝑋1 PROPN cana-2728 298 23 . . PUNCT cana-2728 299 1 let let VERB cana-2728 299 2 πœ™ πœ™ PRON cana-2728 299 3 be be AUX cana-2728 299 4 ( ( PUNCT cana-2728 299 5 1,1)-tensor 1,1)-tensor NUM cana-2728 299 6 field field NOUN cana-2728 299 7 defined define VERB cana-2728 299 8 by by ADP cana-2728 299 9 πœ™πΈ1 πœ™πΈ1 PROPN cana-2728 299 10 = = SYM cana-2728 299 11 0 0 NUM cana-2728 299 12 , , PUNCT cana-2728 299 13 πœ™πΈ2 πœ™πΈ2 PROPN cana-2728 299 14 = = PUNCT cana-2728 299 15 𝐸3 𝐸3 NOUN cana-2728 299 16 , , PUNCT cana-2728 299 17 πœ™πΈ3 πœ™πΈ3 PROPN cana-2728 299 18 = = PUNCT cana-2728 300 1 𝐸2 𝐸2 PROPN cana-2728 300 2 . . PUNCT cana-2728 301 1 then then ADV cana-2728 301 2 we we PRON cana-2728 301 3 have have VERB cana-2728 301 4 πœ‚(𝐸1 πœ‚(𝐸1 PROPN cana-2728 301 5 ) ) PUNCT cana-2728 301 6 = = SYM cana-2728 301 7 βˆ’1 βˆ’1 NOUN cana-2728 301 8 , , PUNCT cana-2728 301 9 πœ™2(𝑋1 πœ™2(𝑋1 PROPN cana-2728 301 10 ) ) PUNCT cana-2728 301 11 = = SYM cana-2728 302 1 𝑋1 𝑋1 PROPN cana-2728 302 2 + + CCONJ cana-2728 302 3 πœ‚(𝑋1)𝐸1 πœ‚(𝑋1)𝐸1 NOUN cana-2728 302 4 and and CCONJ cana-2728 302 5 𝑔(πœ™π‘‹1 𝑔(πœ™π‘‹1 NUM cana-2728 302 6 , , PUNCT cana-2728 302 7 πœ™π‘‹2 πœ™π‘‹2 PROPN cana-2728 302 8 ) ) PUNCT cana-2728 303 1 = = SYM cana-2728 303 2 𝑔(𝑋1 𝑔(𝑋1 PROPN cana-2728 303 3 , , PUNCT cana-2728 303 4 𝑋2 𝑋2 VERB cana-2728 303 5 ) ) PUNCT cana-2728 304 1 + + CCONJ cana-2728 304 2 πœ‚(𝑋1)πœ‚(𝑋2 πœ‚(𝑋1)πœ‚(𝑋2 NOUN cana-2728 304 3 ) ) PUNCT cana-2728 304 4 . . PUNCT cana-2728 305 1 thus thus ADV cana-2728 305 2 for for ADP cana-2728 305 3 πœ‰ πœ‰ PROPN cana-2728 305 4 = = SYM cana-2728 305 5 𝐸1 𝐸1 PROPN cana-2728 305 6 , , PUNCT cana-2728 305 7 ( ( PUNCT cana-2728 305 8 πœ™ πœ™ NOUN cana-2728 305 9 , , PUNCT cana-2728 305 10 πœ‰ πœ‰ X cana-2728 305 11 , , PUNCT cana-2728 305 12 πœ‚ πœ‚ NOUN cana-2728 305 13 , , PUNCT cana-2728 305 14 𝑔 𝑔 NOUN cana-2728 305 15 ) ) PUNCT cana-2728 305 16 defines define VERB cana-2728 305 17 a a DET cana-2728 305 18 lorentzian lorentzian ADJ cana-2728 305 19 almost almost ADV cana-2728 305 20 paracontact paracontact NOUN cana-2728 305 21 metric metric ADJ cana-2728 305 22 structure structure NOUN cana-2728 305 23 on on ADP cana-2728 305 24 𝑀. 𝑀. PROPN cana-2728 305 25 let let VERB cana-2728 305 26 βˆ‡ βˆ‡ X cana-2728 305 27 be be AUX cana-2728 305 28 the the DET cana-2728 305 29 levi levi PROPN cana-2728 305 30 - - PUNCT cana-2728 305 31 civita civita PROPN cana-2728 305 32 connection connection NOUN cana-2728 305 33 of of ADP cana-2728 305 34 the the DET cana-2728 305 35 lorentzian lorentzian ADJ cana-2728 305 36 metric metric PROPN cana-2728 305 37 𝑔. 𝑔. NOUN cana-2728 305 38 then then ADV cana-2728 305 39 using use VERB cana-2728 305 40 koszul koszul PROPN cana-2728 305 41 's 's PART cana-2728 305 42 formula formula NOUN cana-2728 305 43 , , PUNCT cana-2728 305 44 we we PRON cana-2728 305 45 obtain obtain VERB cana-2728 305 46 βˆ‡πΈ1 βˆ‡πΈ1 ADJ cana-2728 305 47 𝐸1 𝐸1 NOUN cana-2728 305 48 = = SYM cana-2728 305 49 0 0 NUM cana-2728 305 50 , , PUNCT cana-2728 305 51 βˆ‡πΈ1 βˆ‡πΈ1 ADJ cana-2728 305 52 𝐸2 𝐸2 PROPN cana-2728 305 53 = = SYM cana-2728 305 54 0 0 NUM cana-2728 305 55 , , PUNCT cana-2728 305 56 βˆ‡πΈ1 βˆ‡πΈ1 ADJ cana-2728 305 57 𝐸3 𝐸3 NOUN cana-2728 305 58 = = SYM cana-2728 305 59 0 0 NUM cana-2728 305 60 , , PUNCT cana-2728 305 61 βˆ‡πΈ2 βˆ‡πΈ2 NOUN cana-2728 305 62 𝐸1 𝐸1 PROPN cana-2728 305 63 = = SYM cana-2728 305 64 βˆ’πΈ2 βˆ’πΈ2 PROPN cana-2728 305 65 , , PUNCT cana-2728 305 66 βˆ‡πΈ2 βˆ‡πΈ2 NOUN cana-2728 305 67 𝐸2 𝐸2 PROPN cana-2728 305 68 = = PROPN cana-2728 305 69 βˆ’πΈ1 βˆ’πΈ1 PROPN cana-2728 305 70 , , PUNCT cana-2728 305 71 βˆ‡πΈ2 βˆ‡πΈ2 NOUN cana-2728 305 72 𝐸3 𝐸3 PROPN cana-2728 305 73 = = SYM cana-2728 305 74 0 0 NUM cana-2728 305 75 , , PUNCT cana-2728 305 76 βˆ‡πΈ3 βˆ‡πΈ3 ADJ cana-2728 305 77 𝐸1 𝐸1 NOUN cana-2728 305 78 = = PUNCT cana-2728 305 79 βˆ’πΈ3 βˆ’πΈ3 NOUN cana-2728 305 80 , , PUNCT cana-2728 305 81 βˆ‡πΈ3 βˆ‡πΈ3 ADJ cana-2728 305 82 𝐸2 𝐸2 PROPN cana-2728 305 83 = = SYM cana-2728 305 84 0 0 NUM cana-2728 305 85 , , PUNCT cana-2728 305 86 βˆ‡πΈ3 βˆ‡πΈ3 ADJ cana-2728 305 87 𝐸3 𝐸3 NOUN cana-2728 305 88 = = SYM cana-2728 305 89 βˆ’πΈ1 βˆ’πΈ1 PROPN cana-2728 305 90 . . PUNCT cana-2728 306 1 from from ADP cana-2728 306 2 the the DET cana-2728 306 3 above above ADJ cana-2728 306 4 calculation calculation NOUN cana-2728 306 5 , , PUNCT cana-2728 306 6 one one PRON cana-2728 306 7 can can AUX cana-2728 306 8 easily easily ADV cana-2728 306 9 verify verify VERB cana-2728 306 10 that that SCONJ cana-2728 306 11 βˆ‡π‘‹1 βˆ‡π‘‹1 NOUN cana-2728 306 12 πœ‰ πœ‰ ADP cana-2728 306 13 = = SYM cana-2728 306 14 βˆ’πœ™2𝑋1 βˆ’πœ™2𝑋1 PROPN cana-2728 306 15 , , PUNCT cana-2728 306 16 and and CCONJ cana-2728 306 17 ( ( PUNCT cana-2728 306 18 βˆ‡π‘‹1 βˆ‡π‘‹1 NOUN cana-2728 306 19 πœ™)𝑋2 πœ™)𝑋2 NOUN cana-2728 306 20 = = SYM cana-2728 306 21 βˆ’π‘”(πœ™x1 βˆ’π‘”(πœ™x1 PROPN cana-2728 306 22 , , PUNCT cana-2728 306 23 x2 x2 PROPN cana-2728 306 24 ) ) PUNCT cana-2728 306 25 βˆ’ βˆ’ PROPN cana-2728 306 26 πœ‚(𝑋2)πœ™π‘‹1 πœ‚(𝑋2)πœ™π‘‹1 PROPN cana-2728 306 27 . . PUNCT cana-2728 307 1 therefore therefore ADV cana-2728 307 2 , , PUNCT cana-2728 307 3 the the DET cana-2728 307 4 manifold manifold ADJ cana-2728 307 5 ( ( PUNCT cana-2728 307 6 𝑀 𝑀 PROPN cana-2728 307 7 , , PUNCT cana-2728 307 8 𝑔 𝑔 PROPN cana-2728 307 9 , , PUNCT cana-2728 307 10 πœ‰ πœ‰ X cana-2728 307 11 , , PUNCT cana-2728 307 12 πœ™ πœ™ NOUN cana-2728 307 13 , , PUNCT cana-2728 307 14 πœ‚ πœ‚ NOUN cana-2728 307 15 ) ) PUNCT cana-2728 307 16 is be AUX cana-2728 307 17 a a DET cana-2728 307 18 lorentzian lorentzian ADJ cana-2728 307 19 para para NOUN cana-2728 307 20 - - PUNCT cana-2728 307 21 kenmotsu kenmotsu NOUN cana-2728 307 22 manifold manifold ADJ cana-2728 307 23 . . PUNCT cana-2728 308 1 on on ADP cana-2728 308 2 this this DET cana-2728 308 3 manifold manifold ADJ cana-2728 308 4 ( ( PUNCT cana-2728 308 5 𝑀 𝑀 PROPN cana-2728 308 6 , , PUNCT cana-2728 308 7 𝑔 𝑔 PROPN cana-2728 308 8 , , PUNCT cana-2728 308 9 πœ‰ πœ‰ X cana-2728 308 10 , , PUNCT cana-2728 308 11 πœ™ πœ™ NOUN cana-2728 308 12 , , PUNCT cana-2728 308 13 πœ‚ πœ‚ NOUN cana-2728 308 14 ) ) PUNCT cana-2728 308 15 , , PUNCT cana-2728 308 16 we we PRON cana-2728 308 17 can can AUX cana-2728 308 18 easily easily ADV cana-2728 308 19 verify verify VERB cana-2728 308 20 our our PRON cana-2728 308 21 results result NOUN cana-2728 308 22 . . PUNCT cana-2728 309 1 communications communication NOUN cana-2728 309 2 on on ADP cana-2728 309 3 applied apply VERB cana-2728 309 4 nonlinear nonlinear ADJ cana-2728 309 5 analysis analysis NOUN cana-2728 309 6 issn issn NOUN cana-2728 309 7 : : PUNCT cana-2728 309 8 1074 1074 NUM cana-2728 309 9 - - PUNCT cana-2728 309 10 133x 133x NUM cana-2728 309 11 vol vol NOUN cana-2728 309 12 32 32 NUM cana-2728 309 13 no no NOUN cana-2728 309 14 . . PUNCT cana-2728 310 1 3s 3s NUM cana-2728 310 2 ( ( PUNCT cana-2728 310 3 2025 2025 NUM cana-2728 310 4 ) ) PUNCT cana-2728 310 5 710 710 NUM cana-2728 310 6 https://internationalpubls.com https://internationalpubls.com X cana-2728 310 7 conclusion conclusion NOUN cana-2728 310 8 : : PUNCT cana-2728 310 9 the the DET cana-2728 310 10 study study NOUN cana-2728 310 11 suggests suggest VERB cana-2728 310 12 that that SCONJ cana-2728 310 13 in in ADP cana-2728 310 14 a a DET cana-2728 310 15 lorentzian lorentzian ADJ cana-2728 310 16 para para NOUN cana-2728 310 17 - - PUNCT cana-2728 310 18 kenmotsu kenmotsu NOUN cana-2728 310 19 manifold manifold NOUN cana-2728 310 20 , , PUNCT cana-2728 310 21 a a DET cana-2728 310 22 symmetric symmetric ADJ cana-2728 310 23 parallel parallel ADJ cana-2728 310 24 second second ADJ cana-2728 310 25 - - PUNCT cana-2728 310 26 order order NOUN cana-2728 310 27 covariant covariant ADJ cana-2728 310 28 tensor tensor NOUN cana-2728 310 29 is be AUX cana-2728 310 30 proportional proportional ADJ cana-2728 310 31 to to ADP cana-2728 310 32 the the DET cana-2728 310 33 metric metric ADJ cana-2728 310 34 tensor tensor NOUN cana-2728 310 35 . . PUNCT cana-2728 311 1 additionally additionally ADV cana-2728 311 2 , , PUNCT cana-2728 311 3 if if SCONJ cana-2728 311 4 ℒ𝒱𝑔 ℒ𝒱𝑔 PROPN cana-2728 311 5 + + CCONJ cana-2728 311 6 2𝑆 2𝑆 PROPN cana-2728 311 7 is be AUX cana-2728 311 8 parallel parallel ADJ cana-2728 311 9 , , PUNCT cana-2728 311 10 where where SCONJ cana-2728 311 11 𝑉 𝑉 PROPN cana-2728 311 12 is be AUX cana-2728 311 13 a a DET cana-2728 311 14 vector vector NOUN cana-2728 311 15 field field NOUN cana-2728 311 16 , , PUNCT cana-2728 311 17 then then ADV cana-2728 311 18 ( ( PUNCT cana-2728 311 19 𝑔 𝑔 X cana-2728 311 20 , , PUNCT cana-2728 311 21 𝑉 𝑉 PROPN cana-2728 311 22 , , PUNCT cana-2728 311 23 Ξ» Ξ» NOUN cana-2728 311 24 ) ) PUNCT cana-2728 311 25 is be AUX cana-2728 311 26 a a DET cana-2728 311 27 ricci ricci PROPN cana-2728 311 28 soliton soliton NOUN cana-2728 311 29 . . PUNCT cana-2728 312 1 the the DET cana-2728 312 2 study study NOUN cana-2728 312 3 further further ADJ cana-2728 312 4 states state VERB cana-2728 312 5 that that SCONJ cana-2728 312 6 a a DET cana-2728 312 7 ricci ricci PROPN cana-2728 312 8 soliton soliton NOUN cana-2728 312 9 in in ADP cana-2728 312 10 a a DET cana-2728 312 11 π‘Š2 π‘Š2 ADJ cana-2728 312 12 - - PUNCT cana-2728 312 13 semi semi ADJ cana-2728 312 14 - - ADJ cana-2728 312 15 symmetric symmetric ADJ cana-2728 312 16 lorentzian lorentzian ADJ cana-2728 312 17 para para NOUN cana-2728 312 18 - - PUNCT cana-2728 312 19 kenmotsu kenmotsu PROPN cana-2728 312 20 manifold manifold NOUN cana-2728 312 21 is be AUX cana-2728 312 22 shrinking shrink VERB cana-2728 312 23 , , PUNCT cana-2728 312 24 whereas whereas SCONJ cana-2728 312 25 a a DET cana-2728 312 26 ricci ricci PROPN cana-2728 312 27 soliton soliton NOUN cana-2728 312 28 satisfying satisfy VERB cana-2728 312 29 the the DET cana-2728 312 30 condition condition NOUN cana-2728 312 31 π‘Š2(πœ‰ π‘Š2(πœ‰ NOUN cana-2728 312 32 , , PUNCT cana-2728 312 33 𝑋1 𝑋1 PROPN cana-2728 312 34 ) ) PUNCT cana-2728 312 35 ∘ ∘ NOUN cana-2728 312 36 𝑆 𝑆 PROPN cana-2728 312 37 = = SYM cana-2728 312 38 0 0 NUM cana-2728 312 39 in in ADP cana-2728 312 40 a a DET cana-2728 312 41 lorentzian lorentzian ADJ cana-2728 312 42 para para NOUN cana-2728 312 43 - - PUNCT cana-2728 312 44 kenmotsu kenmotsu PROPN cana-2728 312 45 manifold manifold NOUN cana-2728 312 46 is be AUX cana-2728 312 47 steady steady ADJ cana-2728 312 48 . . PUNCT cana-2728 313 1 the the DET cana-2728 313 2 study study NOUN cana-2728 313 3 concludes conclude VERB cana-2728 313 4 by by ADP cana-2728 313 5 analyzing analyze VERB cana-2728 313 6 certain certain ADJ cana-2728 313 7 curvature curvature NOUN cana-2728 313 8 properties property NOUN cana-2728 313 9 of of ADP cana-2728 313 10 lorentzian lorentzian ADJ cana-2728 313 11 para para PROPN cana-2728 313 12 - - PUNCT cana-2728 313 13 kenmotsu kenmotsu NOUN cana-2728 313 14 manifolds manifold NOUN cana-2728 313 15 admitting admit VERB cana-2728 313 16 ricci ricci PROPN cana-2728 313 17 soliton soliton NOUN cana-2728 313 18 and and CCONJ cana-2728 313 19 provides provide VERB cana-2728 313 20 an an DET cana-2728 313 21 example example NOUN cana-2728 313 22 of of ADP cana-2728 313 23 a a DET cana-2728 313 24 3 3 NUM cana-2728 313 25 - - PUNCT cana-2728 313 26 dimensional dimensional ADJ cana-2728 313 27 lorentzian lorentzian ADJ cana-2728 313 28 para para NOUN cana-2728 313 29 - - PUNCT cana-2728 313 30 kenmotsu kenmotsu NOUN cana-2728 313 31 manifold manifold ADJ cana-2728 313 32 . . PUNCT cana-2728 314 1 references reference NOUN cana-2728 314 2 [ [ X cana-2728 314 3 1 1 NUM cana-2728 314 4 ] ] PUNCT cana-2728 314 5 g. g. NOUN cana-2728 314 6 ayar ayar PROPN cana-2728 314 7 and and CCONJ cana-2728 314 8 d. d. PROPN cana-2728 314 9 demirhan demirhan PROPN cana-2728 314 10 , , PUNCT cana-2728 314 11 ricci ricci PROPN cana-2728 314 12 soliton soliton NOUN cana-2728 314 13 on on ADP cana-2728 314 14 nearly nearly ADV cana-2728 314 15 kenmotsu kenmotsu NOUN cana-2728 314 16 manifolds manifold NOUN cana-2728 314 17 with with ADP cana-2728 314 18 semi semi ADJ cana-2728 314 19 - - ADJ cana-2728 314 20 symmetric symmetric ADJ cana-2728 314 21 metric metric ADJ cana-2728 314 22 connection connection NOUN cana-2728 314 23 , , PUNCT cana-2728 314 24 j. j. PROPN cana-2728 314 25 eng eng PROPN cana-2728 314 26 . . PROPN cana-2728 314 27 technol technol PROPN cana-2728 314 28 . . PUNCT cana-2728 314 29 appl appl PROPN cana-2728 314 30 . . PUNCT cana-2728 315 1 sci sci PROPN cana-2728 315 2 . . PROPN cana-2728 315 3 , , PUNCT cana-2728 315 4 4(3 4(3 NUM cana-2728 315 5 ) ) PUNCT cana-2728 315 6 ( ( PUNCT cana-2728 315 7 2019 2019 NUM cana-2728 315 8 ) ) PUNCT cana-2728 315 9 , , PUNCT cana-2728 315 10 131–-140 131–-140 NUM cana-2728 315 11 . . PUNCT cana-2728 315 12 doi:10.30931 doi:10.30931 PROPN cana-2728 315 13 / / SYM cana-2728 315 14 jetas.643643 jetas.643643 NOUN cana-2728 316 1 [ [ X cana-2728 316 2 2 2 NUM cana-2728 316 3 ] ] PUNCT cana-2728 316 4 z. z. PROPN cana-2728 316 5 chen chen PROPN cana-2728 316 6 , , PUNCT cana-2728 316 7 y. y. PROPN cana-2728 316 8 li li PROPN cana-2728 316 9 , , PUNCT cana-2728 316 10 s. s. PROPN cana-2728 316 11 sarkar sarkar PROPN cana-2728 316 12 , , PUNCT cana-2728 316 13 s. s. PROPN cana-2728 316 14 dey dey PROPN cana-2728 316 15 and and CCONJ cana-2728 316 16 a. a. NOUN cana-2728 316 17 bhattacharyya bhattacharyya PROPN cana-2728 316 18 , , PUNCT cana-2728 316 19 ricci ricci PROPN cana-2728 316 20 soliton soliton NOUN cana-2728 316 21 and and CCONJ cana-2728 316 22 certain certain ADJ cana-2728 316 23 related related ADJ cana-2728 316 24 metrics metric NOUN cana-2728 316 25 on on ADP cana-2728 316 26 a a DET cana-2728 316 27 threedimensional threedimensional ADJ cana-2728 316 28 trans trans ADJ cana-2728 316 29 - - ADJ cana-2728 316 30 sasakian sasakian ADJ cana-2728 316 31 manifold manifold ADJ cana-2728 316 32 , , PUNCT cana-2728 316 33 universe universe NOUN cana-2728 316 34 , , PUNCT cana-2728 316 35 8(11 8(11 NUM cana-2728 316 36 ) ) PUNCT cana-2728 316 37 ( ( PUNCT cana-2728 316 38 2022 2022 NUM cana-2728 316 39 ) ) PUNCT cana-2728 316 40 , , PUNCT cana-2728 316 41 595 595 NUM cana-2728 316 42 . . PUNCT cana-2728 317 1 doi:10.3390 doi:10.3390 PROPN cana-2728 317 2 / / SYM cana-2728 317 3 universe8110595 universe8110595 PROPN cana-2728 318 1 [ [ X cana-2728 318 2 3 3 X cana-2728 318 3 ] ] X cana-2728 318 4 u. u. PROPN cana-2728 318 5 c. c. PROPN cana-2728 318 6 de de PROPN cana-2728 318 7 and and CCONJ cana-2728 318 8 s. s. PROPN cana-2728 318 9 biswas biswas PROPN cana-2728 318 10 , , PUNCT cana-2728 318 11 a a DET cana-2728 318 12 note note NOUN cana-2728 318 13 on on ADP cana-2728 318 14 πœ‰-conformally πœ‰-conformally ADV cana-2728 318 15 flat flat ADJ cana-2728 318 16 contact contact NOUN cana-2728 318 17 manifolds manifold NOUN cana-2728 318 18 , , PUNCT cana-2728 318 19 bull bull NOUN cana-2728 318 20 . . PUNCT cana-2728 319 1 malays malays PROPN cana-2728 319 2 . . PUNCT cana-2728 320 1 math math NOUN cana-2728 320 2 . . PUNCT cana-2728 321 1 sci sci PROPN cana-2728 321 2 . . PROPN cana-2728 321 3 soc soc PROPN cana-2728 321 4 . . PROPN cana-2728 321 5 , , PUNCT cana-2728 321 6 29(1 29(1 NUM cana-2728 321 7 ) ) PUNCT cana-2728 321 8 ( ( PUNCT cana-2728 321 9 2006 2006 NUM cana-2728 321 10 ) ) PUNCT cana-2728 321 11 , , PUNCT cana-2728 321 12 51 51 NUM cana-2728 321 13 - - NUM cana-2728 321 14 -57 -57 PROPN cana-2728 321 15 . . PUNCT cana-2728 322 1 [ [ X cana-2728 322 2 4 4 X cana-2728 322 3 ] ] PUNCT cana-2728 322 4 m. m. NOUN cana-2728 322 5 s. s. PROPN cana-2728 322 6 devi devi PROPN cana-2728 322 7 and and CCONJ cana-2728 322 8 j. j. PROPN cana-2728 322 9 p. p. PROPN cana-2728 322 10 singh singh PROPN cana-2728 322 11 , , PUNCT cana-2728 322 12 on on ADP cana-2728 322 13 a a DET cana-2728 322 14 type type NOUN cana-2728 322 15 of of ADP cana-2728 322 16 m‐projective m‐projective PUNCT cana-2728 322 17 curvature curvature NOUN cana-2728 322 18 tensor tensor NOUN cana-2728 322 19 on on ADP cana-2728 322 20 kenmotsu kenmotsu PROPN cana-2728 322 21 manifolds manifolds PROPN cana-2728 322 22 , , PUNCT cana-2728 322 23 international international ADJ cana-2728 322 24 j. j. PROPN cana-2728 322 25 of of ADP cana-2728 322 26 math math PROPN cana-2728 322 27 . . PUNCT cana-2728 323 1 sci sci PROPN cana-2728 323 2 . . PROPN cana-2728 323 3 and and CCONJ cana-2728 323 4 engg engg PROPN cana-2728 323 5 . . PUNCT cana-2728 324 1 appls appls PROPN cana-2728 324 2 . . PUNCT cana-2728 324 3 , , PUNCT cana-2728 324 4 9 9 NUM cana-2728 324 5 ( ( PUNCT cana-2728 324 6 2015 2015 NUM cana-2728 324 7 ) ) PUNCT cana-2728 324 8 , , PUNCT cana-2728 324 9 37 37 NUM cana-2728 324 10 - - NUM cana-2728 324 11 -48 -48 NOUN cana-2728 324 12 . . PUNCT cana-2728 325 1 [ [ X cana-2728 325 2 5 5 NUM cana-2728 325 3 ] ] PUNCT cana-2728 325 4 r. r. PROPN cana-2728 325 5 s. s. PROPN cana-2728 325 6 hamilton hamilton PROPN cana-2728 325 7 , , PUNCT cana-2728 325 8 three three NUM cana-2728 325 9 manifolds manifold NOUN cana-2728 325 10 with with ADP cana-2728 325 11 positive positive ADJ cana-2728 325 12 ricci ricci PROPN cana-2728 325 13 curvature curvature NOUN cana-2728 325 14 , , PUNCT cana-2728 325 15 j. j. PROPN cana-2728 325 16 differ differ VERB cana-2728 325 17 . . PUNCT cana-2728 326 1 geom geom PROPN cana-2728 326 2 . . PROPN cana-2728 326 3 , , PUNCT cana-2728 326 4 17(2 17(2 NUM cana-2728 326 5 ) ) PUNCT cana-2728 326 6 ( ( PUNCT cana-2728 326 7 1982 1982 NUM cana-2728 326 8 ) ) PUNCT cana-2728 326 9 , , PUNCT cana-2728 326 10 225 225 NUM cana-2728 326 11 - - PUNCT cana-2728 326 12 -306 -306 PROPN cana-2728 326 13 . . PUNCT cana-2728 326 14 doi:10.4310 doi:10.4310 PROPN cana-2728 326 15 / / SYM cana-2728 326 16 jdg/1214436922 jdg/1214436922 PROPN cana-2728 327 1 [ [ X cana-2728 327 2 6 6 NUM cana-2728 327 3 ] ] PUNCT cana-2728 327 4 a. a. NOUN cana-2728 327 5 haseeb haseeb PROPN cana-2728 327 6 and and CCONJ cana-2728 327 7 r. r. PROPN cana-2728 327 8 prasad prasad PROPN cana-2728 327 9 , , PUNCT cana-2728 327 10 certain certain ADJ cana-2728 327 11 results result NOUN cana-2728 327 12 on on ADP cana-2728 327 13 lorentzian lorentzian ADJ cana-2728 327 14 para para PROPN cana-2728 327 15 - - PUNCT cana-2728 327 16 kenmotsu kenmotsu NOUN cana-2728 327 17 manifolds manifold NOUN cana-2728 327 18 , , PUNCT cana-2728 327 19 bol bol NOUN cana-2728 327 20 . . PUNCT cana-2728 328 1 soc soc PROPN cana-2728 328 2 . . PUNCT cana-2728 329 1 parana parana PROPN cana-2728 329 2 . . PUNCT cana-2728 330 1 mat mat PROPN cana-2728 330 2 . . PROPN cana-2728 330 3 , , PUNCT cana-2728 330 4 39(3 39(3 NUM cana-2728 330 5 ) ) PUNCT cana-2728 330 6 ( ( PUNCT cana-2728 330 7 2021 2021 NUM cana-2728 330 8 ) ) PUNCT cana-2728 330 9 , , PUNCT cana-2728 330 10 201 201 NUM cana-2728 330 11 - - SYM cana-2728 330 12 -220 -220 PROPN cana-2728 330 13 . . PROPN cana-2728 330 14 doi:10.5269 doi:10.5269 PROPN cana-2728 330 15 / / SYM cana-2728 331 1 bspm.40607 bspm.40607 PROPN cana-2728 332 1 [ [ X cana-2728 332 2 7 7 NUM cana-2728 332 3 ] ] PUNCT cana-2728 332 4 a. a. NOUN cana-2728 332 5 haseeb haseeb PROPN cana-2728 332 6 and and CCONJ cana-2728 332 7 r. r. PROPN cana-2728 332 8 prasad prasad PROPN cana-2728 332 9 , , PUNCT cana-2728 332 10 some some PRON cana-2728 332 11 results result NOUN cana-2728 332 12 on on ADP cana-2728 332 13 lorentzian lorentzian ADJ cana-2728 332 14 para para PROPN cana-2728 332 15 - - PUNCT cana-2728 332 16 kenmotsu kenmotsu NOUN cana-2728 332 17 manifolds manifold NOUN cana-2728 332 18 , , PUNCT cana-2728 332 19 bull bull NOUN cana-2728 332 20 . . PUNCT cana-2728 333 1 transilv transilv PROPN cana-2728 333 2 . . PUNCT cana-2728 334 1 univ univ PROPN cana-2728 334 2 . . PUNCT cana-2728 334 3 bras bras PROPN cana-2728 334 4 . . PUNCT cana-2728 335 1 iii iii X cana-2728 335 2 : : PUNCT cana-2728 335 3 math math NOUN cana-2728 335 4 . . PUNCT cana-2728 336 1 inform inform NOUN cana-2728 336 2 . . PUNCT cana-2728 337 1 phys phy NOUN cana-2728 337 2 . . PUNCT cana-2728 337 3 , , PUNCT cana-2728 337 4 13(62 13(62 NUM cana-2728 337 5 ) ) PUNCT cana-2728 337 6 ( ( PUNCT cana-2728 337 7 2020 2020 NUM cana-2728 337 8 ) ) PUNCT cana-2728 337 9 , , PUNCT cana-2728 337 10 185 185 NUM cana-2728 337 11 - - PUNCT cana-2728 337 12 -198 -198 PROPN cana-2728 337 13 . . PROPN cana-2728 337 14 doi:10.31926 doi:10.31926 PROPN cana-2728 337 15 / / SYM cana-2728 337 16 but.mif.2020.13.62.1.14 but.mif.2020.13.62.1.14 NOUN cana-2728 338 1 [ [ X cana-2728 338 2 8 8 NUM cana-2728 338 3 ] ] X cana-2728 338 4 g. g. PROPN cana-2728 338 5 ingalahalli ingalahalli PROPN cana-2728 338 6 and and CCONJ cana-2728 338 7 c. c. PROPN cana-2728 338 8 s. s. PROPN cana-2728 338 9 bagewadi bagewadi PROPN cana-2728 338 10 , , PUNCT cana-2728 338 11 ricci ricci PROPN cana-2728 338 12 soliton soliton NOUN cana-2728 338 13 in in ADP cana-2728 338 14 𝛼-sasakian 𝛼-sasakian PROPN cana-2728 338 15 manifolds manifold NOUN cana-2728 338 16 , , PUNCT cana-2728 338 17 isrn isrn NOUN cana-2728 338 18 geometry geometry NOUN cana-2728 338 19 2012 2012 NUM cana-2728 338 20 , , PUNCT cana-2728 338 21 ( ( PUNCT cana-2728 338 22 2012 2012 NUM cana-2728 338 23 ) ) PUNCT cana-2728 338 24 , , PUNCT cana-2728 338 25 1 1 NUM cana-2728 338 26 - - NUM cana-2728 338 27 -13 -13 NUM cana-2728 338 28 . . PUNCT cana-2728 339 1 doi:10.5402/2012/421384 doi:10.5402/2012/421384 PROPN cana-2728 339 2 [ [ X cana-2728 339 3 9 9 NUM cana-2728 339 4 ] ] PUNCT cana-2728 339 5 k. k. PROPN cana-2728 339 6 matsumoto matsumoto PROPN cana-2728 339 7 , , PUNCT cana-2728 339 8 on on ADP cana-2728 339 9 lorentzian lorentzian ADJ cana-2728 339 10 paracontact paracontact PROPN cana-2728 339 11 manifolds manifold NOUN cana-2728 339 12 , , PUNCT cana-2728 339 13 bull bull NOUN cana-2728 339 14 . . PUNCT cana-2728 340 1 yamagata yamagata PROPN cana-2728 340 2 univ univ PROPN cana-2728 340 3 . . PUNCT cana-2728 341 1 natur natur PROPN cana-2728 341 2 . . PUNCT cana-2728 342 1 sci sci PROPN cana-2728 342 2 . . PROPN cana-2728 342 3 , , PUNCT cana-2728 342 4 12(2 12(2 NUM cana-2728 342 5 ) ) PUNCT cana-2728 342 6 ( ( PUNCT cana-2728 342 7 1989 1989 NUM cana-2728 342 8 ) ) PUNCT cana-2728 342 9 , , PUNCT cana-2728 342 10 151 151 NUM cana-2728 342 11 - - SYM cana-2728 342 12 -156 -156 PROPN cana-2728 342 13 . . PUNCT cana-2728 343 1 url= url= ADV cana-2728 343 2 { { PUNCT cana-2728 343 3 https://api.semanticscholar.org/corpusid:117311696 https://api.semanticscholar.org/corpusid:117311696 NOUN cana-2728 343 4 } } PUNCT cana-2728 343 5 [ [ X cana-2728 343 6 10 10 NUM cana-2728 343 7 ] ] PUNCT cana-2728 343 8 k. k. PROPN cana-2728 343 9 matsumoto matsumoto PROPN cana-2728 343 10 and and CCONJ cana-2728 343 11 i. i. PROPN cana-2728 343 12 mihai mihai PROPN cana-2728 343 13 , , PUNCT cana-2728 343 14 on on ADP cana-2728 343 15 a a DET cana-2728 343 16 certain certain ADJ cana-2728 343 17 transformation transformation NOUN cana-2728 343 18 in in ADP cana-2728 343 19 a a DET cana-2728 343 20 lorentzian lorentzian ADJ cana-2728 343 21 parasasakian parasasakian NOUN cana-2728 343 22 manifold manifold NOUN cana-2728 343 23 , , PUNCT cana-2728 343 24 tensor tensor NOUN cana-2728 343 25 n.s n.s PROPN cana-2728 343 26 . . PROPN cana-2728 343 27 , , PUNCT cana-2728 343 28 47(2 47(2 PROPN cana-2728 343 29 ) ) PUNCT cana-2728 343 30 ( ( PUNCT cana-2728 343 31 1988 1988 NUM cana-2728 343 32 ) ) PUNCT cana-2728 343 33 , , PUNCT cana-2728 343 34 189 189 NUM cana-2728 343 35 - - PUNCT cana-2728 343 36 -197 -197 NOUN cana-2728 343 37 . . PUNCT cana-2728 344 1 url={https://api.semanticscholar.org url={https://api.semanticscholar.org PROPN cana-2728 344 2 / / SYM cana-2728 344 3 corpusid:116985612 corpusid:116985612 NOUN cana-2728 344 4 } } PUNCT cana-2728 344 5 [ [ X cana-2728 344 6 11 11 NUM cana-2728 344 7 ] ] PUNCT cana-2728 344 8 s. s. PROPN cana-2728 344 9 pandey pandey PROPN cana-2728 344 10 , , PUNCT cana-2728 344 11 a. a. PROPN cana-2728 344 12 singh singh PROPN cana-2728 344 13 and and CCONJ cana-2728 344 14 v. v. ADP cana-2728 344 15 n. n. PROPN cana-2728 344 16 mishra mishra PROPN cana-2728 344 17 , , PUNCT cana-2728 344 18 πœ‚-ricci πœ‚-ricci NOUN cana-2728 344 19 soliton soliton NOUN cana-2728 344 20 on on ADP cana-2728 344 21 lorentzian lorentzian ADJ cana-2728 344 22 - - PUNCT cana-2728 344 23 para para NOUN cana-2728 344 24 kenmotsu kenmotsu NOUN cana-2728 344 25 manifolds manifold NOUN cana-2728 344 26 , , PUNCT cana-2728 344 27 facta facta NOUN cana-2728 344 28 universitasis universitasis NOUN cana-2728 344 29 ( ( PUNCT cana-2728 344 30 nis nis NOUN cana-2728 344 31 ) ) PUNCT cana-2728 344 32 , , PUNCT cana-2728 344 33 36(2 36(2 NUM cana-2728 344 34 ) ) PUNCT cana-2728 344 35 ( ( PUNCT cana-2728 344 36 2021 2021 NUM cana-2728 344 37 ) ) PUNCT cana-2728 344 38 , , PUNCT cana-2728 344 39 419 419 NUM cana-2728 344 40 - - SYM cana-2728 344 41 -434 -434 PROPN cana-2728 344 42 . . PUNCT cana-2728 344 43 doi:10.22190 doi:10.22190 NOUN cana-2728 344 44 / / SYM cana-2728 344 45 fumi200923031p fumi200923031p PROPN cana-2728 344 46 [ [ X cana-2728 344 47 12 12 NUM cana-2728 344 48 ] ] PUNCT cana-2728 344 49 g. g. PROPN cana-2728 344 50 p. p. PROPN cana-2728 344 51 pokhariyal pokhariyal PROPN cana-2728 344 52 , , PUNCT cana-2728 344 53 y. y. PROPN cana-2728 344 54 yadav yadav PROPN cana-2728 344 55 and and CCONJ cana-2728 344 56 s. s. PROPN cana-2728 344 57 k. k. PROPN cana-2728 344 58 chaubey chaubey PROPN cana-2728 344 59 , , PUNCT cana-2728 344 60 ricci ricci PROPN cana-2728 344 61 solitons soliton NOUN cana-2728 344 62 on on ADP cana-2728 344 63 trans trans PROPN cana-2728 344 64 - - ADJ cana-2728 344 65 sasakian sasakian ADJ cana-2728 344 66 manifolds manifold NOUN cana-2728 344 67 , , PUNCT cana-2728 344 68 differ differ VERB cana-2728 344 69 . . PUNCT cana-2728 345 1 geom.-dyn geom.-dyn ADV cana-2728 345 2 . . PUNCT cana-2728 346 1 syst syst PROPN cana-2728 346 2 . . PROPN cana-2728 346 3 , , PUNCT cana-2728 346 4 20 20 NUM cana-2728 346 5 ( ( PUNCT cana-2728 346 6 2018 2018 NUM cana-2728 346 7 ) ) PUNCT cana-2728 346 8 , , PUNCT cana-2728 346 9 138 138 NUM cana-2728 346 10 - - PUNCT cana-2728 346 11 -158 -158 PROPN cana-2728 346 12 . . PUNCT cana-2728 347 1 [ [ X cana-2728 347 2 13 13 NUM cana-2728 347 3 ] ] PUNCT cana-2728 347 4 g. g. PROPN cana-2728 347 5 p. p. PROPN cana-2728 347 6 pokhariyal pokhariyal PROPN cana-2728 347 7 and and CCONJ cana-2728 347 8 r. r. PROPN cana-2728 347 9 s. s. PROPN cana-2728 347 10 mishra mishra PROPN cana-2728 347 11 , , PUNCT cana-2728 347 12 the the DET cana-2728 347 13 curvature curvature NOUN cana-2728 347 14 tensor tensor NOUN cana-2728 347 15 and and CCONJ cana-2728 347 16 their their PRON cana-2728 347 17 relativistic relativistic ADJ cana-2728 347 18 significance significance NOUN cana-2728 347 19 , , PUNCT cana-2728 347 20 yokohoma yokohoma ADJ cana-2728 347 21 math math NOUN cana-2728 347 22 . . PUNCT cana-2728 348 1 j. j. PROPN cana-2728 348 2 , , PUNCT cana-2728 348 3 18 18 NUM cana-2728 348 4 ( ( PUNCT cana-2728 348 5 1970 1970 NUM cana-2728 348 6 ) ) PUNCT cana-2728 348 7 , , PUNCT cana-2728 348 8 305 305 NUM cana-2728 348 9 - - SYM cana-2728 348 10 -314 -314 NOUN cana-2728 348 11 . . PUNCT cana-2728 349 1 [ [ X cana-2728 349 2 14 14 NUM cana-2728 349 3 ] ] X cana-2728 349 4 i. i. NOUN cana-2728 349 5 sato sato PROPN cana-2728 349 6 , , PUNCT cana-2728 349 7 on on ADP cana-2728 349 8 a a DET cana-2728 349 9 stucture stucture NOUN cana-2728 349 10 similar similar ADJ cana-2728 349 11 to to ADP cana-2728 349 12 the the DET cana-2728 349 13 almost almost ADV cana-2728 349 14 contact contact NOUN cana-2728 349 15 structure structure NOUN cana-2728 349 16 , , PUNCT cana-2728 349 17 tensor tensor NOUN cana-2728 349 18 n.s n.s PROPN cana-2728 349 19 . . PROPN cana-2728 349 20 , , PUNCT cana-2728 349 21 30 30 NUM cana-2728 349 22 ( ( PUNCT cana-2728 349 23 1976 1976 NUM cana-2728 349 24 ) ) PUNCT cana-2728 349 25 , , PUNCT cana-2728 349 26 752 752 NUM cana-2728 349 27 - - SYM cana-2728 349 28 -783 -783 NOUN cana-2728 349 29 . . PUNCT cana-2728 350 1 [ [ X cana-2728 350 2 15 15 NUM cana-2728 350 3 ] ] X cana-2728 350 4 r. r. PROPN cana-2728 350 5 j. j. PROPN cana-2728 350 6 shah shah PROPN cana-2728 350 7 , , PUNCT cana-2728 350 8 ricci ricci PROPN cana-2728 350 9 solitons soliton NOUN cana-2728 350 10 on on ADP cana-2728 350 11 lorentzian lorentzian ADJ cana-2728 350 12 para para PROPN cana-2728 350 13 - - PUNCT cana-2728 350 14 sasakian sasakian ADJ cana-2728 350 15 manifolds manifold NOUN cana-2728 350 16 , , PUNCT cana-2728 350 17 bibechana bibechana NOUN cana-2728 350 18 , , PUNCT cana-2728 350 19 17 17 NUM cana-2728 350 20 ( ( PUNCT cana-2728 350 21 2020 2020 NUM cana-2728 350 22 ) ) PUNCT cana-2728 350 23 , , PUNCT cana-2728 350 24 110 110 NUM cana-2728 350 25 - - SYM cana-2728 350 26 -116 -116 PROPN cana-2728 350 27 . . PROPN cana-2728 350 28 doi:10.3126 doi:10.3126 PROPN cana-2728 350 29 / / SYM cana-2728 350 30 bibechana.v17i0.24341 bibechana.v17i0.24341 PROPN cana-2728 351 1 [ [ X cana-2728 351 2 16 16 NUM cana-2728 351 3 ] ] PUNCT cana-2728 351 4 a. a. NOUN cana-2728 351 5 a. a. NOUN cana-2728 351 6 shaikh shaikh PROPN cana-2728 351 7 and and CCONJ cana-2728 351 8 s. s. PROPN cana-2728 351 9 biswas biswas PROPN cana-2728 351 10 , , PUNCT cana-2728 351 11 on on ADP cana-2728 351 12 lp lp ADJ cana-2728 351 13 - - ADJ cana-2728 351 14 sasakian sasakian ADJ cana-2728 351 15 manifolds manifold NOUN cana-2728 351 16 , , PUNCT cana-2728 351 17 bull bull NOUN cana-2728 351 18 . . PUNCT cana-2728 352 1 malays malays PROPN cana-2728 352 2 . . PUNCT cana-2728 353 1 math math NOUN cana-2728 353 2 . . PUNCT cana-2728 354 1 sci sci PROPN cana-2728 354 2 . . PROPN cana-2728 354 3 soc soc PROPN cana-2728 354 4 . . PUNCT cana-2728 355 1 ( ( PUNCT cana-2728 355 2 2 2 NUM cana-2728 355 3 ) ) PUNCT cana-2728 355 4 , , PUNCT cana-2728 355 5 27 27 NUM cana-2728 355 6 ( ( PUNCT cana-2728 355 7 2004 2004 NUM cana-2728 355 8 ) ) PUNCT cana-2728 355 9 , , PUNCT cana-2728 355 10 17 17 NUM cana-2728 355 11 - - SYM cana-2728 355 12 -26 -26 NOUN cana-2728 355 13 . . PUNCT cana-2728 356 1 [ [ X cana-2728 356 2 17 17 NUM cana-2728 356 3 ] ] X cana-2728 356 4 b. b. PROPN cana-2728 356 5 b. b. PROPN cana-2728 356 6 sinha sinha PROPN cana-2728 356 7 and and CCONJ cana-2728 356 8 k. k. PROPN cana-2728 356 9 l. l. PROPN cana-2728 356 10 s. s. PROPN cana-2728 356 11 prasad prasad PROPN cana-2728 356 12 , , PUNCT cana-2728 356 13 a a DET cana-2728 356 14 class class NOUN cana-2728 356 15 of of ADP cana-2728 356 16 almost almost ADV cana-2728 356 17 para para NOUN cana-2728 356 18 contact contact NOUN cana-2728 356 19 metric metric ADJ cana-2728 356 20 manifold manifold ADJ cana-2728 356 21 , , PUNCT cana-2728 356 22 bull bull NOUN cana-2728 356 23 . . PUNCT cana-2728 357 1 calcutta calcutta PROPN cana-2728 357 2 math math PROPN cana-2728 357 3 . . PUNCT cana-2728 358 1 soc soc PROPN cana-2728 358 2 , , PUNCT cana-2728 358 3 87 87 NUM cana-2728 358 4 ( ( PUNCT cana-2728 358 5 1995 1995 NUM cana-2728 358 6 ) ) PUNCT cana-2728 358 7 , , PUNCT cana-2728 358 8 307 307 NUM cana-2728 358 9 - - SYM cana-2728 358 10 -312 -312 PROPN cana-2728 358 11 . . PUNCT cana-2728 359 1 [ [ X cana-2728 359 2 18 18 NUM cana-2728 359 3 ] ] X cana-2728 359 4 ventakesha ventakesha PROPN cana-2728 359 5 and and CCONJ cana-2728 359 6 c. c. PROPN cana-2728 359 7 s. s. PROPN cana-2728 359 8 bagewadi bagewadi PROPN cana-2728 359 9 , , PUNCT cana-2728 359 10 on on ADP cana-2728 359 11 circular circular ADJ cana-2728 359 12 πœ™-recurrent πœ™-recurrent NOUN cana-2728 359 13 lp lp ADJ cana-2728 359 14 - - ADJ cana-2728 359 15 sasakian sasakian ADJ cana-2728 359 16 manifolds manifold NOUN cana-2728 359 17 , , PUNCT cana-2728 359 18 differ differ VERB cana-2728 359 19 . . PUNCT cana-2728 360 1 geom geom PROPN cana-2728 360 2 . . PUNCT cana-2728 361 1 dyn dyn PROPN cana-2728 361 2 . . PUNCT cana-2728 362 1 syst syst PROPN cana-2728 362 2 . . PROPN cana-2728 362 3 , , PUNCT cana-2728 362 4 10 10 NUM cana-2728 362 5 ( ( PUNCT cana-2728 362 6 2008 2008 NUM cana-2728 362 7 ) ) PUNCT cana-2728 362 8 , , PUNCT cana-2728 362 9 312β€”319 312β€”319 PROPN cana-2728 362 10 . . PUNCT cana-2728 363 1 [ [ X cana-2728 363 2 19 19 NUM cana-2728 363 3 ] ] PUNCT cana-2728 363 4 k. k. PROPN cana-2728 363 5 yano yano PROPN cana-2728 363 6 and and CCONJ cana-2728 363 7 m. m. PROPN cana-2728 363 8 kon kon PROPN cana-2728 363 9 , , PUNCT cana-2728 363 10 structures structure NOUN cana-2728 363 11 on on ADP cana-2728 363 12 manifolds manifold NOUN cana-2728 363 13 , , PUNCT cana-2728 363 14 series series NOUN cana-2728 363 15 in in ADP cana-2728 363 16 pure pure ADJ cana-2728 363 17 math math NOUN cana-2728 363 18 . . PUNCT cana-2728 364 1 , , PUNCT cana-2728 364 2 3 3 NUM cana-2728 364 3 ( ( PUNCT cana-2728 364 4 1984 1984 NUM cana-2728 364 5 ) ) PUNCT cana-2728 364 6 . . PUNCT cana-2728 365 1 doi:10.1142/0067 doi:10.1142/0067 NOUN