id sid tid token lemma pos cana-2735 1 1 communications communication NOUN cana-2735 1 2 on on ADP cana-2735 1 3 applied apply VERB cana-2735 1 4 nonlinear nonlinear ADJ cana-2735 1 5 analysis analysis NOUN cana-2735 1 6 issn issn NOUN cana-2735 1 7 : : PUNCT cana-2735 1 8 1074 1074 NUM cana-2735 1 9 - - PUNCT cana-2735 1 10 133x 133x NUM cana-2735 1 11 vol vol NOUN cana-2735 1 12 32 32 NUM cana-2735 1 13 no no NOUN cana-2735 1 14 . . PUNCT cana-2735 2 1 4s 4s NUM cana-2735 2 2 ( ( PUNCT cana-2735 2 3 2025 2025 NUM cana-2735 2 4 ) ) PUNCT cana-2735 2 5 1 1 NUM cana-2735 2 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 2 7 ( ( PUNCT cana-2735 2 8 𝜢 𝜢 NOUN cana-2735 2 9 , , PUNCT cana-2735 2 10 𝜷)-metric 𝜷)-metric PUNCT cana-2735 2 11 on on ADP cana-2735 2 12 cartan cartan ADJ cana-2735 2 13 space space NOUN cana-2735 2 14 robin robin PROPN cana-2735 2 15 kumar1 kumar1 PROPN cana-2735 2 16 , , PUNCT cana-2735 2 17 mohammad mohammad PROPN cana-2735 2 18 rafee2 rafee2 PROPN cana-2735 2 19 , , PUNCT cana-2735 2 20 gaurav gaurav PROPN cana-2735 2 21 kumar3 kumar3 PROPN cana-2735 2 22 1,2,3 1,2,3 NUM cana-2735 2 23 department department NOUN cana-2735 2 24 of of ADP cana-2735 2 25 mathematics mathematic NOUN cana-2735 2 26 , , PUNCT cana-2735 2 27 school school NOUN cana-2735 2 28 of of ADP cana-2735 2 29 sciences science NOUN cana-2735 2 30 , , PUNCT cana-2735 2 31 rimt rimt ADJ cana-2735 2 32 university university NOUN cana-2735 2 33 , , PUNCT cana-2735 2 34 punjab punjab PROPN cana-2735 2 35 , , PUNCT cana-2735 2 36 india india PROPN cana-2735 2 37 email email NOUN cana-2735 2 38 i i PROPN cana-2735 2 39 d d PROPN cana-2735 2 40 : : PUNCT cana-2735 2 41 robinkumar15101983@gmail.com1 robinkumar15101983@gmail.com1 PROPN cana-2735 2 42 , , PUNCT cana-2735 2 43 corresponding corresponding ADJ cana-2735 2 44 author author NOUN cana-2735 2 45 : : PUNCT cana-2735 2 46 mohd_rafee60@yahoo.com mohd_rafee60@yahoo.com X cana-2735 2 47 article article NOUN cana-2735 2 48 history history NOUN cana-2735 2 49 : : PUNCT cana-2735 2 50 received receive VERB cana-2735 2 51 : : PUNCT cana-2735 2 52 10 10 NUM cana-2735 2 53 - - SYM cana-2735 2 54 09 09 NUM cana-2735 2 55 - - PUNCT cana-2735 2 56 2024 2024 NUM cana-2735 2 57 revised revise VERB cana-2735 2 58 : : PUNCT cana-2735 2 59 15 15 NUM cana-2735 2 60 - - SYM cana-2735 2 61 11 11 NUM cana-2735 2 62 - - PUNCT cana-2735 2 63 2024 2024 NUM cana-2735 2 64 accepted accept VERB cana-2735 2 65 : : PUNCT cana-2735 2 66 25 25 NUM cana-2735 2 67 - - SYM cana-2735 2 68 11 11 NUM cana-2735 2 69 - - PUNCT cana-2735 2 70 2024 2024 NUM cana-2735 2 71 abstract abstract NOUN cana-2735 2 72 : : PUNCT cana-2735 2 73 in in ADP cana-2735 2 74 finsler finsler NOUN cana-2735 2 75 geometry geometry NOUN cana-2735 2 76 , , PUNCT cana-2735 2 77 a a DET cana-2735 2 78 cartan cartan ADJ cana-2735 2 79 space space NOUN cana-2735 2 80 or or CCONJ cana-2735 2 81 cartan cartan PROPN cana-2735 2 82 manifold manifold PROPN cana-2735 2 83 refers refer VERB cana-2735 2 84 to to ADP cana-2735 2 85 a a DET cana-2735 2 86 special special ADJ cana-2735 2 87 type type NOUN cana-2735 2 88 of of ADP cana-2735 2 89 geometrical geometrical ADJ cana-2735 2 90 structure structure NOUN cana-2735 2 91 where where SCONJ cana-2735 2 92 there there PRON cana-2735 2 93 is be VERB cana-2735 2 94 a a DET cana-2735 2 95 preferred preferred ADJ cana-2735 2 96 connection connection NOUN cana-2735 2 97 or or CCONJ cana-2735 2 98 curvature curvature NOUN cana-2735 2 99 , , PUNCT cana-2735 2 100 often often ADV cana-2735 2 101 associated associate VERB cana-2735 2 102 with with ADP cana-2735 2 103 a a DET cana-2735 2 104 homogeneous homogeneous ADJ cana-2735 2 105 space space NOUN cana-2735 2 106 that that PRON cana-2735 2 107 carries carry VERB cana-2735 2 108 additional additional ADJ cana-2735 2 109 symmetries symmetry NOUN cana-2735 2 110 . . PUNCT cana-2735 3 1 in in ADP cana-2735 3 2 the the DET cana-2735 3 3 present present ADJ cana-2735 3 4 research research NOUN cana-2735 3 5 paper paper NOUN cana-2735 3 6 , , PUNCT cana-2735 3 7 we we PRON cana-2735 3 8 have have AUX cana-2735 3 9 deduced deduce VERB cana-2735 3 10 necessary necessary ADJ cana-2735 3 11 and and CCONJ cana-2735 3 12 sufficient sufficient ADJ cana-2735 3 13 conditions condition NOUN cana-2735 3 14 under under ADP cana-2735 3 15 which which PRON cana-2735 3 16 an an DET cana-2735 3 17 ( ( PUNCT cana-2735 3 18 Ξ± Ξ± NOUN cana-2735 3 19 , , PUNCT cana-2735 3 20 Ξ²)-metric Ξ²)-metric PUNCT cana-2735 3 21 , , PUNCT cana-2735 3 22 k(x k(x PROPN cana-2735 3 23 , , PUNCT cana-2735 3 24 Ο‰)=Ξ±(x Ο‰)=Ξ±(x PROPN cana-2735 3 25 , , PUNCT cana-2735 3 26 Ο‰)+ϡβ(x Ο‰)+ϡβ(x PROPN cana-2735 3 27 , , PUNCT cana-2735 3 28 Ο‰)+2k Ο‰)+2k X cana-2735 3 29 ( ( PUNCT cana-2735 3 30 Ξ²^2 Ξ²^2 PROPN cana-2735 3 31 ( ( PUNCT cana-2735 3 32 x x NOUN cana-2735 3 33 , , PUNCT cana-2735 3 34 Ο‰))/(Ξ±(x Ο‰))/(Ξ±(x PROPN cana-2735 3 35 , , PUNCT cana-2735 3 36 Ο‰))-k^2/3 Ο‰))-k^2/3 PROPN cana-2735 3 37 ( ( PUNCT cana-2735 3 38 Ξ²^4 Ξ²^4 X cana-2735 3 39 ( ( PUNCT cana-2735 3 40 x x NOUN cana-2735 3 41 , , PUNCT cana-2735 3 42 Ο‰))/(Ξ±^3 Ο‰))/(Ξ±^3 NOUN cana-2735 3 43 ( ( PUNCT cana-2735 3 44 x x X cana-2735 3 45 , , PUNCT cana-2735 3 46 Ο‰ Ο‰ NOUN cana-2735 3 47 ) ) PUNCT cana-2735 3 48 ) ) PUNCT cana-2735 3 49 , , PUNCT cana-2735 3 50 on on ADP cana-2735 3 51 a a DET cana-2735 3 52 cartan cartan ADJ cana-2735 3 53 space space NOUN cana-2735 3 54 admitting admit VERB cana-2735 3 55 h h NOUN cana-2735 3 56 - - ADJ cana-2735 3 57 metrical metrical ADJ cana-2735 3 58 d d NOUN cana-2735 3 59 - - PUNCT cana-2735 3 60 connection connection NOUN cana-2735 3 61 becomes become VERB cana-2735 3 62 a a DET cana-2735 3 63 locally locally ADV cana-2735 3 64 minkowski minkowski ADJ cana-2735 3 65 and and CCONJ cana-2735 3 66 conformally conformally ADV cana-2735 3 67 flat flat ADJ cana-2735 3 68 space space NOUN cana-2735 3 69 . . PUNCT cana-2735 4 1 keyword keyword NOUN cana-2735 4 2 : : PUNCT cana-2735 4 3 cartan cartan ADJ cana-2735 4 4 space space NOUN cana-2735 4 5 , , PUNCT cana-2735 4 6 ( ( PUNCT cana-2735 4 7 Ξ± Ξ± NOUN cana-2735 4 8 , , PUNCT cana-2735 4 9 Ξ²)-metric Ξ²)-metric PUNCT cana-2735 4 10 , , PUNCT cana-2735 4 11 minkowski minkowski ADJ cana-2735 4 12 space space NOUN cana-2735 4 13 , , PUNCT cana-2735 4 14 conformally conformally ADV cana-2735 4 15 flat flat ADJ cana-2735 4 16 space space NOUN cana-2735 4 17 . . PUNCT cana-2735 5 1 ams am NOUN cana-2735 5 2 subject subject ADJ cana-2735 5 3 classification classification NOUN cana-2735 5 4 : : PUNCT cana-2735 5 5 53b40 53b40 NUM cana-2735 5 6 , , PUNCT cana-2735 5 7 53c60 53c60 NUM cana-2735 5 8 1 1 NUM cana-2735 5 9 introduction introduction NOUN cana-2735 5 10 the the DET cana-2735 5 11 term term NOUN cana-2735 5 12 " " PUNCT cana-2735 5 13 cartan cartan ADJ cana-2735 5 14 space space NOUN cana-2735 5 15 " " PUNCT cana-2735 5 16 can can AUX cana-2735 5 17 refer refer VERB cana-2735 5 18 to to ADP cana-2735 5 19 spaces space NOUN cana-2735 5 20 that that PRON cana-2735 5 21 have have VERB cana-2735 5 22 a a DET cana-2735 5 23 particular particular ADJ cana-2735 5 24 kind kind NOUN cana-2735 5 25 of of ADP cana-2735 5 26 connection connection NOUN cana-2735 5 27 ( ( PUNCT cana-2735 5 28 typically typically ADV cana-2735 5 29 the the DET cana-2735 5 30 cartan cartan ADJ cana-2735 5 31 connection connection NOUN cana-2735 5 32 ) ) PUNCT cana-2735 5 33 or or CCONJ cana-2735 5 34 spaces space NOUN cana-2735 5 35 that that PRON cana-2735 5 36 have have AUX cana-2735 5 37 been be AUX cana-2735 5 38 generalized generalize VERB cana-2735 5 39 to to PART cana-2735 5 40 accommodate accommodate VERB cana-2735 5 41 more more ADV cana-2735 5 42 complex complex ADJ cana-2735 5 43 geometrical geometrical ADJ cana-2735 5 44 structures structure NOUN cana-2735 5 45 . . PUNCT cana-2735 6 1 this this DET cana-2735 6 2 space space NOUN cana-2735 6 3 was be AUX cana-2735 6 4 founded found VERB cana-2735 6 5 by by ADP cana-2735 6 6 e. e. PROPN cana-2735 6 7 cartan cartan PROPN cana-2735 6 8 a a DET cana-2735 6 9 french french ADJ cana-2735 6 10 mathematician mathematician NOUN cana-2735 6 11 and and CCONJ cana-2735 6 12 geometer geometer NOUN cana-2735 7 1 [ [ X cana-2735 7 2 2 2 NUM cana-2735 7 3 ] ] PUNCT cana-2735 7 4 . . PUNCT cana-2735 8 1 cartan cartan PROPN cana-2735 8 2 space space NOUN cana-2735 8 3 is be AUX cana-2735 8 4 the the DET cana-2735 8 5 dual dual ADJ cana-2735 8 6 of of ADP cana-2735 8 7 a a DET cana-2735 8 8 finsler finsler NOUN cana-2735 8 9 space space NOUN cana-2735 8 10 [ [ X cana-2735 8 11 6 6 NUM cana-2735 8 12 ] ] PUNCT cana-2735 8 13 and and CCONJ cana-2735 8 14 this this DET cana-2735 8 15 dual dual ADJ cana-2735 8 16 space space NOUN cana-2735 8 17 was be AUX cana-2735 8 18 defined define VERB cana-2735 8 19 using use VERB cana-2735 8 20 a a DET cana-2735 8 21 linear linear ADJ cana-2735 8 22 functional functional NOUN cana-2735 8 23 named name VERB cana-2735 8 24 as as ADP cana-2735 8 25 legender legender NOUN cana-2735 8 26 transformation transformation NOUN cana-2735 8 27 . . PUNCT cana-2735 9 1 the the DET cana-2735 9 2 relation relation NOUN cana-2735 9 3 between between ADP cana-2735 9 4 cartan cartan ADJ cana-2735 9 5 space space NOUN cana-2735 9 6 and and CCONJ cana-2735 9 7 finsler finsler NOUN cana-2735 9 8 space space NOUN cana-2735 9 9 has have AUX cana-2735 9 10 been be AUX cana-2735 9 11 studied study VERB cana-2735 9 12 by by ADP cana-2735 9 13 f. f. PROPN cana-2735 9 14 brickell brickell PROPN cana-2735 10 1 [ [ X cana-2735 10 2 1 1 NUM cana-2735 10 3 ] ] PUNCT cana-2735 10 4 , , PUNCT cana-2735 10 5 h. h. PROPN cana-2735 10 6 rund rund PROPN cana-2735 11 1 [ [ X cana-2735 11 2 10 10 NUM cana-2735 11 3 ] ] PUNCT cana-2735 11 4 and and CCONJ cana-2735 11 5 others other NOUN cana-2735 11 6 . . PUNCT cana-2735 12 1 r. r. PROPN cana-2735 12 2 miron miron PROPN cana-2735 13 1 ( ( PUNCT cana-2735 13 2 [ [ X cana-2735 13 3 6 6 NUM cana-2735 13 4 ] ] PUNCT cana-2735 13 5 , , PUNCT cana-2735 13 6 [ [ X cana-2735 13 7 7 7 NUM cana-2735 13 8 ] ] PUNCT cana-2735 13 9 ) ) PUNCT cana-2735 13 10 introduced introduce VERB cana-2735 13 11 the the DET cana-2735 13 12 theory theory NOUN cana-2735 13 13 of of ADP cana-2735 13 14 hamiltonian hamiltonian ADJ cana-2735 13 15 space space NOUN cana-2735 13 16 , , PUNCT cana-2735 13 17 he he PRON cana-2735 13 18 proved prove VERB cana-2735 13 19 that that SCONJ cana-2735 13 20 cartan cartan ADJ cana-2735 13 21 space space NOUN cana-2735 13 22 is be AUX cana-2735 13 23 a a DET cana-2735 13 24 particular particular ADJ cana-2735 13 25 case case NOUN cana-2735 13 26 of of ADP cana-2735 13 27 hamilton hamilton PROPN cana-2735 13 28 space space NOUN cana-2735 13 29 . . PUNCT cana-2735 14 1 the the DET cana-2735 14 2 notion notion NOUN cana-2735 14 3 of of ADP cana-2735 14 4 ( ( PUNCT cana-2735 14 5 𝛼 𝛼 PROPN cana-2735 14 6 , , PUNCT cana-2735 14 7 𝛽)-metric 𝛽)-metric ADJ cana-2735 14 8 in in ADP cana-2735 14 9 cartan cartan ADJ cana-2735 14 10 space space NOUN cana-2735 14 11 was be AUX cana-2735 14 12 introduced introduce VERB cana-2735 14 13 by by ADP cana-2735 14 14 t. t. PROPN cana-2735 14 15 igrashi igrashi PROPN cana-2735 14 16 ( ( PUNCT cana-2735 14 17 [ [ X cana-2735 14 18 4 4 NUM cana-2735 14 19 ] ] PUNCT cana-2735 14 20 , , PUNCT cana-2735 14 21 [ [ X cana-2735 14 22 3 3 NUM cana-2735 14 23 ] ] PUNCT cana-2735 14 24 . . PUNCT cana-2735 15 1 he he PRON cana-2735 15 2 obtained obtain VERB cana-2735 15 3 the the DET cana-2735 15 4 metric metric ADJ cana-2735 15 5 tensors tensor NOUN cana-2735 15 6 and and CCONJ cana-2735 15 7 some some DET cana-2735 15 8 invariants invariant NOUN cana-2735 15 9 which which PRON cana-2735 15 10 characterize characterize VERB cana-2735 15 11 the the DET cana-2735 15 12 special special ADJ cana-2735 15 13 class class NOUN cana-2735 15 14 of of ADP cana-2735 15 15 cartan cartan PROPN cana-2735 15 16 spaces space NOUN cana-2735 15 17 with with ADP cana-2735 15 18 ( ( PUNCT cana-2735 15 19 𝛼 𝛼 NOUN cana-2735 15 20 , , PUNCT cana-2735 15 21 𝛽)-metric 𝛽)-metric ADJ cana-2735 15 22 . . PUNCT cana-2735 16 1 h.g h.g PROPN cana-2735 16 2 . . PROPN cana-2735 16 3 nagaraja nagaraja PROPN cana-2735 17 1 [ [ X cana-2735 17 2 8 8 NUM cana-2735 17 3 ] ] PUNCT cana-2735 17 4 , , PUNCT cana-2735 17 5 g. g. PROPN cana-2735 17 6 shanker shanker PROPN cana-2735 18 1 [ [ X cana-2735 18 2 11 11 NUM cana-2735 18 3 ] ] PUNCT cana-2735 18 4 , , PUNCT cana-2735 18 5 m. m. NOUN cana-2735 18 6 rafee rafee NOUN cana-2735 18 7 ( ( PUNCT cana-2735 18 8 [ [ X cana-2735 18 9 9 9 NUM cana-2735 18 10 ] ] PUNCT cana-2735 18 11 , , PUNCT cana-2735 18 12 [ [ X cana-2735 18 13 15 15 NUM cana-2735 18 14 ] ] PUNCT cana-2735 18 15 ) ) PUNCT cana-2735 18 16 and and CCONJ cana-2735 18 17 tripathi tripathi NOUN cana-2735 19 1 [ [ X cana-2735 19 2 13 13 NUM cana-2735 19 3 ] ] PUNCT cana-2735 19 4 have have AUX cana-2735 19 5 also also ADV cana-2735 19 6 made make VERB cana-2735 19 7 significant significant ADJ cana-2735 19 8 development development NOUN cana-2735 19 9 in in ADP cana-2735 19 10 the the DET cana-2735 19 11 theory theory NOUN cana-2735 19 12 of of ADP cana-2735 19 13 cartan cartan PROPN cana-2735 19 14 spaces space NOUN cana-2735 19 15 with with ADP cana-2735 19 16 ( ( PUNCT cana-2735 19 17 𝛼 𝛼 NOUN cana-2735 19 18 , , PUNCT cana-2735 19 19 𝛽)-metric 𝛽)-metric ADJ cana-2735 19 20 . . PUNCT cana-2735 20 1 the the DET cana-2735 20 2 paper paper NOUN cana-2735 20 3 is be AUX cana-2735 20 4 organized organize VERB cana-2735 20 5 as as SCONJ cana-2735 20 6 follows follow VERB cana-2735 20 7 : : PUNCT cana-2735 20 8 in in ADP cana-2735 20 9 section section NOUN cana-2735 20 10 2 2 NUM cana-2735 20 11 , , PUNCT cana-2735 20 12 we we PRON cana-2735 20 13 give give VERB cana-2735 20 14 basic basic ADJ cana-2735 20 15 definitions definition NOUN cana-2735 20 16 and and CCONJ cana-2735 20 17 results result NOUN cana-2735 20 18 required require VERB cana-2735 20 19 for for ADP cana-2735 20 20 subsequent subsequent ADJ cana-2735 20 21 sections section NOUN cana-2735 20 22 . . PUNCT cana-2735 21 1 in in ADP cana-2735 21 2 section section NOUN cana-2735 21 3 3 3 NUM cana-2735 21 4 , , PUNCT cana-2735 21 5 we we PRON cana-2735 21 6 deal deal VERB cana-2735 21 7 with with ADP cana-2735 21 8 cartan cartan ADJ cana-2735 21 9 space space NOUN cana-2735 21 10 with with ADP cana-2735 21 11 an an DET cana-2735 21 12 ( ( PUNCT cana-2735 21 13 𝛼 𝛼 PROPN cana-2735 21 14 , , PUNCT cana-2735 21 15 𝛽 𝛽 NOUN cana-2735 21 16 ) ) PUNCT cana-2735 21 17 -metric -metric NOUN cana-2735 21 18 , , PUNCT cana-2735 21 19 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 21 20 , , PUNCT cana-2735 21 21 πœ” πœ” ADJ cana-2735 21 22 ) ) PUNCT cana-2735 21 23 = = SYM cana-2735 22 1 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 22 2 , , PUNCT cana-2735 22 3 πœ” πœ” PRON cana-2735 22 4 ) ) PUNCT cana-2735 22 5 + + NUM cana-2735 22 6 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 22 7 , , PUNCT cana-2735 22 8 πœ” πœ” PRON cana-2735 22 9 ) ) PUNCT cana-2735 22 10 + + CCONJ cana-2735 22 11 2π‘˜ 2π‘˜ NUM cana-2735 22 12 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 22 13 ) ) PUNCT cana-2735 22 14 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 22 15 ) ) PUNCT cana-2735 22 16 βˆ’ βˆ’ PROPN cana-2735 23 1 π‘˜2 π‘˜2 PROPN cana-2735 23 2 3 3 NUM cana-2735 23 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 23 4 ) ) PUNCT cana-2735 23 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 23 6 ) ) PUNCT cana-2735 23 7 , , PUNCT cana-2735 23 8 admitting admit VERB cana-2735 23 9 β„Ž-metrical β„Ž-metrical PROPN cana-2735 23 10 𝑑-connection 𝑑-connection PROPN cana-2735 23 11 . . PUNCT cana-2735 24 1 in in ADP cana-2735 24 2 section section NOUN cana-2735 24 3 4 4 NUM cana-2735 24 4 , , PUNCT cana-2735 24 5 we we PRON cana-2735 24 6 study study VERB cana-2735 24 7 the the DET cana-2735 24 8 conformal conformal ADJ cana-2735 24 9 change change NOUN cana-2735 24 10 of of ADP cana-2735 24 11 cartan cartan ADJ cana-2735 24 12 space space NOUN cana-2735 24 13 and and CCONJ cana-2735 24 14 find find VERB cana-2735 24 15 some some DET cana-2735 24 16 important important ADJ cana-2735 24 17 results result NOUN cana-2735 24 18 . . PUNCT cana-2735 25 1 2 2 NUM cana-2735 25 2 preliminaries preliminary NOUN cana-2735 25 3 we we PRON cana-2735 25 4 recall recall VERB cana-2735 25 5 some some DET cana-2735 25 6 important important ADJ cana-2735 25 7 definitions definition NOUN cana-2735 25 8 like like ADP cana-2735 25 9 finsler finsler NOUN cana-2735 25 10 metric metric ADJ cana-2735 25 11 in in ADP cana-2735 25 12 cotangent cotangent NOUN cana-2735 25 13 bundle bundle NOUN cana-2735 25 14 , , PUNCT cana-2735 25 15 cartan cartan PROPN cana-2735 25 16 space space NOUN cana-2735 25 17 etc etc X cana-2735 25 18 : : PUNCT cana-2735 25 19 definition definition NOUN cana-2735 25 20 2.1 2.1 NUM cana-2735 25 21 ( ( PUNCT cana-2735 25 22 finsler finsler NOUN cana-2735 25 23 metric metric NOUN cana-2735 25 24 of of ADP cana-2735 25 25 cotangent cotangent NOUN cana-2735 25 26 bundle bundle NOUN cana-2735 25 27 ) ) PUNCT cana-2735 25 28 let let VERB cana-2735 25 29 𝑀 𝑀 PRON cana-2735 25 30 be be AUX cana-2735 25 31 a a DET cana-2735 25 32 smooth smooth ADJ cana-2735 25 33 manifold manifold NOUN cana-2735 25 34 and and CCONJ cana-2735 25 35 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 25 36 be be VERB cana-2735 25 37 its its PRON cana-2735 25 38 cotangent cotangent NOUN cana-2735 25 39 bundle bundle NOUN cana-2735 25 40 . . PUNCT cana-2735 26 1 a a DET cana-2735 26 2 𝐢∞ 𝐢∞ PROPN cana-2735 26 3 function function NOUN cana-2735 26 4 𝐾 𝐾 PROPN cana-2735 26 5 : : PUNCT cana-2735 26 6 π‘‡βˆ—π‘€\{0 π‘‡βˆ—π‘€\{0 NOUN cana-2735 26 7 } } PUNCT cana-2735 26 8 β†’ β†’ SYM cana-2735 26 9 𝑅 𝑅 PROPN cana-2735 26 10 is be AUX cana-2735 26 11 mailto:robinkumar15101983@gmail.com mailto:robinkumar15101983@gmail.com X cana-2735 26 12 mailto:mohd_rafee60@yahoo.com mailto:mohd_rafee60@yahoo.com X cana-2735 26 13 communications communication NOUN cana-2735 26 14 on on ADP cana-2735 26 15 applied apply VERB cana-2735 26 16 nonlinear nonlinear ADJ cana-2735 26 17 analysis analysis NOUN cana-2735 26 18 issn issn NOUN cana-2735 26 19 : : PUNCT cana-2735 26 20 1074 1074 NUM cana-2735 26 21 - - PUNCT cana-2735 26 22 133x 133x NUM cana-2735 26 23 vol vol NOUN cana-2735 26 24 32 32 NUM cana-2735 26 25 no no NOUN cana-2735 26 26 . . PUNCT cana-2735 27 1 4s 4s NUM cana-2735 27 2 ( ( PUNCT cana-2735 27 3 2025 2025 NUM cana-2735 27 4 ) ) PUNCT cana-2735 27 5 2 2 NUM cana-2735 27 6 https://internationalpubls.com https://internationalpubls.com NUM cana-2735 27 7 called call VERB cana-2735 27 8 finsler finsler NOUN cana-2735 27 9 metric metric ADJ cana-2735 27 10 or or CCONJ cana-2735 27 11 finsler finsler NOUN cana-2735 27 12 fundamental fundamental ADJ cana-2735 27 13 function function NOUN cana-2735 27 14 on on ADP cana-2735 27 15 the the DET cana-2735 27 16 cotangent cotangent NOUN cana-2735 27 17 bundle bundle NOUN cana-2735 27 18 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 27 19 if if SCONJ cana-2735 27 20 it it PRON cana-2735 27 21 satisfies satisfy VERB cana-2735 27 22 the the DET cana-2735 27 23 following follow VERB cana-2735 27 24 properties property NOUN cana-2735 27 25 : : PUNCT cana-2735 27 26 ( ( PUNCT cana-2735 27 27 1 1 X cana-2735 27 28 ) ) PUNCT cana-2735 27 29 positivity positivity NOUN cana-2735 27 30 : : PUNCT cana-2735 27 31 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 27 32 , , PUNCT cana-2735 27 33 πœ” πœ” ADJ cana-2735 27 34 ) ) PUNCT cana-2735 27 35 β‰₯ β‰₯ NOUN cana-2735 27 36 0 0 NUM cana-2735 27 37 for for ADP cana-2735 27 38 all all PRON cana-2735 27 39 πœ” πœ” PART cana-2735 27 40 ∈ ∈ NOUN cana-2735 27 41 𝑇𝑝 𝑇𝑝 PROPN cana-2735 27 42 βˆ—π‘€. βˆ—π‘€. PROPN cana-2735 27 43 ( ( PUNCT cana-2735 27 44 2 2 NUM cana-2735 27 45 ) ) PUNCT cana-2735 27 46 positive positive ADJ cana-2735 27 47 homogeneity homogeneity NOUN cana-2735 27 48 : : PUNCT cana-2735 27 49 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 27 50 , , PUNCT cana-2735 27 51 πœ” πœ” PRON cana-2735 27 52 ) ) PUNCT cana-2735 27 53 is be AUX cana-2735 27 54 + + VERB cana-2735 27 55 ve ve VERB cana-2735 27 56 1 1 NUM cana-2735 27 57 - - PUNCT cana-2735 27 58 homogeneous homogeneous ADJ cana-2735 27 59 on on ADP cana-2735 27 60 the the DET cana-2735 27 61 fibers fiber NOUN cana-2735 27 62 of of ADP cana-2735 27 63 the the DET cana-2735 27 64 cotangent cotangent NOUN cana-2735 27 65 bundle bundle NOUN cana-2735 27 66 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 27 67 , , PUNCT cana-2735 27 68 i.e. i.e. X cana-2735 27 69 , , PUNCT cana-2735 27 70 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 27 71 , , PUNCT cana-2735 27 72 πœ†πœ” πœ†πœ” NOUN cana-2735 27 73 ) ) PUNCT cana-2735 27 74 = = SYM cana-2735 28 1 πœ†πΎ(π‘₯ πœ†πΎ(π‘₯ NOUN cana-2735 28 2 , , PUNCT cana-2735 28 3 πœ” πœ” NOUN cana-2735 28 4 ) ) PUNCT cana-2735 28 5 , , PUNCT cana-2735 28 6 βˆ€ βˆ€ X cana-2735 29 1 πœ† πœ† X cana-2735 29 2 > > X cana-2735 29 3 0 0 NUM cana-2735 29 4 ; ; PUNCT cana-2735 29 5 for for ADP cana-2735 29 6 any any DET cana-2735 29 7 π‘₯ π‘₯ PRON cana-2735 29 8 ∈ ∈ PROPN cana-2735 29 9 𝑀 𝑀 PROPN cana-2735 29 10 , , PUNCT cana-2735 29 11 πœ” πœ” PROPN cana-2735 29 12 ∈ ∈ NOUN cana-2735 29 13 𝑇π‘₯ 𝑇π‘₯ ADP cana-2735 29 14 βˆ—π‘€. βˆ—π‘€. PROPN cana-2735 29 15 ( ( PUNCT cana-2735 29 16 3 3 NUM cana-2735 29 17 ) ) PUNCT cana-2735 29 18 strict strict ADJ cana-2735 29 19 convexity convexity NOUN cana-2735 29 20 of of ADP cana-2735 29 21 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 29 22 , , PUNCT cana-2735 29 23 πœ” πœ” VERB cana-2735 29 24 ): ): PUNCT cana-2735 29 25 the the DET cana-2735 29 26 hessian hessian ADJ cana-2735 29 27 matrix matrix NOUN cana-2735 29 28 defined define VERB cana-2735 29 29 by by ADP cana-2735 29 30 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ PROPN cana-2735 29 31 , , PUNCT cana-2735 29 32 πœ” πœ” ADJ cana-2735 29 33 ) ) PUNCT cana-2735 29 34 = = SYM cana-2735 29 35 1 1 NUM cana-2735 29 36 2 2 NUM cana-2735 29 37 πœ•2𝐾2 πœ•2𝐾2 ADJ cana-2735 29 38 πœ•πœ”π‘–πœ•πœ”π‘— πœ•πœ”π‘–πœ•πœ”π‘— NOUN cana-2735 29 39 ( ( PUNCT cana-2735 29 40 π‘₯ π‘₯ NOUN cana-2735 29 41 , , PUNCT cana-2735 29 42 πœ” πœ” VERB cana-2735 29 43 ) ) PUNCT cana-2735 29 44 is be AUX cana-2735 29 45 positive positive ADJ cana-2735 29 46 definite definite ADJ cana-2735 29 47 for for ADP cana-2735 29 48 all all DET cana-2735 29 49 ( ( PUNCT cana-2735 29 50 π‘₯ π‘₯ NOUN cana-2735 29 51 , , PUNCT cana-2735 29 52 πœ” πœ” NOUN cana-2735 29 53 ) ) PUNCT cana-2735 29 54 ∈ ∈ NOUN cana-2735 29 55 π‘‡βˆ—π‘€\{0 π‘‡βˆ—π‘€\{0 NOUN cana-2735 29 56 } } PUNCT cana-2735 29 57 . . PUNCT cana-2735 30 1 definition definition NOUN cana-2735 30 2 2.2 2.2 NUM cana-2735 30 3 ( ( PUNCT cana-2735 30 4 cartan cartan ADJ cana-2735 30 5 space space NOUN cana-2735 30 6 ) ) PUNCT cana-2735 30 7 a a DET cana-2735 30 8 differentiable differentiable ADJ cana-2735 30 9 manifold manifold ADJ cana-2735 30 10 𝑀 𝑀 PROPN cana-2735 30 11 equipped equip VERB cana-2735 30 12 with with ADP cana-2735 30 13 a a DET cana-2735 30 14 finsler finsler NOUN cana-2735 30 15 metric metric ADJ cana-2735 30 16 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 30 17 , , PUNCT cana-2735 30 18 πœ” πœ” PRON cana-2735 30 19 ) ) PUNCT cana-2735 30 20 defined define VERB cana-2735 30 21 on on ADP cana-2735 30 22 the the DET cana-2735 30 23 cotangent cotangent NOUN cana-2735 30 24 bundle bundle NOUN cana-2735 30 25 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 30 26 is be AUX cana-2735 30 27 called call VERB cana-2735 30 28 a a DET cana-2735 30 29 cartan cartan ADJ cana-2735 30 30 space space NOUN cana-2735 30 31 .. .. PUNCT cana-2735 31 1 cartan cartan ADJ cana-2735 31 2 space space NOUN cana-2735 31 3 is be AUX cana-2735 31 4 denoted denote VERB cana-2735 31 5 by by ADP cana-2735 31 6 𝐢 𝐢 PROPN cana-2735 31 7 = = SYM cana-2735 31 8 ( ( PUNCT cana-2735 31 9 𝑀 𝑀 PROPN cana-2735 31 10 , , PUNCT cana-2735 31 11 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 31 12 , , PUNCT cana-2735 31 13 πœ” πœ” NOUN cana-2735 31 14 ) ) PUNCT cana-2735 31 15 ) ) PUNCT cana-2735 31 16 , , PUNCT cana-2735 31 17 where where SCONJ cana-2735 31 18 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 31 19 , , PUNCT cana-2735 31 20 πœ” πœ” PRON cana-2735 31 21 ) ) PUNCT cana-2735 31 22 represents represent VERB cana-2735 31 23 norm norm NOUN cana-2735 31 24 of of ADP cana-2735 31 25 the the DET cana-2735 31 26 differential differential ADJ cana-2735 31 27 one one NUM cana-2735 31 28 form form NOUN cana-2735 31 29 πœ” πœ” ADP cana-2735 31 30 ∈ ∈ NOUN cana-2735 31 31 𝑇π‘₯ 𝑇π‘₯ NOUN cana-2735 31 32 βˆ—π‘€ βˆ—π‘€ PROPN cana-2735 31 33 based base VERB cana-2735 31 34 at at ADP cana-2735 31 35 any any DET cana-2735 31 36 point point NOUN cana-2735 31 37 π‘₯ π‘₯ PRON cana-2735 31 38 ∈ ∈ NOUN cana-2735 31 39 𝑀. 𝑀. PROPN cana-2735 31 40 the the DET cana-2735 31 41 function function NOUN cana-2735 31 42 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 31 43 , , PUNCT cana-2735 31 44 πœ” πœ” PRON cana-2735 31 45 ) ) PUNCT cana-2735 31 46 is be AUX cana-2735 31 47 called call VERB cana-2735 31 48 the the DET cana-2735 31 49 fundamental fundamental ADJ cana-2735 31 50 function function NOUN cana-2735 31 51 and and CCONJ cana-2735 31 52 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ PROPN cana-2735 31 53 , , PUNCT cana-2735 31 54 πœ” πœ” ADJ cana-2735 31 55 ) ) PUNCT cana-2735 31 56 = = SYM cana-2735 31 57 1 1 NUM cana-2735 31 58 2 2 NUM cana-2735 31 59 πœ•2𝐾2 πœ•2𝐾2 ADJ cana-2735 31 60 πœ•πœ”π‘–πœ•πœ”π‘— πœ•πœ”π‘–πœ•πœ”π‘— NOUN cana-2735 31 61 ( ( PUNCT cana-2735 31 62 π‘₯ π‘₯ NOUN cana-2735 31 63 , , PUNCT cana-2735 31 64 πœ” πœ” VERB cana-2735 31 65 ) ) PUNCT cana-2735 31 66 is be AUX cana-2735 31 67 called call VERB cana-2735 31 68 the the DET cana-2735 31 69 fundamental fundamental ADJ cana-2735 31 70 metric metric ADJ cana-2735 31 71 tensor tensor NOUN cana-2735 31 72 of of ADP cana-2735 31 73 the the DET cana-2735 31 74 cartan cartan PROPN cana-2735 31 75 space space NOUN cana-2735 31 76 𝐢 𝐢 PROPN cana-2735 31 77 . . PUNCT cana-2735 32 1 in in ADP cana-2735 32 2 cartan cartan ADJ cana-2735 32 3 space space NOUN cana-2735 32 4 the the DET cana-2735 32 5 metric metric ADJ cana-2735 32 6 𝐾 𝐾 PROPN cana-2735 32 7 : : PUNCT cana-2735 32 8 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 32 9 β†’ β†’ PUNCT cana-2735 32 10 [ [ X cana-2735 32 11 0 0 NUM cana-2735 32 12 , , PUNCT cana-2735 32 13 ∞ ∞ NUM cana-2735 32 14 ) ) PUNCT cana-2735 32 15 is be AUX cana-2735 32 16 defined define VERB cana-2735 32 17 from from ADP cana-2735 32 18 cotangent cotangent NOUN cana-2735 32 19 bundle bundle NOUN cana-2735 32 20 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 32 21 to to ADP cana-2735 32 22 non non ADJ cana-2735 32 23 - - ADJ cana-2735 32 24 negative negative ADJ cana-2735 32 25 real real ADJ cana-2735 32 26 numbers number NOUN cana-2735 32 27 , , PUNCT cana-2735 32 28 so so SCONJ cana-2735 32 29 at at ADP cana-2735 32 30 a a DET cana-2735 32 31 point point NOUN cana-2735 32 32 π‘₯ π‘₯ DET cana-2735 32 33 ∈ ∈ PROPN cana-2735 32 34 𝑀 𝑀 PROPN cana-2735 32 35 , , PUNCT cana-2735 32 36 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 32 37 , , PUNCT cana-2735 32 38 βˆ’ βˆ’ NOUN cana-2735 32 39 ) ) PUNCT cana-2735 32 40 eats eat VERB cana-2735 32 41 one one NUM cana-2735 32 42 - - PUNCT cana-2735 32 43 form form NOUN cana-2735 32 44 πœ” πœ” PART cana-2735 32 45 ∈ ∈ NOUN cana-2735 32 46 𝑇𝑝 𝑇𝑝 VERB cana-2735 32 47 βˆ—π‘€ βˆ—π‘€ NOUN cana-2735 32 48 and and CCONJ cana-2735 32 49 spits spit VERB cana-2735 32 50 non non ADJ cana-2735 32 51 - - ADJ cana-2735 32 52 negative negative ADJ cana-2735 32 53 reals real NOUN cana-2735 32 54 , , PUNCT cana-2735 32 55 amounts amount VERB cana-2735 32 56 to to ADP cana-2735 32 57 saying say VERB cana-2735 32 58 that that SCONJ cana-2735 32 59 cartan cartan ADJ cana-2735 32 60 space space NOUN cana-2735 32 61 is be AUX cana-2735 32 62 constructed construct VERB cana-2735 32 63 on on ADP cana-2735 32 64 the the DET cana-2735 32 65 cotangent cotangent NOUN cana-2735 32 66 bundle bundle NOUN cana-2735 32 67 π‘‡βˆ—π‘€ π‘‡βˆ—π‘€ NOUN cana-2735 32 68 in in ADP cana-2735 32 69 the the DET cana-2735 32 70 same same ADJ cana-2735 32 71 way way NOUN cana-2735 32 72 a a DET cana-2735 32 73 finsler finsler NOUN cana-2735 32 74 space space NOUN cana-2735 32 75 ( ( PUNCT cana-2735 32 76 𝑀 𝑀 PROPN cana-2735 32 77 , , PUNCT cana-2735 32 78 𝐹(π‘₯ 𝐹(π‘₯ NUM cana-2735 32 79 , , PUNCT cana-2735 32 80 𝑦 𝑦 NOUN cana-2735 32 81 ) ) PUNCT cana-2735 32 82 ) ) PUNCT cana-2735 32 83 , , PUNCT cana-2735 32 84 where where SCONJ cana-2735 32 85 𝐹 𝐹 PROPN cana-2735 32 86 : : PUNCT cana-2735 32 87 𝑇𝑀 𝑇𝑀 PROPN cana-2735 32 88 β†’ β†’ SYM cana-2735 32 89 [ [ X cana-2735 32 90 0 0 NUM cana-2735 32 91 , , PUNCT cana-2735 32 92 ∞ ∞ PROPN cana-2735 32 93 ) ) PUNCT cana-2735 32 94 , , PUNCT cana-2735 32 95 is be AUX cana-2735 32 96 constructed construct VERB cana-2735 32 97 on on ADP cana-2735 32 98 the the DET cana-2735 32 99 tangent tangent NOUN cana-2735 32 100 bundle bundle NOUN cana-2735 32 101 𝑇𝑀. 𝑇𝑀. VERB cana-2735 32 102 next next ADV cana-2735 32 103 we we PRON cana-2735 32 104 define define VERB cana-2735 32 105 the the DET cana-2735 32 106 norm norm NOUN cana-2735 32 107 of of ADP cana-2735 32 108 a a DET cana-2735 32 109 differential differential ADJ cana-2735 32 110 one one NUM cana-2735 32 111 form form NOUN cana-2735 33 1 πœ” πœ” PART cana-2735 33 2 ∈ ∈ NOUN cana-2735 33 3 𝑇𝑝 𝑇𝑝 NOUN cana-2735 33 4 βˆ—π‘€ βˆ—π‘€ VERB cana-2735 33 5 in in ADP cana-2735 33 6 local local ADJ cana-2735 33 7 coordinates coordinate NOUN cana-2735 33 8 or or CCONJ cana-2735 33 9 in in ADP cana-2735 33 10 terms term NOUN cana-2735 33 11 of of ADP cana-2735 33 12 fundamental fundamental ADJ cana-2735 33 13 metric metric ADJ cana-2735 33 14 tensor tensor NOUN cana-2735 33 15 𝑔𝑖𝑗 𝑔𝑖𝑗 NOUN cana-2735 33 16 of of ADP cana-2735 33 17 the the DET cana-2735 33 18 corresponding corresponding ADJ cana-2735 33 19 cartan cartan ADJ cana-2735 33 20 space space NOUN cana-2735 33 21 ( ( PUNCT cana-2735 33 22 𝑀 𝑀 PROPN cana-2735 33 23 , , PUNCT cana-2735 33 24 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 33 25 , , PUNCT cana-2735 33 26 πœ” πœ” NOUN cana-2735 33 27 ) ) PUNCT cana-2735 33 28 ) ) PUNCT cana-2735 33 29 . . PUNCT cana-2735 34 1 definition definition NOUN cana-2735 34 2 2.3 2.3 NUM cana-2735 34 3 ( ( PUNCT cana-2735 34 4 finsler finsler NOUN cana-2735 34 5 norm norm NOUN cana-2735 34 6 of of ADP cana-2735 34 7 a a DET cana-2735 34 8 differential differential ADJ cana-2735 34 9 one one NUM cana-2735 34 10 form form NOUN cana-2735 34 11 ) ) PUNCT cana-2735 34 12 let let VERB cana-2735 34 13 ( ( PUNCT cana-2735 34 14 𝑀 𝑀 PROPN cana-2735 34 15 , , PUNCT cana-2735 34 16 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 34 17 , , PUNCT cana-2735 34 18 πœ” πœ” NOUN cana-2735 34 19 ) ) PUNCT cana-2735 34 20 ) ) PUNCT cana-2735 34 21 be be AUX cana-2735 34 22 a a DET cana-2735 34 23 cartan cartan ADJ cana-2735 34 24 space space NOUN cana-2735 34 25 , , PUNCT cana-2735 34 26 where where SCONJ cana-2735 34 27 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 34 28 , , PUNCT cana-2735 34 29 πœ” πœ” PRON cana-2735 34 30 ) ) PUNCT cana-2735 34 31 is be AUX cana-2735 34 32 a a DET cana-2735 34 33 finsler finsler NOUN cana-2735 34 34 metric metric ADJ cana-2735 34 35 on on ADP cana-2735 34 36 the the DET cana-2735 34 37 cotangent cotangent NOUN cana-2735 34 38 bundle bundle NOUN cana-2735 34 39 π‘‡βˆ—π‘€. π‘‡βˆ—π‘€. PUNCT cana-2735 34 40 then then ADV cana-2735 34 41 the the DET cana-2735 34 42 norm norm NOUN cana-2735 34 43 of of ADP cana-2735 34 44 a a DET cana-2735 34 45 differential differential ADJ cana-2735 34 46 one one NUM cana-2735 34 47 form form NOUN cana-2735 34 48 πœ” πœ” PART cana-2735 34 49 ∈ ∈ NOUN cana-2735 34 50 𝑇𝑝 𝑇𝑝 NOUN cana-2735 34 51 βˆ—π‘€ βˆ—π‘€ NOUN cana-2735 34 52 at at ADP cana-2735 34 53 any any DET cana-2735 34 54 fixed fixed ADJ cana-2735 34 55 point point NOUN cana-2735 34 56 π‘₯ π‘₯ PRON cana-2735 34 57 ∈ ∈ PROPN cana-2735 34 58 𝑀 𝑀 PROPN cana-2735 34 59 is be AUX cana-2735 34 60 denoted denote VERB cana-2735 34 61 by by ADP cana-2735 34 62 𝐾π‘₯(πœ” 𝐾π‘₯(πœ” NOUN cana-2735 34 63 ) ) PUNCT cana-2735 34 64 and and CCONJ cana-2735 34 65 defined define VERB cana-2735 34 66 by by ADP cana-2735 34 67 𝐾π‘₯ 𝐾π‘₯ PROPN cana-2735 34 68 2(πœ” 2(πœ” NUM cana-2735 34 69 ) ) PUNCT cana-2735 34 70 = = SYM cana-2735 34 71 1 1 NUM cana-2735 34 72 2 2 NUM cana-2735 34 73 πœ•2𝐾2 πœ•2𝐾2 ADJ cana-2735 34 74 πœ•πœ”π‘–πœ•πœ”π‘— πœ•πœ”π‘–πœ•πœ”π‘— NOUN cana-2735 34 75 ( ( PUNCT cana-2735 34 76 π‘₯ π‘₯ NOUN cana-2735 34 77 , , PUNCT cana-2735 34 78 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— NOUN cana-2735 34 79 = = SYM cana-2735 34 80 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ PROPN cana-2735 34 81 , , PUNCT cana-2735 34 82 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— PRON cana-2735 34 83 , , PUNCT cana-2735 34 84 where where SCONJ cana-2735 34 85 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ NOUN cana-2735 34 86 , , PUNCT cana-2735 34 87 πœ” πœ” ADJ cana-2735 34 88 ) ) PUNCT cana-2735 34 89 = = SYM cana-2735 34 90 1 1 NUM cana-2735 34 91 2 2 NUM cana-2735 34 92 πœ•2𝐾2 πœ•2𝐾2 ADJ cana-2735 34 93 πœ•πœ”π‘–πœ•πœ”π‘— πœ•πœ”π‘–πœ•πœ”π‘— NOUN cana-2735 34 94 ( ( PUNCT cana-2735 34 95 π‘₯ π‘₯ NOUN cana-2735 34 96 , , PUNCT cana-2735 34 97 πœ” πœ” VERB cana-2735 34 98 ) ) PUNCT cana-2735 34 99 is be AUX cana-2735 34 100 the the DET cana-2735 34 101 fundamental fundamental ADJ cana-2735 34 102 metric metric ADJ cana-2735 34 103 tensor tensor NOUN cana-2735 34 104 of of ADP cana-2735 34 105 the the DET cana-2735 34 106 finsler finsler NOUN cana-2735 34 107 metric metric PROPN cana-2735 34 108 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 34 109 , , PUNCT cana-2735 34 110 πœ” πœ” NOUN cana-2735 34 111 ) ) PUNCT cana-2735 34 112 of of ADP cana-2735 34 113 cotangent cotangent NOUN cana-2735 34 114 bundle bundle NOUN cana-2735 34 115 . . PUNCT cana-2735 35 1 moreover moreover ADV cana-2735 35 2 , , PUNCT cana-2735 35 3 when when SCONJ cana-2735 35 4 referring refer VERB cana-2735 35 5 to to ADP cana-2735 35 6 a a DET cana-2735 35 7 cartan cartan ADJ cana-2735 35 8 space space NOUN cana-2735 35 9 with with ADP cana-2735 35 10 an an DET cana-2735 35 11 ( ( PUNCT cana-2735 35 12 𝛼 𝛼 NOUN cana-2735 35 13 , , PUNCT cana-2735 35 14 𝛽)-metric 𝛽)-metric ADJ cana-2735 35 15 , , PUNCT cana-2735 35 16 this this PRON cana-2735 35 17 could could AUX cana-2735 35 18 mean mean VERB cana-2735 35 19 the the DET cana-2735 35 20 cartan cartan ADJ cana-2735 35 21 space space NOUN cana-2735 35 22 is be AUX cana-2735 35 23 endowed endow VERB cana-2735 35 24 with with ADP cana-2735 35 25 a a DET cana-2735 35 26 metric metric NOUN cana-2735 35 27 that that PRON cana-2735 35 28 is be AUX cana-2735 35 29 parametrized parametrize VERB cana-2735 35 30 by by ADP cana-2735 35 31 two two NUM cana-2735 35 32 parameters parameter NOUN cana-2735 35 33 , , PUNCT cana-2735 35 34 typically typically ADV cana-2735 35 35 denoted denote VERB cana-2735 35 36 as as ADP cana-2735 35 37 Ξ± Ξ± NOUN cana-2735 35 38 and and CCONJ cana-2735 35 39 𝛽. 𝛽. VERB cana-2735 35 40 this this DET cana-2735 35 41 type type NOUN cana-2735 35 42 of of ADP cana-2735 35 43 metric metric NOUN cana-2735 35 44 is be AUX cana-2735 35 45 used use VERB cana-2735 35 46 in in ADP cana-2735 35 47 various various ADJ cana-2735 35 48 contexts contexts NOUN cana-2735 35 49 , , PUNCT cana-2735 35 50 including include VERB cana-2735 35 51 in in ADP cana-2735 35 52 the the DET cana-2735 35 53 study study NOUN cana-2735 35 54 of of ADP cana-2735 35 55 spacetimes spacetime NOUN cana-2735 35 56 or or CCONJ cana-2735 35 57 in in ADP cana-2735 35 58 certain certain ADJ cana-2735 35 59 models model NOUN cana-2735 35 60 of of ADP cana-2735 35 61 differential differential ADJ cana-2735 35 62 geometry geometry NOUN cana-2735 35 63 and and CCONJ cana-2735 35 64 gravity gravity NOUN cana-2735 35 65 , , PUNCT cana-2735 35 66 such such ADJ cana-2735 35 67 as as ADP cana-2735 35 68 generalized generalized ADJ cana-2735 35 69 theories theory NOUN cana-2735 35 70 of of ADP cana-2735 35 71 relativity relativity NOUN cana-2735 35 72 . . PUNCT cana-2735 36 1 let let VERB cana-2735 36 2 us we PRON cana-2735 36 3 precisely precisely ADV cana-2735 36 4 define define VERB cana-2735 36 5 what what PRON cana-2735 36 6 is be AUX cana-2735 36 7 meant mean VERB cana-2735 36 8 by by ADP cana-2735 36 9 a a DET cana-2735 36 10 cartan cartan ADJ cana-2735 36 11 space space NOUN cana-2735 36 12 with with ADP cana-2735 36 13 ( ( PUNCT cana-2735 36 14 𝛼 𝛼 NOUN cana-2735 36 15 , , PUNCT cana-2735 36 16 𝛽)-metric 𝛽)-metric ADJ cana-2735 36 17 . . PUNCT cana-2735 37 1 communications communication NOUN cana-2735 37 2 on on ADP cana-2735 37 3 applied apply VERB cana-2735 37 4 nonlinear nonlinear ADJ cana-2735 37 5 analysis analysis NOUN cana-2735 37 6 issn issn NOUN cana-2735 37 7 : : PUNCT cana-2735 37 8 1074 1074 NUM cana-2735 37 9 - - PUNCT cana-2735 37 10 133x 133x NUM cana-2735 37 11 vol vol NOUN cana-2735 37 12 32 32 NUM cana-2735 37 13 no no NOUN cana-2735 37 14 . . PUNCT cana-2735 38 1 4s 4s NUM cana-2735 38 2 ( ( PUNCT cana-2735 38 3 2025 2025 NUM cana-2735 38 4 ) ) PUNCT cana-2735 38 5 3 3 NUM cana-2735 38 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 38 7 definition definition NOUN cana-2735 38 8 2.4 2.4 NUM cana-2735 38 9 if if SCONJ cana-2735 38 10 the the DET cana-2735 38 11 fundamental fundamental ADJ cana-2735 38 12 function function NOUN cana-2735 38 13 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 38 14 , , PUNCT cana-2735 38 15 πœ” πœ” NOUN cana-2735 38 16 ) ) PUNCT cana-2735 38 17 of of ADP cana-2735 38 18 a a DET cana-2735 38 19 cartan cartan ADJ cana-2735 38 20 space space NOUN cana-2735 38 21 𝐢 𝐢 PROPN cana-2735 38 22 = = SYM cana-2735 38 23 ( ( PUNCT cana-2735 38 24 𝑀 𝑀 PROPN cana-2735 38 25 , , PUNCT cana-2735 38 26 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 38 27 , , PUNCT cana-2735 38 28 πœ” πœ” NOUN cana-2735 38 29 ) ) PUNCT cana-2735 38 30 ) ) PUNCT cana-2735 38 31 is be AUX cana-2735 38 32 a a DET cana-2735 38 33 function function NOUN cana-2735 38 34 of of ADP cana-2735 38 35 variables variable NOUN cana-2735 38 36 𝛽(π‘₯ 𝛽(π‘₯ PROPN cana-2735 38 37 , , PUNCT cana-2735 38 38 πœ” πœ” NOUN cana-2735 38 39 ) ) PUNCT cana-2735 38 40 = = PUNCT cana-2735 38 41 πœ”π‘–π‘ πœ”π‘–π‘ ADJ cana-2735 38 42 𝑖(π‘₯ 𝑖(π‘₯ PROPN cana-2735 38 43 ) ) PUNCT cana-2735 38 44 , , PUNCT cana-2735 38 45 where where SCONJ cana-2735 38 46 π‘Žπ‘–π‘—(π‘₯ π‘Žπ‘–π‘—(π‘₯ PROPN cana-2735 38 47 ) ) PUNCT cana-2735 38 48 is be AUX cana-2735 38 49 a a DET cana-2735 38 50 riemannian riemannian ADJ cana-2735 38 51 metric metric NOUN cana-2735 38 52 and and CCONJ cana-2735 38 53 𝑏𝑖(π‘₯ 𝑏𝑖(π‘₯ NUM cana-2735 38 54 ) ) PUNCT cana-2735 38 55 is be AUX cana-2735 38 56 a a DET cana-2735 38 57 vector vector NOUN cana-2735 38 58 field field NOUN cana-2735 38 59 depending depend VERB cana-2735 38 60 only only ADV cana-2735 38 61 on on ADP cana-2735 38 62 π‘₯ π‘₯ PROPN cana-2735 38 63 , , PUNCT cana-2735 38 64 then then ADV cana-2735 38 65 𝐢 𝐢 PROPN cana-2735 38 66 is be AUX cana-2735 38 67 called call VERB cana-2735 38 68 cartan cartan ADJ cana-2735 38 69 space space NOUN cana-2735 38 70 with with ADP cana-2735 38 71 ( ( PUNCT cana-2735 38 72 𝛼 𝛼 NOUN cana-2735 38 73 , , PUNCT cana-2735 38 74 𝛽)-metric 𝛽)-metric ADJ cana-2735 38 75 . . PUNCT cana-2735 39 1 here here ADV cana-2735 39 2 it it PRON cana-2735 39 3 is be AUX cana-2735 39 4 to to PART cana-2735 39 5 be be AUX cana-2735 39 6 remarked remark VERB cana-2735 39 7 that that SCONJ cana-2735 39 8 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 39 9 , , PUNCT cana-2735 39 10 πœ” πœ” PRON cana-2735 39 11 ) ) PUNCT cana-2735 39 12 must must AUX cana-2735 39 13 satisfy satisfy VERB cana-2735 39 14 all all DET cana-2735 39 15 the the DET cana-2735 39 16 conditions condition NOUN cana-2735 39 17 imposed impose VERB cana-2735 39 18 on on ADP cana-2735 39 19 the the DET cana-2735 39 20 fundamental fundamental ADJ cana-2735 39 21 function function NOUN cana-2735 39 22 of of ADP cana-2735 39 23 a a DET cana-2735 39 24 cartan cartan ADJ cana-2735 39 25 space space NOUN cana-2735 39 26 . . PUNCT cana-2735 40 1 let let VERB cana-2735 40 2 us we PRON cana-2735 40 3 consider consider VERB cana-2735 40 4 a a DET cana-2735 40 5 cartan cartan ADJ cana-2735 40 6 space space NOUN cana-2735 40 7 𝐢 𝐢 PROPN cana-2735 40 8 = = SYM cana-2735 40 9 ( ( PUNCT cana-2735 40 10 𝑀 𝑀 PROPN cana-2735 40 11 , , PUNCT cana-2735 40 12 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 40 13 , , PUNCT cana-2735 40 14 πœ” πœ” NOUN cana-2735 40 15 ) ) PUNCT cana-2735 40 16 ) ) PUNCT cana-2735 40 17 with with ADP cana-2735 40 18 an an DET cana-2735 40 19 ( ( PUNCT cana-2735 40 20 𝛼 𝛼 PROPN cana-2735 40 21 , , PUNCT cana-2735 40 22 𝛽 𝛽 NOUN cana-2735 40 23 ) ) PUNCT cana-2735 40 24 -metric -metric NOUN cana-2735 40 25 , , PUNCT cana-2735 40 26 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 40 27 , , PUNCT cana-2735 40 28 πœ” πœ” ADJ cana-2735 40 29 ) ) PUNCT cana-2735 40 30 = = SYM cana-2735 41 1 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 41 2 , , PUNCT cana-2735 41 3 πœ” πœ” PRON cana-2735 41 4 ) ) PUNCT cana-2735 41 5 + + NUM cana-2735 41 6 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 41 7 , , PUNCT cana-2735 41 8 πœ” πœ” PRON cana-2735 41 9 ) ) PUNCT cana-2735 41 10 + + CCONJ cana-2735 41 11 2π‘˜ 2π‘˜ NUM cana-2735 41 12 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 41 13 ) ) PUNCT cana-2735 41 14 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 41 15 ) ) PUNCT cana-2735 41 16 βˆ’ βˆ’ PROPN cana-2735 42 1 π‘˜2 π‘˜2 PROPN cana-2735 42 2 3 3 NUM cana-2735 42 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 42 4 ) ) PUNCT cana-2735 42 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 42 6 ) ) PUNCT cana-2735 42 7 , , PUNCT cana-2735 42 8 where where SCONJ cana-2735 42 9 𝛼 𝛼 X cana-2735 42 10 = = SYM cana-2735 42 11 ( ( PUNCT cana-2735 42 12 π‘Žπ‘–π‘—(π‘₯ π‘Žπ‘–π‘—(π‘₯ PROPN cana-2735 42 13 , , PUNCT cana-2735 42 14 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— PRON cana-2735 42 15 ) ) PUNCT cana-2735 42 16 1 1 NUM cana-2735 42 17 2 2 NUM cana-2735 42 18 and and CCONJ cana-2735 42 19 𝛽 𝛽 NOUN cana-2735 42 20 = = PUNCT cana-2735 42 21 πœ”π‘–π‘ πœ”π‘–π‘ ADJ cana-2735 42 22 𝑖(π‘₯ 𝑖(π‘₯ PROPN cana-2735 42 23 ) ) PUNCT cana-2735 42 24 . . PUNCT cana-2735 43 1 the the DET cana-2735 43 2 fundamental fundamental ADJ cana-2735 43 3 tensor tensor NOUN cana-2735 43 4 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ NOUN cana-2735 43 5 , , PUNCT cana-2735 43 6 πœ” πœ” PRON cana-2735 43 7 ) ) PUNCT cana-2735 43 8 and and CCONJ cana-2735 43 9 its its PRON cana-2735 43 10 reciprocal reciprocal ADJ cana-2735 43 11 tensor tensor NOUN cana-2735 43 12 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ PROPN cana-2735 43 13 , , PUNCT cana-2735 43 14 πœ” πœ” NOUN cana-2735 43 15 ) ) PUNCT cana-2735 43 16 of of ADP cana-2735 43 17 the the DET cana-2735 43 18 cartan cartan PROPN cana-2735 43 19 space space NOUN cana-2735 43 20 𝐢 𝐢 PROPN cana-2735 43 21 = = SYM cana-2735 43 22 ( ( PUNCT cana-2735 43 23 𝑀 𝑀 PROPN cana-2735 43 24 , , PUNCT cana-2735 43 25 𝐾(𝛼 𝐾(𝛼 PROPN cana-2735 43 26 , , PUNCT cana-2735 43 27 𝛽 𝛽 NOUN cana-2735 43 28 ) ) PUNCT cana-2735 43 29 ) ) PUNCT cana-2735 43 30 are be AUX cana-2735 43 31 given give VERB cana-2735 43 32 by by ADP cana-2735 43 33 [ [ X cana-2735 43 34 3 3 NUM cana-2735 43 35 ] ] SYM cana-2735 43 36 𝑔𝑖𝑗 𝑔𝑖𝑗 NOUN cana-2735 43 37 = = NOUN cana-2735 43 38 πœŒπ‘Žπ‘–π‘— πœŒπ‘Žπ‘–π‘— NOUN cana-2735 43 39 + + CCONJ cana-2735 43 40 𝜌0𝑏𝑖𝑏𝑗 𝜌0𝑏𝑖𝑏𝑗 ADP cana-2735 43 41 + + CCONJ cana-2735 43 42 πœŒβˆ’1(π‘π‘–πœ”π‘— πœŒβˆ’1(π‘π‘–πœ”π‘— NUM cana-2735 43 43 + + CCONJ cana-2735 43 44 π‘π‘—πœ”π‘– π‘π‘—πœ”π‘– NOUN cana-2735 43 45 ) ) PUNCT cana-2735 43 46 + + X cana-2735 44 1 πœŒβˆ’2πœ”π‘–πœ”π‘— πœŒβˆ’2πœ”π‘–πœ”π‘— X cana-2735 44 2 , , PUNCT cana-2735 44 3 ( ( PUNCT cana-2735 44 4 1 1 X cana-2735 44 5 ) ) PUNCT cana-2735 44 6 where where SCONJ cana-2735 44 7 𝜌 𝜌 X cana-2735 44 8 , , PUNCT cana-2735 44 9 𝜌0 𝜌0 PROPN cana-2735 44 10 , , PUNCT cana-2735 44 11 πœŒβˆ’1 πœŒβˆ’1 PROPN cana-2735 44 12 and and CCONJ cana-2735 44 13 πœŒβˆ’2 πœŒβˆ’2 PROPN cana-2735 44 14 are be AUX cana-2735 44 15 invariants invariant NOUN cana-2735 44 16 which which PRON cana-2735 44 17 are be AUX cana-2735 44 18 defined define VERB cana-2735 44 19 and and CCONJ cana-2735 44 20 calculated calculate VERB cana-2735 44 21 as as SCONJ cana-2735 44 22 follows follow VERB cana-2735 44 23 : : PUNCT cana-2735 44 24 𝜌 𝜌 X cana-2735 44 25 = = SYM cana-2735 44 26 1 1 NUM cana-2735 44 27 2𝛼 2𝛼 NOUN cana-2735 44 28 𝐾𝛼 𝐾𝛼 PROPN cana-2735 44 29 = = PUNCT cana-2735 44 30 𝛼5βˆ’2π‘˜π›Ό2𝛽2+π‘˜2𝛽4 𝛼5βˆ’2π‘˜π›Ό2𝛽2+π‘˜2𝛽4 NOUN cana-2735 44 31 2𝛼5 2𝛼5 NUM cana-2735 45 1 𝜌0 𝜌0 NOUN cana-2735 45 2 = = SYM cana-2735 45 3 1 1 NUM cana-2735 45 4 2 2 NUM cana-2735 45 5 𝐾𝛽𝛽 𝐾𝛽𝛽 PROPN cana-2735 45 6 = = SYM cana-2735 45 7 2π‘˜π›Ό2βˆ’2π‘˜2𝛽2 2π‘˜π›Ό2βˆ’2π‘˜2𝛽2 NUM cana-2735 45 8 𝛼3 𝛼3 NOUN cana-2735 45 9 πœŒβˆ’1 πœŒβˆ’1 PROPN cana-2735 45 10 = = SYM cana-2735 45 11 1 1 NUM cana-2735 45 12 2𝛼 2𝛼 PROPN cana-2735 45 13 𝐾𝛼𝛽 𝐾𝛼𝛽 PROPN cana-2735 45 14 = = PUNCT cana-2735 46 1 βˆ’ βˆ’ PROPN cana-2735 46 2 2π‘˜2𝛽3βˆ’2π‘˜π›Ό2𝛽 2π‘˜2𝛽3βˆ’2π‘˜π›Ό2𝛽 NUM cana-2735 46 3 𝛼5 𝛼5 NOUN cana-2735 46 4 πœŒβˆ’2 πœŒβˆ’2 PROPN cana-2735 46 5 = = SYM cana-2735 46 6 1 1 NUM cana-2735 46 7 2𝛼2 2𝛼2 NUM cana-2735 46 8 ( ( PUNCT cana-2735 46 9 𝐾𝛼𝛼 𝐾𝛼𝛼 NOUN cana-2735 46 10 βˆ’ βˆ’ PROPN cana-2735 46 11 1 1 NUM cana-2735 46 12 𝛼 𝛼 X cana-2735 46 13 𝐾𝛼 𝐾𝛼 PROPN cana-2735 46 14 ) ) PUNCT cana-2735 46 15 = = PUNCT cana-2735 47 1 6π‘˜π›Ό2𝛽2βˆ’5π‘˜2𝛽4βˆ’π›Ό4 6π‘˜π›Ό2𝛽2βˆ’5π‘˜2𝛽4βˆ’π›Ό4 NUM cana-2735 47 2 2𝛼7 2𝛼7 NUM cana-2735 47 3 and and CCONJ cana-2735 47 4 𝑔𝑖𝑗 𝑔𝑖𝑗 VERB cana-2735 47 5 = = NOUN cana-2735 47 6 πœŽπ‘Žπ‘–π‘— πœŽπ‘Žπ‘–π‘— NOUN cana-2735 47 7 βˆ’ βˆ’ NOUN cana-2735 48 1 𝜎0𝑏𝑖𝑏𝑗 𝜎0𝑏𝑖𝑏𝑗 NOUN cana-2735 48 2 + + CCONJ cana-2735 48 3 πœŽβˆ’1(π‘π‘–πœ”π‘— πœŽβˆ’1(π‘π‘–πœ”π‘— NUM cana-2735 48 4 + + CCONJ cana-2735 48 5 π‘π‘—πœ”π‘– π‘π‘—πœ”π‘– NOUN cana-2735 48 6 ) ) PUNCT cana-2735 49 1 + + NUM cana-2735 50 1 πœŽβˆ’2πœ”π‘–πœ”π‘— πœŽβˆ’2πœ”π‘–πœ”π‘— NOUN cana-2735 50 2 , , PUNCT cana-2735 50 3 ( ( PUNCT cana-2735 50 4 2 2 X cana-2735 50 5 ) ) PUNCT cana-2735 51 1 where where SCONJ cana-2735 51 2 𝜎 𝜎 NOUN cana-2735 51 3 = = NOUN cana-2735 51 4 1 1 NUM cana-2735 51 5 𝜌 𝜌 ADP cana-2735 51 6 = = SYM cana-2735 51 7 2𝛼5 2𝛼5 NUM cana-2735 51 8 𝛼5βˆ’2π‘˜π›Ό2𝛽2+π‘˜2𝛽4 𝛼5βˆ’2π‘˜π›Ό2𝛽2+π‘˜2𝛽4 NOUN cana-2735 51 9 𝜎0 𝜎0 NOUN cana-2735 51 10 = = SYM cana-2735 51 11 𝜌0 𝜌0 ADJ cana-2735 51 12 𝜌𝜏 𝜌𝜏 X cana-2735 51 13 𝜏 𝜏 X cana-2735 51 14 = = SYM cana-2735 51 15 𝜎 𝜎 PROPN cana-2735 51 16 + + CCONJ cana-2735 51 17 𝜎0𝐡2 𝜎0𝐡2 PROPN cana-2735 51 18 + + CCONJ cana-2735 51 19 πœŒβˆ’1𝛽 πœŒβˆ’1𝛽 NOUN cana-2735 51 20 πœŽβˆ’1 πœŽβˆ’1 PROPN cana-2735 51 21 = = SYM cana-2735 51 22 πœŒβˆ’1 πœŒβˆ’1 PROPN cana-2735 51 23 𝜌𝜏 𝜌𝜏 X cana-2735 51 24 communications communication NOUN cana-2735 51 25 on on ADP cana-2735 51 26 applied apply VERB cana-2735 51 27 nonlinear nonlinear ADJ cana-2735 51 28 analysis analysis NOUN cana-2735 51 29 issn issn NOUN cana-2735 51 30 : : PUNCT cana-2735 51 31 1074 1074 NUM cana-2735 51 32 - - PUNCT cana-2735 51 33 133x 133x NUM cana-2735 51 34 vol vol NOUN cana-2735 51 35 32 32 NUM cana-2735 51 36 no no NOUN cana-2735 51 37 . . PUNCT cana-2735 52 1 4s 4s NUM cana-2735 52 2 ( ( PUNCT cana-2735 52 3 2025 2025 NUM cana-2735 52 4 ) ) PUNCT cana-2735 52 5 4 4 NUM cana-2735 52 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 52 7 πœŽβˆ’2 πœŽβˆ’2 NUM cana-2735 52 8 = = SYM cana-2735 52 9 πœŒβˆ’2 πœŒβˆ’2 PROPN cana-2735 52 10 𝜌𝜏 𝜌𝜏 X cana-2735 52 11 , , PUNCT cana-2735 52 12 where where SCONJ cana-2735 52 13 𝐡2 𝐡2 NOUN cana-2735 52 14 = = SYM cana-2735 52 15 𝑏𝑖𝑏𝑗 𝑏𝑖𝑏𝑗 PROPN cana-2735 52 16 and and CCONJ cana-2735 52 17 𝐡 𝐡 NOUN cana-2735 52 18 represents represent VERB cana-2735 52 19 the the DET cana-2735 52 20 norm norm NOUN cana-2735 52 21 of of ADP cana-2735 52 22 the the DET cana-2735 52 23 differential differential ADJ cana-2735 52 24 form form NOUN cana-2735 52 25 𝛽(π‘₯ 𝛽(π‘₯ PROPN cana-2735 52 26 , , PUNCT cana-2735 52 27 πœ” πœ” NOUN cana-2735 52 28 ) ) PUNCT cana-2735 52 29 = = SYM cana-2735 52 30 πœ”π‘–π‘ πœ”π‘–π‘ ADJ cana-2735 52 31 𝑖(π‘₯ 𝑖(π‘₯ PROPN cana-2735 52 32 ) ) PUNCT cana-2735 52 33 ∈ ∈ PROPN cana-2735 53 1 𝑇𝑝 𝑇𝑝 PROPN cana-2735 53 2 βˆ—π‘€. βˆ—π‘€. PROPN cana-2735 53 3 morover morover PROPN cana-2735 53 4 , , PUNCT cana-2735 53 5 the the DET cana-2735 53 6 cartan cartan ADJ cana-2735 53 7 connection connection NOUN cana-2735 53 8 , , PUNCT cana-2735 53 9 which which PRON cana-2735 53 10 is be AUX cana-2735 53 11 a a DET cana-2735 53 12 generalized generalized ADJ cana-2735 53 13 connection connection NOUN cana-2735 53 14 often often ADV cana-2735 53 15 used use VERB cana-2735 53 16 to to PART cana-2735 53 17 describe describe VERB cana-2735 53 18 homogeneous homogeneous ADJ cana-2735 53 19 spaces space NOUN cana-2735 53 20 or or CCONJ cana-2735 53 21 spaces space NOUN cana-2735 53 22 with with ADP cana-2735 53 23 symmetries symmetry NOUN cana-2735 53 24 , , PUNCT cana-2735 53 25 the the DET cana-2735 53 26 torsion torsion NOUN cana-2735 53 27 tensor tensor NOUN cana-2735 53 28 is be AUX cana-2735 53 29 defined define VERB cana-2735 53 30 in in ADP cana-2735 53 31 a a DET cana-2735 53 32 similar similar ADJ cana-2735 53 33 way way NOUN cana-2735 53 34 , , PUNCT cana-2735 53 35 but but CCONJ cana-2735 53 36 it it PRON cana-2735 53 37 incorporates incorporate VERB cana-2735 53 38 the the DET cana-2735 53 39 structure structure NOUN cana-2735 53 40 of of ADP cana-2735 53 41 the the DET cana-2735 53 42 cartan cartan ADJ cana-2735 53 43 connection connection NOUN cana-2735 53 44 itself itself PRON cana-2735 53 45 . . PUNCT cana-2735 54 1 a a DET cana-2735 54 2 cartan cartan ADJ cana-2735 54 3 connection connection NOUN cana-2735 54 4 provides provide VERB cana-2735 54 5 a a DET cana-2735 54 6 framework framework NOUN cana-2735 54 7 to to PART cana-2735 54 8 study study VERB cana-2735 54 9 affine affine NOUN cana-2735 54 10 connections connection NOUN cana-2735 54 11 on on ADP cana-2735 54 12 a a DET cana-2735 54 13 space space NOUN cana-2735 54 14 that that PRON cana-2735 54 15 is be AUX cana-2735 54 16 often often ADV cana-2735 54 17 non non ADJ cana-2735 54 18 - - ADJ cana-2735 54 19 riemannian riemannian ADJ cana-2735 54 20 . . PUNCT cana-2735 55 1 the the DET cana-2735 55 2 cartan cartan ADJ cana-2735 55 3 connection connection NOUN cana-2735 55 4 is be AUX cana-2735 55 5 particularly particularly ADV cana-2735 55 6 useful useful ADJ cana-2735 55 7 for for ADP cana-2735 55 8 describing describe VERB cana-2735 55 9 curved curved ADJ cana-2735 55 10 spaces space NOUN cana-2735 55 11 with with ADP cana-2735 55 12 torsion torsion NOUN cana-2735 55 13 and and CCONJ cana-2735 55 14 other other ADJ cana-2735 55 15 structural structural ADJ cana-2735 55 16 properties property NOUN cana-2735 55 17 , , PUNCT cana-2735 55 18 especially especially ADV cana-2735 55 19 in in ADP cana-2735 55 20 the the DET cana-2735 55 21 study study NOUN cana-2735 55 22 of of ADP cana-2735 55 23 lie lie NOUN cana-2735 55 24 groups group NOUN cana-2735 55 25 , , PUNCT cana-2735 55 26 affine affine NOUN cana-2735 55 27 manifolds manifold NOUN cana-2735 55 28 , , PUNCT cana-2735 55 29 and and CCONJ cana-2735 55 30 non non ADJ cana-2735 55 31 - - ADJ cana-2735 55 32 metric metric ADJ cana-2735 55 33 connections connection NOUN cana-2735 55 34 . . PUNCT cana-2735 56 1 the the DET cana-2735 56 2 cartan cartan PROPN cana-2735 56 3 torsion torsion NOUN cana-2735 56 4 tensor tensor NOUN cana-2735 56 5 can can AUX cana-2735 56 6 be be AUX cana-2735 56 7 thought think VERB cana-2735 56 8 of of ADP cana-2735 56 9 as as ADP cana-2735 56 10 a a DET cana-2735 56 11 measure measure NOUN cana-2735 56 12 of of ADP cana-2735 56 13 the the DET cana-2735 56 14 failure failure NOUN cana-2735 56 15 of of ADP cana-2735 56 16 the the DET cana-2735 56 17 cartan cartan ADJ cana-2735 56 18 connection connection NOUN cana-2735 56 19 to to PART cana-2735 56 20 satisfy satisfy VERB cana-2735 56 21 the the DET cana-2735 56 22 torsion torsion NOUN cana-2735 56 23 - - PUNCT cana-2735 56 24 free free ADJ cana-2735 56 25 condition condition NOUN cana-2735 56 26 in in ADP cana-2735 56 27 a a DET cana-2735 56 28 way way NOUN cana-2735 56 29 similar similar ADJ cana-2735 56 30 to to ADP cana-2735 56 31 the the DET cana-2735 56 32 standard standard ADJ cana-2735 56 33 torsion torsion NOUN cana-2735 56 34 tensor tensor NOUN cana-2735 56 35 in in ADP cana-2735 56 36 riemannian riemannian ADJ cana-2735 56 37 geometry geometry NOUN cana-2735 56 38 . . PUNCT cana-2735 57 1 the the DET cana-2735 57 2 cartan cartan PROPN cana-2735 57 3 torsion torsion NOUN cana-2735 57 4 tensor tensor NOUN cana-2735 57 5 πΆπ‘–π‘—π‘˜ πΆπ‘–π‘—π‘˜ PROPN cana-2735 57 6 [ [ NOUN cana-2735 57 7 5 5 NUM cana-2735 57 8 ] ] PUNCT cana-2735 57 9 in in ADP cana-2735 57 10 the the DET cana-2735 57 11 cartan cartan ADJ cana-2735 57 12 space space NOUN cana-2735 57 13 with with ADP cana-2735 57 14 an an DET cana-2735 57 15 ( ( PUNCT cana-2735 57 16 𝛼 𝛼 PROPN cana-2735 57 17 , , PUNCT cana-2735 57 18 𝛽)-metric 𝛽)-metric ADJ cana-2735 57 19 is be AUX cana-2735 57 20 given give VERB cana-2735 57 21 by by ADP cana-2735 57 22 πΆπ‘–π‘—π‘˜ πΆπ‘–π‘—π‘˜ PROPN cana-2735 57 23 = = SYM cana-2735 57 24 βˆ’ βˆ’ PROPN cana-2735 57 25 1 1 NUM cana-2735 57 26 2 2 NUM cana-2735 57 27 [ [ X cana-2735 57 28 π‘Ÿβˆ’1π‘π‘–π‘π‘—π‘π‘˜ π‘Ÿβˆ’1π‘π‘–π‘π‘—π‘π‘˜ NOUN cana-2735 57 29 + + CCONJ cana-2735 57 30 { { PUNCT cana-2735 57 31 πœŒβˆ’1π‘Žπ‘–π‘—π‘π‘˜ πœŒβˆ’1π‘Žπ‘–π‘—π‘π‘˜ NOUN cana-2735 57 32 + + NUM cana-2735 57 33 πœŒβˆ’2π‘Žπ‘–π‘—πœ”π‘˜ πœŒβˆ’2π‘Žπ‘–π‘—πœ”π‘˜ NOUN cana-2735 57 34 + + NUM cana-2735 57 35 π‘Ÿβˆ’2π‘π‘–π‘π‘—πœ”π‘˜ π‘Ÿβˆ’2π‘π‘–π‘π‘—πœ”π‘˜ NOUN cana-2735 57 36 + + CCONJ cana-2735 57 37 π‘Ÿβˆ’3π‘π‘–πœ”π‘—πœ”π‘˜ π‘Ÿβˆ’3π‘π‘–πœ”π‘—πœ”π‘˜ NOUN cana-2735 57 38 + + X cana-2735 57 39 𝑖|𝑗|π‘˜ 𝑖|𝑗|π‘˜ NOUN cana-2735 57 40 } } PUNCT cana-2735 57 41 + + CCONJ cana-2735 57 42 π‘Ÿβˆ’4πœ”π‘–πœ”π‘—πœ”π‘˜ π‘Ÿβˆ’4πœ”π‘–πœ”π‘—πœ”π‘˜ NOUN cana-2735 57 43 ] ] X cana-2735 57 44 , , PUNCT cana-2735 57 45 ( ( PUNCT cana-2735 57 46 3 3 X cana-2735 57 47 ) ) PUNCT cana-2735 57 48 where where SCONJ cana-2735 57 49 its its PRON cana-2735 57 50 coefficients coefficient NOUN cana-2735 57 51 π‘Ÿβˆ’1 π‘Ÿβˆ’1 PROPN cana-2735 57 52 , , PUNCT cana-2735 57 53 π‘Ÿβˆ’2 π‘Ÿβˆ’2 PROPN cana-2735 57 54 , , PUNCT cana-2735 57 55 π‘Ÿβˆ’3 π‘Ÿβˆ’3 PROPN cana-2735 57 56 and and CCONJ cana-2735 57 57 π‘Ÿβˆ’4 π‘Ÿβˆ’4 PROPN cana-2735 57 58 are be AUX cana-2735 57 59 defined define VERB cana-2735 57 60 and and CCONJ cana-2735 57 61 calculated calculate VERB cana-2735 57 62 as as SCONJ cana-2735 57 63 follows follow VERB cana-2735 57 64 : : PUNCT cana-2735 57 65 π‘Ÿβˆ’1 π‘Ÿβˆ’1 PROPN cana-2735 57 66 = = SYM cana-2735 57 67 1 1 NUM cana-2735 57 68 2 2 NUM cana-2735 57 69 𝐾𝛽𝛽𝛽 𝐾𝛽𝛽𝛽 PROPN cana-2735 57 70 = = PUNCT cana-2735 57 71 βˆ’4π‘˜2𝛽2 βˆ’4π‘˜2𝛽2 NOUN cana-2735 57 72 𝛼3 𝛼3 VERB cana-2735 57 73 π‘Ÿβˆ’2 π‘Ÿβˆ’2 PROPN cana-2735 57 74 = = SYM cana-2735 57 75 1 1 NUM cana-2735 57 76 2𝛼 2𝛼 PROPN cana-2735 57 77 𝐾𝛼𝛽𝛽 𝐾𝛼𝛽𝛽 PROPN cana-2735 57 78 = = PUNCT cana-2735 58 1 6π‘˜2𝛽2βˆ’2π‘˜π›Ό2 6π‘˜2𝛽2βˆ’2π‘˜π›Ό2 NOUN cana-2735 58 2 𝛼5 𝛼5 ADP cana-2735 58 3 π‘Ÿβˆ’3 π‘Ÿβˆ’3 NOUN cana-2735 58 4 = = SYM cana-2735 58 5 1 1 NUM cana-2735 58 6 2𝛼2 2𝛼2 NUM cana-2735 58 7 ( ( PUNCT cana-2735 58 8 𝐾𝛼𝛼𝛽 𝐾𝛼𝛼𝛽 PROPN cana-2735 58 9 βˆ’ βˆ’ PROPN cana-2735 58 10 1 1 NUM cana-2735 58 11 𝛼 𝛼 PROPN cana-2735 58 12 𝐾𝛼𝛽 𝐾𝛼𝛽 PROPN cana-2735 58 13 ) ) PUNCT cana-2735 58 14 = = SYM cana-2735 59 1 6π‘˜π›Ό2π›½βˆ’10π‘˜2𝛽3 6π‘˜π›Ό2π›½βˆ’10π‘˜2𝛽3 NUM cana-2735 60 1 𝛼7 𝛼7 VERB cana-2735 60 2 π‘Ÿβˆ’4 π‘Ÿβˆ’4 PROPN cana-2735 60 3 = = SYM cana-2735 60 4 1 1 NUM cana-2735 60 5 2𝛼3 2𝛼3 NUM cana-2735 60 6 ( ( PUNCT cana-2735 60 7 𝐾𝛼𝛼𝛼 𝐾𝛼𝛼𝛼 PROPN cana-2735 60 8 βˆ’ βˆ’ PROPN cana-2735 60 9 3 3 NUM cana-2735 60 10 𝛼 𝛼 NOUN cana-2735 60 11 𝐾𝛼𝛼 𝐾𝛼𝛼 PROPN cana-2735 60 12 + + NOUN cana-2735 60 13 3 3 NUM cana-2735 60 14 𝛼2 𝛼2 PROPN cana-2735 60 15 𝐾𝛼 𝐾𝛼 PROPN cana-2735 60 16 ) ) PUNCT cana-2735 60 17 = = PUNCT cana-2735 61 1 35π‘˜2𝛽4βˆ’30π‘˜π›Ό2𝛽2 35π‘˜2𝛽4βˆ’30π‘˜π›Ό2𝛽2 NUM cana-2735 61 2 + + NOUN cana-2735 61 3 3𝛼4 3𝛼4 NUM cana-2735 61 4 2𝛼9 2𝛼9 NUM cana-2735 61 5 . . PUNCT cana-2735 62 1 we we PRON cana-2735 62 2 use use VERB cana-2735 62 3 the the DET cana-2735 62 4 symole symole NOUN cana-2735 62 5 ’ ' PUNCT cana-2735 62 6 : : PUNCT cana-2735 62 7 ’ ' PUNCT cana-2735 62 8 to to PART cana-2735 62 9 denote denote VERB cana-2735 62 10 the the DET cana-2735 62 11 covariant covariant ADJ cana-2735 62 12 differentiation differentiation NOUN cana-2735 62 13 with with ADP cana-2735 62 14 respect respect NOUN cana-2735 62 15 to to ADP cana-2735 62 16 christoffel christoffel ADJ cana-2735 62 17 symbols symbol NOUN cana-2735 62 18 π›Ύπ‘—π‘˜ π›Ύπ‘—π‘˜ X cana-2735 62 19 𝑖 𝑖 PUNCT cana-2735 62 20 constructed construct VERB cana-2735 62 21 from from ADP cana-2735 62 22 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-2735 62 23 . . PUNCT cana-2735 63 1 whenever whenever SCONJ cana-2735 63 2 we we PRON cana-2735 63 3 talk talk VERB cana-2735 63 4 about about ADP cana-2735 63 5 christoffel christoffel NOUN cana-2735 63 6 symbols symbol NOUN cana-2735 63 7 π›Ύπ‘—π‘˜ π›Ύπ‘—π‘˜ X cana-2735 63 8 𝑖 𝑖 PUNCT cana-2735 63 9 constructed construct VERB cana-2735 63 10 from from ADP cana-2735 63 11 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN cana-2735 63 12 , , PUNCT cana-2735 63 13 we we PRON cana-2735 63 14 mean mean VERB cana-2735 63 15 π›Ύπ‘—π‘˜ π›Ύπ‘—π‘˜ NOUN cana-2735 63 16 𝑖 𝑖 SYM cana-2735 63 17 = = SYM cana-2735 63 18 1 1 NUM cana-2735 63 19 2 2 NUM cana-2735 63 20 π‘Žπ‘™π‘– π‘Žπ‘™π‘– NOUN cana-2735 63 21 ( ( PUNCT cana-2735 63 22 πœ•π‘Žπ‘˜π‘™ πœ•π‘Žπ‘˜π‘™ NOUN cana-2735 63 23 πœ•π‘₯𝑗 πœ•π‘₯𝑗 PUNCT cana-2735 64 1 + + CCONJ cana-2735 64 2 πœ•π‘Žπ‘™π‘— πœ•π‘Žπ‘™π‘— PROPN cana-2735 64 3 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 64 4 βˆ’ βˆ’ PROPN cana-2735 64 5 πœ•π‘Žπ‘—π‘˜ πœ•π‘Žπ‘—π‘˜ PROPN cana-2735 64 6 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 64 7 ) ) PUNCT cana-2735 64 8 . . PUNCT cana-2735 65 1 since since SCONJ cana-2735 65 2 πœ”π‘–:π‘˜ πœ”π‘–:π‘˜ PROPN cana-2735 65 3 = = SYM cana-2735 65 4 0 0 NUM cana-2735 65 5 and and CCONJ cana-2735 65 6 from from ADP cana-2735 65 7 ricci ricci PROPN cana-2735 65 8 ’s ’s PART cana-2735 65 9 theorem theorem NOUN cana-2735 65 10 of of ADP cana-2735 65 11 tensor tensor NOUN cana-2735 65 12 calculus calculus NOUN cana-2735 65 13 [ [ X cana-2735 65 14 14 14 NUM cana-2735 65 15 ] ] X cana-2735 65 16 we we PRON cana-2735 65 17 have have VERB cana-2735 65 18 π‘Ž:π‘˜ π‘Ž:π‘˜ NOUN cana-2735 65 19 𝑖𝑗 𝑖𝑗 ADP cana-2735 66 1 = = NOUN cana-2735 66 2 0 0 PUNCT cana-2735 66 3 , , PUNCT cana-2735 66 4 if if SCONJ cana-2735 66 5 𝑏:π‘˜ 𝑏:π‘˜ NOUN cana-2735 66 6 𝑖 𝑖 X cana-2735 66 7 = = NOUN cana-2735 66 8 0 0 NUM cana-2735 66 9 , , PUNCT cana-2735 66 10 then then ADV cana-2735 66 11 𝑔:π‘˜ 𝑔:π‘˜ ADP cana-2735 66 12 𝑖𝑗 𝑖𝑗 X cana-2735 67 1 = = PUNCT cana-2735 67 2 0 0 X cana-2735 67 3 . . PUNCT cana-2735 68 1 also also ADV cana-2735 68 2 , , PUNCT cana-2735 68 3 let let VERB cana-2735 68 4 Ξ³π‘—π‘˜ Ξ³π‘—π‘˜ NOUN cana-2735 68 5 𝑖 𝑖 NOUN cana-2735 68 6 ( ( PUNCT cana-2735 68 7 𝑝 𝑝 NOUN cana-2735 68 8 ) ) PUNCT cana-2735 68 9 = = SYM cana-2735 68 10 1 1 NUM cana-2735 68 11 2 2 NUM cana-2735 68 12 π‘”π‘–π‘Ÿ(πœ•π‘—π‘”π‘Ÿπ‘˜ π‘”π‘–π‘Ÿ(πœ•π‘—π‘”π‘Ÿπ‘˜ NOUN cana-2735 68 13 + + CCONJ cana-2735 68 14 πœ•π‘˜π‘”π‘—π‘Ÿ πœ•π‘˜π‘”π‘—π‘Ÿ PROPN cana-2735 68 15 βˆ’ βˆ’ ADP cana-2735 68 16 πœ•π‘Ÿπ‘”π‘—π‘˜ πœ•π‘Ÿπ‘”π‘—π‘˜ NOUN cana-2735 68 17 ) ) PUNCT cana-2735 68 18 be be VERB cana-2735 68 19 the the DET cana-2735 68 20 christoffel christoffel ADJ cana-2735 68 21 symbols symbol NOUN cana-2735 68 22 constructed construct VERB cana-2735 68 23 from from ADP cana-2735 68 24 fundamental fundamental ADJ cana-2735 68 25 metric metric ADJ cana-2735 68 26 tensor tensor NOUN cana-2735 68 27 𝑔𝑖𝑗(π‘₯ 𝑔𝑖𝑗(π‘₯ NOUN cana-2735 68 28 , , PUNCT cana-2735 68 29 πœ” πœ” NOUN cana-2735 68 30 ) ) PUNCT cana-2735 68 31 of of ADP cana-2735 68 32 the the DET cana-2735 68 33 cartan cartan ADJ cana-2735 68 34 space space NOUN cana-2735 68 35 ( ( PUNCT cana-2735 68 36 𝑀 𝑀 PROPN cana-2735 68 37 , , PUNCT cana-2735 68 38 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 68 39 , , PUNCT cana-2735 68 40 πœ” πœ” NOUN cana-2735 68 41 ) ) PUNCT cana-2735 68 42 ) ) PUNCT cana-2735 68 43 . . PUNCT cana-2735 69 1 now now ADV cana-2735 69 2 , , PUNCT cana-2735 69 3 for for ADP cana-2735 69 4 the the DET cana-2735 69 5 cartan cartan ADJ cana-2735 69 6 space space NOUN cana-2735 69 7 ( ( PUNCT cana-2735 69 8 𝑀 𝑀 PROPN cana-2735 69 9 , , PUNCT cana-2735 69 10 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 69 11 , , PUNCT cana-2735 69 12 πœ” πœ” NOUN cana-2735 69 13 ) ) PUNCT cana-2735 69 14 ) ) PUNCT cana-2735 69 15 , , PUNCT cana-2735 69 16 we we PRON cana-2735 69 17 state state VERB cana-2735 69 18 canonical canonical ADJ cana-2735 69 19 𝑑-connection 𝑑-connection NOUN cana-2735 69 20 is be AUX cana-2735 69 21 a a DET cana-2735 69 22 triplet triplet NOUN cana-2735 69 23 communications communication NOUN cana-2735 69 24 on on ADP cana-2735 69 25 applied apply VERB cana-2735 69 26 nonlinear nonlinear ADJ cana-2735 69 27 analysis analysis NOUN cana-2735 69 28 issn issn NOUN cana-2735 69 29 : : PUNCT cana-2735 69 30 1074 1074 NUM cana-2735 69 31 - - PUNCT cana-2735 69 32 133x 133x NUM cana-2735 69 33 vol vol NOUN cana-2735 69 34 32 32 NUM cana-2735 69 35 no no NOUN cana-2735 69 36 . . PUNCT cana-2735 70 1 4s 4s NUM cana-2735 70 2 ( ( PUNCT cana-2735 70 3 2025 2025 NUM cana-2735 70 4 ) ) PUNCT cana-2735 70 5 5 5 NUM cana-2735 70 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 70 7 given give VERB cana-2735 70 8 by by ADP cana-2735 70 9 𝐷γ 𝐷γ PROPN cana-2735 70 10 = = SYM cana-2735 70 11 ( ( PUNCT cana-2735 70 12 π‘π‘—π‘˜ π‘π‘—π‘˜ PROPN cana-2735 70 13 , , PUNCT cana-2735 70 14 π»π‘—π‘˜ π»π‘—π‘˜ PROPN cana-2735 70 15 𝑖 𝑖 SYM cana-2735 70 16 , , PUNCT cana-2735 70 17 𝐢𝑖 𝐢𝑖 PROPN cana-2735 70 18 π‘—π‘˜ π‘—π‘˜ PROPN cana-2735 70 19 ) ) PUNCT cana-2735 70 20 , , PUNCT cana-2735 70 21 where where SCONJ cana-2735 70 22 𝑁𝑖𝑗 𝑁𝑖𝑗 NOUN cana-2735 70 23 = = NOUN cana-2735 70 24 γ𝑖𝑗 γ𝑖𝑗 VERB cana-2735 70 25 π‘˜ π‘˜ PRON cana-2735 70 26 πœ”π‘˜ πœ”π‘˜ ADP cana-2735 70 27 βˆ’ βˆ’ NUM cana-2735 70 28 1 1 NUM cana-2735 70 29 2 2 NUM cana-2735 70 30 Ξ³β„Žπ‘Ÿ Ξ³β„Žπ‘Ÿ PROPN cana-2735 70 31 π‘˜ π‘˜ DET cana-2735 70 32 πœ”π‘˜πœ”π‘Ÿ πœ”π‘˜πœ”π‘Ÿ PROPN cana-2735 70 33 οΏ½ οΏ½ PROPN cana-2735 70 34 Μ‡ Μ‡ NOUN cana-2735 70 35 οΏ½ οΏ½ PROPN cana-2735 70 36 β„Žπ‘”π‘–π‘— β„Žπ‘”π‘–π‘— NOUN cana-2735 70 37 ( ( PUNCT cana-2735 70 38 4 4 NUM cana-2735 70 39 ) ) PUNCT cana-2735 71 1 π»π‘—π‘˜ π»π‘—π‘˜ PROPN cana-2735 71 2 𝑖 𝑖 SYM cana-2735 71 3 = = NOUN cana-2735 71 4 1 1 NUM cana-2735 71 5 2 2 NUM cana-2735 71 6 π‘”π‘–π‘Ÿ(πœ•π‘—π‘”π‘Ÿπ‘˜ π‘”π‘–π‘Ÿ(πœ•π‘—π‘”π‘Ÿπ‘˜ NOUN cana-2735 71 7 + + CCONJ cana-2735 71 8 πœ•π‘˜π‘”π‘—π‘Ÿ πœ•π‘˜π‘”π‘—π‘Ÿ PROPN cana-2735 71 9 βˆ’ βˆ’ ADP cana-2735 71 10 πœ•π‘Ÿπ‘”π‘—π‘˜ πœ•π‘Ÿπ‘”π‘—π‘˜ NOUN cana-2735 71 11 ) ) PUNCT cana-2735 71 12 ( ( PUNCT cana-2735 71 13 5 5 X cana-2735 71 14 ) ) PUNCT cana-2735 71 15 𝐢𝑖 𝐢𝑖 NOUN cana-2735 71 16 π‘—π‘˜ π‘—π‘˜ PROPN cana-2735 71 17 ( ( PUNCT cana-2735 71 18 π‘₯ π‘₯ NOUN cana-2735 71 19 , , PUNCT cana-2735 71 20 πœ” πœ” ADJ cana-2735 71 21 ) ) PUNCT cana-2735 71 22 = = SYM cana-2735 72 1 βˆ’ βˆ’ PROPN cana-2735 72 2 1 1 NUM cana-2735 72 3 2 2 NUM cana-2735 72 4 π‘”π‘–π‘Ÿ(π‘₯ π‘”π‘–π‘Ÿ(π‘₯ NOUN cana-2735 72 5 , , PUNCT cana-2735 72 6 πœ” πœ” NOUN cana-2735 72 7 ) ) PUNCT cana-2735 72 8 πœ•π‘”π‘—π‘˜(π‘₯,πœ” πœ•π‘”π‘—π‘˜(π‘₯,πœ” NOUN cana-2735 72 9 ) ) PUNCT cana-2735 72 10 πœ•πœ”π‘Ÿ πœ•πœ”π‘Ÿ NOUN cana-2735 73 1 = = PUNCT cana-2735 73 2 π‘”π‘–π‘Ÿ(π‘₯ π‘”π‘–π‘Ÿ(π‘₯ NOUN cana-2735 73 3 , , PUNCT cana-2735 73 4 πœ”)πΆπ‘Ÿπ‘—π‘˜(π‘₯ πœ”)πΆπ‘Ÿπ‘—π‘˜(π‘₯ NOUN cana-2735 73 5 , , PUNCT cana-2735 73 6 πœ” πœ” NOUN cana-2735 73 7 ) ) PUNCT cana-2735 73 8 . . PUNCT cana-2735 74 1 ( ( PUNCT cana-2735 74 2 6 6 X cana-2735 74 3 ) ) PUNCT cana-2735 74 4 are be AUX cana-2735 74 5 respectively respectively ADV cana-2735 74 6 called call VERB cana-2735 74 7 canonical canonical ADJ cana-2735 74 8 𝑁-connection 𝑁-connection NOUN cana-2735 74 9 , , PUNCT cana-2735 74 10 christoffel christoffel ADJ cana-2735 74 11 symbols symbol NOUN cana-2735 74 12 and and CCONJ cana-2735 74 13 𝑑-tensor 𝑑-tensor NOUN cana-2735 74 14 field field NOUN cana-2735 74 15 of of ADP cana-2735 74 16 type type NOUN cana-2735 74 17 ( ( PUNCT cana-2735 74 18 2,1 2,1 NUM cana-2735 74 19 ) ) PUNCT cana-2735 74 20 . . PUNCT cana-2735 75 1 let let VERB cana-2735 75 2 β„Ž β„Ž PRON cana-2735 75 3 -covariant -covariant VERB cana-2735 75 4 derivative derivative NOUN cana-2735 75 5 with with ADP cana-2735 75 6 respect respect NOUN cana-2735 75 7 to to AUX cana-2735 75 8 𝐷γ 𝐷γ ADV cana-2735 75 9 be be AUX cana-2735 75 10 denoted denote VERB cana-2735 75 11 by by ADP cana-2735 75 12 the the DET cana-2735 75 13 symbol symbol NOUN cana-2735 75 14 β€²|β„Žβ€² β€²|β„Žβ€² NUM cana-2735 75 15 . . PUNCT cana-2735 76 1 then then ADV cana-2735 76 2 , , PUNCT cana-2735 76 3 we we PRON cana-2735 76 4 have have VERB cana-2735 76 5 the the DET cana-2735 76 6 following follow VERB cana-2735 76 7 definition definition NOUN cana-2735 76 8 for for ADP cana-2735 76 9 later later ADJ cana-2735 76 10 use use NOUN cana-2735 76 11 . . PUNCT cana-2735 77 1 definition definition NOUN cana-2735 77 2 2.5 2.5 NUM cana-2735 78 1 [ [ X cana-2735 78 2 9 9 NUM cana-2735 78 3 ] ] PUNCT cana-2735 78 4 an an DET cana-2735 78 5 β„Ž-metrical β„Ž-metrical ADJ cana-2735 78 6 𝑑-connection 𝑑-connection NOUN cana-2735 78 7 on on ADP cana-2735 78 8 a a DET cana-2735 78 9 cartan cartan ADJ cana-2735 78 10 space space NOUN cana-2735 78 11 𝐢 𝐢 PROPN cana-2735 78 12 = = SYM cana-2735 78 13 ( ( PUNCT cana-2735 78 14 𝑀 𝑀 PROPN cana-2735 78 15 , , PUNCT cana-2735 78 16 𝐾(𝛼(π‘₯ 𝐾(𝛼(π‘₯ PROPN cana-2735 78 17 , , PUNCT cana-2735 78 18 πœ” πœ” NOUN cana-2735 78 19 ) ) PUNCT cana-2735 78 20 , , PUNCT cana-2735 78 21 𝛽(πœ” 𝛽(πœ” PROPN cana-2735 78 22 ) ) PUNCT cana-2735 78 23 ) ) PUNCT cana-2735 78 24 with with ADP cana-2735 78 25 ( ( PUNCT cana-2735 78 26 𝛼 𝛼 PROPN cana-2735 78 27 , , PUNCT cana-2735 78 28 𝛽)-metric 𝛽)-metric ADJ cana-2735 78 29 is be AUX cana-2735 78 30 a a DET cana-2735 78 31 𝑑-connection 𝑑-connection NOUN cana-2735 78 32 , , PUNCT cana-2735 78 33 𝐷𝛀 𝐷𝛀 PROPN cana-2735 78 34 on on ADP cana-2735 78 35 𝐢 𝐢 PROPN cana-2735 78 36 , , PUNCT cana-2735 78 37 satisfying satisfy VERB cana-2735 78 38 the the DET cana-2735 78 39 following follow VERB cana-2735 78 40 properties property NOUN cana-2735 78 41 : : PUNCT cana-2735 78 42 ( ( PUNCT cana-2735 78 43 1 1 X cana-2735 78 44 ) ) PUNCT cana-2735 78 45 β„Ž-deflection β„Ž-deflection NOUN cana-2735 78 46 tensor tensor NOUN cana-2735 78 47 𝐷𝑖𝑗(= 𝐷𝑖𝑗(= PROPN cana-2735 78 48 πœ”π‘–|𝑗 πœ”π‘–|𝑗 NOUN cana-2735 78 49 ) ) PUNCT cana-2735 79 1 = = SYM cana-2735 79 2 0 0 PUNCT cana-2735 80 1 ( ( PUNCT cana-2735 80 2 2 2 NUM cana-2735 80 3 ) ) PUNCT cana-2735 80 4 π‘Ž|β„Ž π‘Ž|β„Ž X cana-2735 81 1 𝑖𝑗 𝑖𝑗 X cana-2735 81 2 = = SYM cana-2735 81 3 0 0 PUNCT cana-2735 81 4 ( ( PUNCT cana-2735 81 5 3 3 NUM cana-2735 81 6 ) ) PUNCT cana-2735 81 7 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 82 1 𝑖𝑗 𝑖𝑗 ADP cana-2735 83 1 = = PUNCT cana-2735 83 2 0 0 X cana-2735 83 3 . . NOUN cana-2735 83 4 3 3 NUM cana-2735 83 5 cartan cartan PROPN cana-2735 83 6 spaces space VERB cana-2735 83 7 with with ADP cana-2735 83 8 an an DET cana-2735 83 9 ( ( PUNCT cana-2735 83 10 𝜢 𝜢 NOUN cana-2735 83 11 , , PUNCT cana-2735 83 12 𝜷)-metric 𝜷)-metric PUNCT cana-2735 83 13 with with ADP cana-2735 83 14 𝒉-metrical 𝒉-metrical ADJ cana-2735 83 15 𝒅-connection 𝒅-connection NOUN cana-2735 83 16 in in ADP cana-2735 83 17 this this DET cana-2735 83 18 section section NOUN cana-2735 83 19 we we PRON cana-2735 83 20 impose impose VERB cana-2735 83 21 the the DET cana-2735 83 22 condition condition NOUN cana-2735 83 23 of of ADP cana-2735 83 24 𝑑 𝑑 PROPN cana-2735 83 25 -connection -connection NOUN cana-2735 83 26 𝐷γ 𝐷γ ADV cana-2735 83 27 on on ADP cana-2735 83 28 the the DET cana-2735 83 29 cartan cartan ADJ cana-2735 83 30 space space NOUN cana-2735 83 31 with with ADP cana-2735 83 32 an an DET cana-2735 83 33 ( ( PUNCT cana-2735 83 34 𝛼 𝛼 PROPN cana-2735 83 35 , , PUNCT cana-2735 83 36 𝛽 𝛽 NOUN cana-2735 83 37 ) ) PUNCT cana-2735 83 38 metric metric NOUN cana-2735 83 39 to to PART cana-2735 83 40 be be AUX cana-2735 83 41 β„Ž-metrical β„Ž-metrical ADJ cana-2735 83 42 and and CCONJ cana-2735 83 43 in in ADP cana-2735 83 44 consequence consequence NOUN cana-2735 83 45 we we PRON cana-2735 83 46 analyse analyse VERB cana-2735 83 47 what what PRON cana-2735 83 48 shapes shape VERB cana-2735 83 49 the the DET cana-2735 83 50 corresponding corresponding ADJ cana-2735 83 51 cartan cartan ADJ cana-2735 83 52 space space NOUN cana-2735 83 53 assumes assume VERB cana-2735 83 54 . . PUNCT cana-2735 84 1 first first ADV cana-2735 84 2 we we PRON cana-2735 84 3 take take VERB cana-2735 84 4 the the DET cana-2735 84 5 β„Ž-covariant β„Ž-covariant ADJ cana-2735 84 6 derivative derivative NOUN cana-2735 84 7 of of ADP cana-2735 84 8 the the DET cana-2735 84 9 given give VERB cana-2735 84 10 ( ( PUNCT cana-2735 84 11 𝛼 𝛼 NOUN cana-2735 84 12 , , PUNCT cana-2735 84 13 𝛽)-metric 𝛽)-metric ADJ cana-2735 84 14 as as SCONJ cana-2735 84 15 follows follow VERB cana-2735 84 16 : : PUNCT cana-2735 84 17 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 84 18 , , PUNCT cana-2735 84 19 πœ” πœ” ADJ cana-2735 84 20 ) ) PUNCT cana-2735 84 21 = = SYM cana-2735 84 22 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 84 23 , , PUNCT cana-2735 84 24 πœ” πœ” PRON cana-2735 84 25 ) ) PUNCT cana-2735 84 26 + + NUM cana-2735 84 27 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 84 28 , , PUNCT cana-2735 84 29 πœ” πœ” PRON cana-2735 84 30 ) ) PUNCT cana-2735 84 31 + + CCONJ cana-2735 84 32 2π‘˜ 2π‘˜ NUM cana-2735 84 33 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 84 34 ) ) PUNCT cana-2735 84 35 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 84 36 ) ) PUNCT cana-2735 84 37 βˆ’ βˆ’ PROPN cana-2735 85 1 π‘˜2 π‘˜2 PROPN cana-2735 85 2 3 3 NUM cana-2735 85 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 85 4 ) ) PUNCT cana-2735 85 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 85 6 ) ) PUNCT cana-2735 85 7 𝑔𝑖𝑗(πœ”π‘–πœ”π‘—|β„Ž 𝑔𝑖𝑗(πœ”π‘–πœ”π‘—|β„Ž NOUN cana-2735 85 8 + + CCONJ cana-2735 85 9 πœ”π‘—πœ”π‘–|β„Ž πœ”π‘—πœ”π‘–|β„Ž NOUN cana-2735 85 10 ) ) PUNCT cana-2735 86 1 + + CCONJ cana-2735 86 2 πœ”π‘–πœ”π‘—π‘”|β„Ž πœ”π‘–πœ”π‘—π‘”|β„Ž ADV cana-2735 86 3 𝑖𝑗 𝑖𝑗 ADP cana-2735 86 4 = = PUNCT cana-2735 86 5 𝛼|β„Ž 𝛼|β„Ž ADJ cana-2735 86 6 + + CCONJ cana-2735 86 7 πœ–π›½|β„Ž πœ–π›½|β„Ž VERB cana-2735 86 8 + + CCONJ cana-2735 86 9 2π‘˜ 2π‘˜ NUM cana-2735 86 10 ( ( PUNCT cana-2735 86 11 2𝛼𝛽𝛽|β„Ž 2𝛼𝛽𝛽|β„Ž NUM cana-2735 86 12 βˆ’ βˆ’ PROPN cana-2735 86 13 𝛽2𝛼|β„Ž 𝛽2𝛼|β„Ž NOUN cana-2735 86 14 𝛼2 𝛼2 PROPN cana-2735 86 15 ) ) PUNCT cana-2735 86 16 βˆ’ βˆ’ PROPN cana-2735 87 1 π‘˜3 π‘˜3 PROPN cana-2735 87 2 3 3 NUM cana-2735 87 3 ( ( PUNCT cana-2735 87 4 4𝛼3𝛽3𝛽|β„Ž 4𝛼3𝛽3𝛽|β„Ž NOUN cana-2735 87 5 βˆ’ βˆ’ PROPN cana-2735 87 6 3𝛼2𝛽4𝛼|β„Ž 3𝛼2𝛽4𝛼|β„Ž NUM cana-2735 87 7 𝛼6 𝛼6 NOUN cana-2735 87 8 ) ) PUNCT cana-2735 87 9 as as SCONJ cana-2735 87 10 we we PRON cana-2735 87 11 have have AUX cana-2735 87 12 stipulated stipulate VERB cana-2735 87 13 the the DET cana-2735 87 14 𝑑-connection 𝑑-connection NOUN cana-2735 87 15 𝐷γ 𝐷γ PROPN cana-2735 87 16 of of ADP cana-2735 87 17 the the DET cana-2735 87 18 cartan cartan ADJ cana-2735 87 19 space space NOUN cana-2735 87 20 is be AUX cana-2735 87 21 β„Ž-metrical β„Ž-metrical PROPN cana-2735 87 22 , , PUNCT cana-2735 87 23 therefore therefore ADV cana-2735 87 24 by by ADP cana-2735 87 25 definition definition NOUN cana-2735 87 26 2.5 2.5 NUM cana-2735 87 27 , , PUNCT cana-2735 87 28 we we PRON cana-2735 87 29 have have VERB cana-2735 87 30 πœ”π‘—|β„Ž πœ”π‘—|β„Ž X cana-2735 87 31 = = SYM cana-2735 87 32 0 0 NUM cana-2735 87 33 , , PUNCT cana-2735 87 34 πœ”π‘–|β„Ž πœ”π‘–|β„Ž PUNCT cana-2735 88 1 = = SYM cana-2735 88 2 0 0 NUM cana-2735 88 3 , , PUNCT cana-2735 88 4 𝛼|β„Ž 𝛼|β„Ž NOUN cana-2735 88 5 = = SYM cana-2735 88 6 0 0 NUM cana-2735 88 7 , , PUNCT cana-2735 88 8 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 89 1 𝑖𝑗 𝑖𝑗 ADP cana-2735 89 2 = = SYM cana-2735 89 3 0 0 PUNCT cana-2735 89 4 using use VERB cana-2735 89 5 these these DET cana-2735 89 6 values value NOUN cana-2735 89 7 in in ADP cana-2735 89 8 above above ADP cana-2735 89 9 expression expression NOUN cana-2735 89 10 , , PUNCT cana-2735 89 11 we we PRON cana-2735 89 12 get get VERB cana-2735 89 13 𝑔𝑖𝑗(πœ”π‘– 𝑔𝑖𝑗(πœ”π‘– ADJ cana-2735 89 14 Γ— Γ— NOUN cana-2735 89 15 0 0 NUM cana-2735 90 1 + + CCONJ cana-2735 90 2 πœ”π‘— πœ”π‘— PRON cana-2735 90 3 Γ— Γ— NOUN cana-2735 90 4 0 0 NUM cana-2735 90 5 ) ) PUNCT cana-2735 91 1 + + NUM cana-2735 91 2 πœ”π‘–πœ”π‘— πœ”π‘–πœ”π‘— NOUN cana-2735 91 3 Γ— Γ— NOUN cana-2735 91 4 0 0 PUNCT cana-2735 92 1 = = SYM cana-2735 92 2 0 0 NUM cana-2735 93 1 + + CCONJ cana-2735 93 2 πœ–π›½|β„Ž πœ–π›½|β„Ž NOUN cana-2735 93 3 + + CCONJ cana-2735 93 4 2π‘˜ 2π‘˜ NUM cana-2735 93 5 ( ( PUNCT cana-2735 93 6 2𝛼𝛽𝛽|β„Žβˆ’π›½2Γ—0 2𝛼𝛽𝛽|β„Žβˆ’π›½2Γ—0 PROPN cana-2735 93 7 𝛼2 𝛼2 PROPN cana-2735 93 8 ) ) PUNCT cana-2735 94 1 βˆ’ βˆ’ PROPN cana-2735 95 1 π‘˜3 π‘˜3 PROPN cana-2735 95 2 3 3 NUM cana-2735 95 3 ( ( PUNCT cana-2735 95 4 4𝛼3𝛽3𝛽|β„Žβˆ’3𝛼2𝛽4Γ—0 4𝛼3𝛽3𝛽|β„Žβˆ’3𝛼2𝛽4Γ—0 NUM cana-2735 95 5 𝛼6 𝛼6 NOUN cana-2735 95 6 ) ) PUNCT cana-2735 95 7 communications communication NOUN cana-2735 95 8 on on ADP cana-2735 95 9 applied apply VERB cana-2735 95 10 nonlinear nonlinear ADJ cana-2735 95 11 analysis analysis NOUN cana-2735 95 12 issn issn NOUN cana-2735 95 13 : : PUNCT cana-2735 95 14 1074 1074 NUM cana-2735 95 15 - - PUNCT cana-2735 95 16 133x 133x NUM cana-2735 95 17 vol vol NOUN cana-2735 95 18 32 32 NUM cana-2735 95 19 no no NOUN cana-2735 95 20 . . PUNCT cana-2735 96 1 4s 4s NUM cana-2735 96 2 ( ( PUNCT cana-2735 96 3 2025 2025 NUM cana-2735 96 4 ) ) PUNCT cana-2735 96 5 6 6 NUM cana-2735 96 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 96 7 𝛽|β„Ž 𝛽|β„Ž NOUN cana-2735 96 8 = = SYM cana-2735 96 9 0 0 NUM cana-2735 96 10 ( ( PUNCT cana-2735 96 11 ∡ ∡ NOUN cana-2735 96 12 Ξ± Ξ± PRON cana-2735 96 13 β‰  β‰  PROPN cana-2735 96 14 0 0 NUM cana-2735 96 15 , , PUNCT cana-2735 96 16 Ξ² Ξ² X cana-2735 96 17 β‰  β‰  PROPN cana-2735 96 18 0 0 NUM cana-2735 96 19 ) ) PUNCT cana-2735 96 20 ( ( PUNCT cana-2735 96 21 7 7 NUM cana-2735 96 22 ) ) PUNCT cana-2735 96 23 ( ( PUNCT cana-2735 96 24 πœ”π‘–π‘ πœ”π‘–π‘ NOUN cana-2735 96 25 𝑖(π‘₯))|β„Ž 𝑖(π‘₯))|β„Ž NOUN cana-2735 96 26 = = SYM cana-2735 96 27 0 0 NUM cana-2735 97 1 ( ( PUNCT cana-2735 97 2 ∡ ∡ PROPN cana-2735 97 3 Ξ²(x Ξ²(x NOUN cana-2735 97 4 , , PUNCT cana-2735 97 5 Ο‰ Ο‰ NUM cana-2735 97 6 ) ) PUNCT cana-2735 97 7 = = SYM cana-2735 97 8 Ο‰ib Ο‰ib NOUN cana-2735 97 9 i(x i(x NOUN cana-2735 97 10 ) ) PUNCT cana-2735 97 11 ) ) PUNCT cana-2735 98 1 πœ”π‘–π‘ πœ”π‘–π‘ ADP cana-2735 98 2 𝑖(π‘₯)|β„Ž 𝑖(π‘₯)|β„Ž PROPN cana-2735 98 3 + + CCONJ cana-2735 98 4 𝑏𝑖(π‘₯)πœ”π‘–|β„Ž 𝑏𝑖(π‘₯)πœ”π‘–|β„Ž PROPN cana-2735 98 5 = = SYM cana-2735 98 6 0 0 PUNCT cana-2735 99 1 as as SCONJ cana-2735 99 2 we we PRON cana-2735 99 3 have have AUX cana-2735 99 4 stipulated stipulate VERB cana-2735 99 5 the the DET cana-2735 99 6 𝑑-connection 𝑑-connection NOUN cana-2735 99 7 𝐷γ 𝐷γ PROPN cana-2735 99 8 of of ADP cana-2735 99 9 the the DET cana-2735 99 10 cartan cartan ADJ cana-2735 99 11 space space NOUN cana-2735 99 12 is be AUX cana-2735 99 13 β„Ž-metrical β„Ž-metrical PROPN cana-2735 99 14 , , PUNCT cana-2735 99 15 therefore therefore ADV cana-2735 99 16 by by ADP cana-2735 99 17 definition definition NOUN cana-2735 99 18 2.5 2.5 NUM cana-2735 99 19 , , PUNCT cana-2735 99 20 we we PRON cana-2735 99 21 have have VERB cana-2735 99 22 πœ”π‘–|β„Ž πœ”π‘–|β„Ž PUNCT cana-2735 99 23 = = SYM cana-2735 99 24 0 0 NUM cana-2735 99 25 , , PUNCT cana-2735 99 26 using use VERB cana-2735 99 27 these these DET cana-2735 99 28 values value NOUN cana-2735 99 29 in in ADP cana-2735 99 30 above above ADP cana-2735 99 31 expression expression NOUN cana-2735 99 32 , , PUNCT cana-2735 99 33 we we PRON cana-2735 99 34 get get VERB cana-2735 99 35 πœ”π‘–π‘ πœ”π‘–π‘ ADJ cana-2735 99 36 𝑖(π‘₯)|β„Ž 𝑖(π‘₯)|β„Ž PROPN cana-2735 99 37 + + PRON cana-2735 99 38 𝑏𝑖(π‘₯ 𝑏𝑖(π‘₯ NUM cana-2735 99 39 ) ) PUNCT cana-2735 99 40 Γ— Γ— NOUN cana-2735 99 41 0 0 NUM cana-2735 99 42 = = SYM cana-2735 99 43 0 0 NUM cana-2735 100 1 πœ”π‘–π‘ πœ”π‘–π‘ NOUN cana-2735 100 2 𝑖(π‘₯)|β„Ž 𝑖(π‘₯)|β„Ž PROPN cana-2735 100 3 = = SYM cana-2735 100 4 0 0 NUM cana-2735 100 5 𝑏𝑖(π‘₯)|β„Ž 𝑏𝑖(π‘₯)|β„Ž NOUN cana-2735 100 6 = = SYM cana-2735 100 7 0 0 PUNCT cana-2735 101 1 ( ( PUNCT cana-2735 101 2 8) 8) NUM cana-2735 101 3 now now ADV cana-2735 101 4 , , PUNCT cana-2735 101 5 we we PRON cana-2735 101 6 find find VERB cana-2735 101 7 β„Ž-covariant β„Ž-covariant ADJ cana-2735 101 8 derivatives derivative NOUN cana-2735 101 9 of of ADP cana-2735 101 10 the the DET cana-2735 101 11 coefficients coefficient NOUN cana-2735 101 12 of of ADP cana-2735 101 13 metric metric ADJ cana-2735 101 14 tensor tensor NOUN cana-2735 101 15 𝑔𝑖𝑗 𝑔𝑖𝑗 NOUN cana-2735 101 16 and and CCONJ cana-2735 101 17 then then ADV cana-2735 101 18 use use VERB cana-2735 101 19 conditions condition NOUN cana-2735 101 20 of of ADP cana-2735 101 21 β„Ž-metrical β„Ž-metrical ADJ cana-2735 101 22 𝑑-connection 𝑑-connection NOUN cana-2735 101 23 𝐷γ 𝐷γ PROPN cana-2735 101 24 of of ADP cana-2735 101 25 cartan cartan ADJ cana-2735 101 26 space space NOUN cana-2735 101 27 as as SCONJ cana-2735 101 28 follows follow VERB cana-2735 101 29 , , PUNCT cana-2735 101 30 we we PRON cana-2735 101 31 get get VERB cana-2735 101 32 ∡ ∡ ADJ cana-2735 101 33 𝜌 𝜌 X cana-2735 101 34 = = SYM cana-2735 101 35 𝛼5βˆ’2π‘˜π›Ό2𝛽2+π‘˜2𝛽4 𝛼5βˆ’2π‘˜π›Ό2𝛽2+π‘˜2𝛽4 PROPN cana-2735 101 36 2𝛼5 2𝛼5 NUM cana-2735 101 37 ∴ ∴ PROPN cana-2735 101 38 𝜌|β„Ž 𝜌|β„Ž PUNCT cana-2735 102 1 = = SYM cana-2735 102 2 0 0 X cana-2735 102 3 . . PUNCT cana-2735 103 1 ( ( PUNCT cana-2735 103 2 9 9 X cana-2735 103 3 ) ) PUNCT cana-2735 103 4 ∡ ∡ NOUN cana-2735 103 5 𝜌0 𝜌0 NOUN cana-2735 103 6 = = SYM cana-2735 103 7 2π‘˜π›Ό2βˆ’2π‘˜2𝛽2 2π‘˜π›Ό2βˆ’2π‘˜2𝛽2 NUM cana-2735 103 8 𝛼3 𝛼3 NOUN cana-2735 103 9 ∴ ∴ VERB cana-2735 103 10 𝜌0|β„Ž 𝜌0|β„Ž ADJ cana-2735 103 11 = = SYM cana-2735 103 12 0 0 NUM cana-2735 103 13 . . PUNCT cana-2735 104 1 ( ( PUNCT cana-2735 104 2 10 10 NUM cana-2735 104 3 ) ) PUNCT cana-2735 104 4 ∡ ∡ NOUN cana-2735 104 5 πœŒβˆ’1 πœŒβˆ’1 PROPN cana-2735 104 6 = = SYM cana-2735 104 7 βˆ’4π‘˜2𝛽2 βˆ’4π‘˜2𝛽2 NOUN cana-2735 104 8 𝛼3 𝛼3 ADV cana-2735 104 9 ∴ ∴ PROPN cana-2735 104 10 πœŒβˆ’1|β„Ž πœŒβˆ’1|β„Ž PROPN cana-2735 104 11 = = SYM cana-2735 104 12 0 0 NUM cana-2735 104 13 . . PUNCT cana-2735 105 1 ( ( PUNCT cana-2735 105 2 11 11 NUM cana-2735 105 3 ) ) PUNCT cana-2735 105 4 ∡ ∡ NOUN cana-2735 105 5 πœŒβˆ’2 πœŒβˆ’2 PROPN cana-2735 105 6 = = PUNCT cana-2735 106 1 6π‘˜2𝛽2βˆ’2π‘˜π›Ό2 6π‘˜2𝛽2βˆ’2π‘˜π›Ό2 NOUN cana-2735 106 2 𝛼5 𝛼5 ADP cana-2735 106 3 ∴ ∴ NOUN cana-2735 106 4 πœŒβˆ’2|β„Ž πœŒβˆ’2|β„Ž NOUN cana-2735 106 5 = = SYM cana-2735 106 6 0 0 NUM cana-2735 106 7 . . PUNCT cana-2735 107 1 ( ( PUNCT cana-2735 107 2 12 12 NUM cana-2735 107 3 ) ) PUNCT cana-2735 107 4 the the DET cana-2735 107 5 β„Ž-covariant β„Ž-covariant ADJ cana-2735 107 6 differentiation differentiation NOUN cana-2735 107 7 of of ADP cana-2735 107 8 the the DET cana-2735 107 9 equation equation NOUN cana-2735 107 10 ( ( PUNCT cana-2735 107 11 1 1 X cana-2735 107 12 ) ) PUNCT cana-2735 107 13 gives give VERB cana-2735 107 14 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 108 1 𝑖𝑗 𝑖𝑗 ADP cana-2735 108 2 = = PUNCT cana-2735 108 3 πœŒπ‘Ž|β„Ž πœŒπ‘Ž|β„Ž X cana-2735 109 1 𝑖𝑗 𝑖𝑗 X cana-2735 109 2 + + PUNCT cana-2735 109 3 π‘Žπ‘–π‘—πœŒ|β„Ž π‘Žπ‘–π‘—πœŒ|β„Ž PROPN cana-2735 109 4 + + CCONJ cana-2735 109 5 𝜌0(𝑏𝑖𝑏𝑗)|β„Ž 𝜌0(𝑏𝑖𝑏𝑗)|β„Ž PROPN cana-2735 109 6 + + NUM cana-2735 109 7 π‘π‘–π‘π‘—πœŒ0 π‘π‘–π‘π‘—πœŒ0 NOUN cana-2735 110 1 + + CCONJ cana-2735 110 2 πœŒβˆ’1(π‘π‘–πœ”π‘— πœŒβˆ’1(π‘π‘–πœ”π‘— NUM cana-2735 110 3 + + X cana-2735 110 4 π‘π‘—πœ”π‘–)|β„Ž π‘π‘—πœ”π‘–)|β„Ž NOUN cana-2735 110 5 + + CCONJ cana-2735 110 6 ( ( PUNCT cana-2735 110 7 π‘π‘–πœ”π‘— π‘π‘–πœ”π‘— NOUN cana-2735 110 8 + + CCONJ cana-2735 110 9 π‘π‘—πœ”π‘–)πœŒβˆ’1|β„Ž π‘π‘—πœ”π‘–)πœŒβˆ’1|β„Ž PROPN cana-2735 110 10 + + CCONJ cana-2735 110 11 πœŒβˆ’2(πœ”π‘–πœ”π‘—)|β„Ž πœŒβˆ’2(πœ”π‘–πœ”π‘—)|β„Ž X cana-2735 110 12 + + X cana-2735 110 13 πœ”π‘–πœ”π‘—πœŒβˆ’2|β„Ž πœ”π‘–πœ”π‘—πœŒβˆ’2|β„Ž NOUN cana-2735 110 14 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 110 15 𝑖𝑗 𝑖𝑗 ADP cana-2735 111 1 = = PUNCT cana-2735 111 2 πœŒπ‘Ž|β„Ž πœŒπ‘Ž|β„Ž X cana-2735 112 1 𝑖𝑗 𝑖𝑗 X cana-2735 112 2 + + PUNCT cana-2735 112 3 π‘Žπ‘–π‘—πœŒ|β„Ž π‘Žπ‘–π‘—πœŒ|β„Ž PROPN cana-2735 112 4 + + X cana-2735 112 5 𝜌0(𝑏𝑖𝑏|β„Ž 𝜌0(𝑏𝑖𝑏|β„Ž VERB cana-2735 112 6 𝑗 𝑗 NOUN cana-2735 113 1 + + X cana-2735 113 2 𝑏𝑗𝑏|β„Ž 𝑏𝑗𝑏|β„Ž ADJ cana-2735 113 3 𝑖 𝑖 X cana-2735 113 4 ) ) PUNCT cana-2735 114 1 + + CCONJ cana-2735 114 2 π‘π‘–π‘π‘—πœŒ0|β„Ž π‘π‘–π‘π‘—πœŒ0|β„Ž NUM cana-2735 114 3 + + CCONJ cana-2735 114 4 πœŒβˆ’1(π‘π‘–πœ”|β„Ž πœŒβˆ’1(π‘π‘–πœ”|β„Ž PROPN cana-2735 114 5 𝑗 𝑗 NOUN cana-2735 114 6 + + X cana-2735 114 7 πœ”π‘–π‘|β„Ž πœ”π‘–π‘|β„Ž ADJ cana-2735 114 8 𝑖 𝑖 X cana-2735 115 1 + + CCONJ cana-2735 115 2 π‘π‘—πœ”|β„Ž π‘π‘—πœ”|β„Ž PROPN cana-2735 115 3 𝑖 𝑖 PROPN cana-2735 116 1 + + X cana-2735 116 2 πœ”π‘–π‘β„Ž πœ”π‘–π‘β„Ž X cana-2735 116 3 𝑗 𝑗 INTJ cana-2735 116 4 ) ) PUNCT cana-2735 116 5 πœŒβˆ’1|β„Ž(π‘π‘–πœ”π‘— πœŒβˆ’1|β„Ž(π‘π‘–πœ”π‘— PROPN cana-2735 116 6 + + CCONJ cana-2735 116 7 π‘π‘—πœ”π‘– π‘π‘—πœ”π‘– NOUN cana-2735 116 8 ) ) PUNCT cana-2735 117 1 + + CCONJ cana-2735 118 1 πœŒβˆ’2|β„Ž(πœ”π‘–πœ”|β„Ž πœŒβˆ’2|β„Ž(πœ”π‘–πœ”|β„Ž PUNCT cana-2735 118 2 𝑗 𝑗 VERB cana-2735 118 3 + + NOUN cana-2735 118 4 πœ”π‘—πœ”|β„Ž πœ”π‘—πœ”|β„Ž PROPN cana-2735 118 5 𝑖 𝑖 X cana-2735 118 6 ) ) PUNCT cana-2735 119 1 + + CCONJ cana-2735 119 2 πœ”π‘–πœ”π‘—πœŒβˆ’2|β„Ž. πœ”π‘–πœ”π‘—πœŒβˆ’2|β„Ž. ADV cana-2735 119 3 using use VERB cana-2735 119 4 the the DET cana-2735 119 5 conditions condition NOUN cana-2735 119 6 of of ADP cana-2735 119 7 β„Ž-metrical β„Ž-metrical ADJ cana-2735 119 8 𝑑-connection 𝑑-connection NOUN cana-2735 119 9 𝐷γ 𝐷γ PROPN cana-2735 119 10 of of ADP cana-2735 119 11 cartan cartan ADJ cana-2735 119 12 space space NOUN cana-2735 119 13 and and CCONJ cana-2735 119 14 equations equation NOUN cana-2735 119 15 ( ( PUNCT cana-2735 119 16 8) 8) NUM cana-2735 119 17 , , PUNCT cana-2735 119 18 ( ( PUNCT cana-2735 119 19 9 9 NUM cana-2735 119 20 ) ) PUNCT cana-2735 119 21 , , PUNCT cana-2735 119 22 ( ( PUNCT cana-2735 119 23 10 10 NUM cana-2735 119 24 ) ) PUNCT cana-2735 119 25 , , PUNCT cana-2735 119 26 ( ( PUNCT cana-2735 119 27 11 11 NUM cana-2735 119 28 ) ) PUNCT cana-2735 119 29 and and CCONJ cana-2735 119 30 ( ( PUNCT cana-2735 119 31 12 12 NUM cana-2735 119 32 ) ) PUNCT cana-2735 119 33 , , PUNCT cana-2735 119 34 above above ADP cana-2735 119 35 equation equation NOUN cana-2735 119 36 reduces reduce VERB cana-2735 119 37 to to ADP cana-2735 119 38 communications communication NOUN cana-2735 119 39 on on ADP cana-2735 119 40 applied apply VERB cana-2735 119 41 nonlinear nonlinear ADJ cana-2735 119 42 analysis analysis NOUN cana-2735 119 43 issn issn NOUN cana-2735 119 44 : : PUNCT cana-2735 119 45 1074 1074 NUM cana-2735 119 46 - - PUNCT cana-2735 119 47 133x 133x NUM cana-2735 119 48 vol vol NOUN cana-2735 119 49 32 32 NUM cana-2735 119 50 no no NOUN cana-2735 119 51 . . PUNCT cana-2735 120 1 4s 4s NUM cana-2735 120 2 ( ( PUNCT cana-2735 120 3 2025 2025 NUM cana-2735 120 4 ) ) PUNCT cana-2735 120 5 7 7 NUM cana-2735 120 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 120 7 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 121 1 𝑖𝑗 𝑖𝑗 ADP cana-2735 121 2 = = NOUN cana-2735 121 3 0 0 X cana-2735 121 4 . . PUNCT cana-2735 122 1 thus thus ADV cana-2735 122 2 , , PUNCT cana-2735 122 3 allowing allow VERB cana-2735 122 4 𝑑 𝑑 PRON cana-2735 122 5 -connection -connection NOUN cana-2735 122 6 𝐷γ 𝐷γ ADV cana-2735 122 7 of of ADP cana-2735 122 8 the the DET cana-2735 122 9 cartan cartan ADJ cana-2735 122 10 space space NOUN cana-2735 122 11 to to PART cana-2735 122 12 be be AUX cana-2735 122 13 β„Ž β„Ž X cana-2735 122 14 -metrical -metrical ADJ cana-2735 122 15 , , PUNCT cana-2735 122 16 it it PRON cana-2735 122 17 gives give VERB cana-2735 122 18 two two NUM cana-2735 122 19 important important ADJ cana-2735 122 20 quantities quantity NOUN cana-2735 122 21 namely namely ADV cana-2735 122 22 π‘Ž|β„Ž π‘Ž|β„Ž PUNCT cana-2735 123 1 𝑖𝑗 𝑖𝑗 X cana-2735 123 2 = = PUNCT cana-2735 123 3 0 0 PUNCT cana-2735 124 1 ( ( PUNCT cana-2735 124 2 by by ADP cana-2735 124 3 definition definition NOUN cana-2735 124 4 of of ADP cana-2735 124 5 β„Ž-metrical β„Ž-metrical PROPN cana-2735 124 6 𝑑-connection 𝑑-connection PROPN cana-2735 124 7 ) ) PUNCT cana-2735 124 8 and and CCONJ cana-2735 124 9 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 125 1 𝑖𝑗 𝑖𝑗 ADP cana-2735 125 2 = = SYM cana-2735 125 3 0 0 NUM cana-2735 125 4 , , PUNCT cana-2735 125 5 i.e. i.e. X cana-2735 125 6 , , PUNCT cana-2735 125 7 β„Ž-covariant β„Ž-covariant ADJ cana-2735 125 8 derivatives derivative NOUN cana-2735 125 9 of of ADP cana-2735 125 10 fundamental fundamental ADJ cana-2735 125 11 metric metric ADJ cana-2735 125 12 tensors tensor NOUN cana-2735 125 13 of of ADP cana-2735 125 14 associated associated ADJ cana-2735 125 15 riemannian riemannian ADJ cana-2735 125 16 space space NOUN cana-2735 125 17 and and CCONJ cana-2735 125 18 cartan cartan ADJ cana-2735 125 19 space space NOUN cana-2735 125 20 vanishes vanish VERB cana-2735 125 21 . . PUNCT cana-2735 126 1 now now ADV cana-2735 126 2 , , PUNCT cana-2735 126 3 since since SCONJ cana-2735 126 4 π‘Ž|β„Ž π‘Ž|β„Ž X cana-2735 126 5 𝑖𝑗 𝑖𝑗 X cana-2735 126 6 = = SYM cana-2735 126 7 0 0 PUNCT cana-2735 126 8 and and CCONJ cana-2735 126 9 𝑔|β„Ž 𝑔|β„Ž NOUN cana-2735 127 1 𝑖𝑗 𝑖𝑗 ADP cana-2735 127 2 = = SYM cana-2735 127 3 0 0 NUM cana-2735 127 4 , , PUNCT cana-2735 127 5 therefore therefore ADV cana-2735 127 6 there there ADV cana-2735 127 7 corresponding correspond VERB cana-2735 127 8 chritoffel chritoffel PROPN cana-2735 127 9 symbols symbol NOUN cana-2735 127 10 will will AUX cana-2735 127 11 also also ADV cana-2735 127 12 be be AUX cana-2735 127 13 same same ADJ cana-2735 127 14 , , PUNCT cana-2735 127 15 i.e. i.e. X cana-2735 127 16 , , PUNCT cana-2735 127 17 π»π‘—β„Ž π»π‘—β„Ž PROPN cana-2735 127 18 𝑖 𝑖 NOUN cana-2735 127 19 = = NOUN cana-2735 127 20 π›Ύπ‘—β„Ž π›Ύπ‘—β„Ž NOUN cana-2735 127 21 𝑖 𝑖 PRON cana-2735 127 22 and and CCONJ cana-2735 127 23 its its PRON cana-2735 127 24 equivalent equivalent ADJ cana-2735 127 25 condition condition NOUN cana-2735 127 26 is be AUX cana-2735 127 27 given give VERB cana-2735 127 28 by by ADP cana-2735 127 29 𝑏:π‘˜ 𝑏:π‘˜ NOUN cana-2735 127 30 𝑖 𝑖 SYM cana-2735 127 31 = = SYM cana-2735 127 32 0 0 NUM cana-2735 127 33 ( ( PUNCT cana-2735 127 34 13 13 NUM cana-2735 127 35 ) ) PUNCT cana-2735 127 36 now now ADV cana-2735 127 37 , , PUNCT cana-2735 127 38 since since SCONJ cana-2735 127 39 π»π‘—β„Ž π»π‘—β„Ž PROPN cana-2735 127 40 𝑖 𝑖 NOUN cana-2735 127 41 = = NOUN cana-2735 127 42 π›Ύπ‘—β„Ž π›Ύπ‘—β„Ž NOUN cana-2735 128 1 𝑖 𝑖 X cana-2735 128 2 therefore therefore ADV cana-2735 128 3 the the DET cana-2735 128 4 curvature curvature NOUN cana-2735 128 5 tensor tensor NOUN cana-2735 128 6 π·β„Žπ‘—π‘˜ π·β„Žπ‘—π‘˜ PROPN cana-2735 128 7 𝑖 𝑖 NUM cana-2735 128 8 of of ADP cana-2735 128 9 𝐷γ 𝐷γ PRON cana-2735 128 10 coincides coincide VERB cana-2735 128 11 with with ADP cana-2735 128 12 the the DET cana-2735 128 13 curvature curvature NOUN cana-2735 128 14 tensor tensor NOUN cana-2735 128 15 π‘…β„Žπ‘—π‘˜ π‘…β„Žπ‘—π‘˜ PROPN cana-2735 128 16 𝑖 𝑖 NUM cana-2735 128 17 of of ADP cana-2735 128 18 riemannian riemannian ADJ cana-2735 128 19 connection connection NOUN cana-2735 128 20 𝑅γ 𝑅γ PROPN cana-2735 128 21 = = PUNCT cana-2735 128 22 ( ( PUNCT cana-2735 128 23 π›Ύπ‘—π‘˜ π›Ύπ‘—π‘˜ NOUN cana-2735 128 24 𝑖 𝑖 SYM cana-2735 128 25 , , PUNCT cana-2735 128 26 π›Ύπ‘—π‘˜ π›Ύπ‘—π‘˜ PROPN cana-2735 128 27 𝑖 𝑖 SYM cana-2735 128 28 𝑦𝑖 𝑦𝑖 PROPN cana-2735 128 29 , , PUNCT cana-2735 128 30 0 0 NUM cana-2735 128 31 ) ) PUNCT cana-2735 128 32 , , PUNCT cana-2735 128 33 i.e. i.e. X cana-2735 128 34 , , PUNCT cana-2735 128 35 π·β„Žπ‘—π‘˜ π·β„Žπ‘—π‘˜ NOUN cana-2735 128 36 𝑖 𝑖 NOUN cana-2735 129 1 = = PUNCT cana-2735 130 1 π‘…β„Žπ‘—π‘˜ π‘…β„Žπ‘—π‘˜ PROPN cana-2735 130 2 𝑖 𝑖 NOUN cana-2735 130 3 if if SCONJ cana-2735 130 4 the the DET cana-2735 130 5 riemannian riemannian ADJ cana-2735 130 6 curvature curvature NOUN cana-2735 130 7 tensor tensor NOUN cana-2735 130 8 vanishes vanish VERB cana-2735 130 9 , , PUNCT cana-2735 130 10 i.e. i.e. X cana-2735 130 11 , , PUNCT cana-2735 130 12 π‘…β„Žπ‘—π‘˜ π‘…β„Žπ‘—π‘˜ PROPN cana-2735 130 13 𝑖 𝑖 SYM cana-2735 130 14 = = SYM cana-2735 130 15 0 0 NUM cana-2735 130 16 , , PUNCT cana-2735 130 17 the the DET cana-2735 130 18 curvature curvature NOUN cana-2735 130 19 tensor tensor NOUN cana-2735 130 20 of of ADP cana-2735 130 21 𝑑-connection 𝑑-connection NOUN cana-2735 130 22 also also ADV cana-2735 130 23 vanishes vanish VERB cana-2735 130 24 , , PUNCT cana-2735 130 25 i.e. i.e. X cana-2735 130 26 , , PUNCT cana-2735 130 27 π·β„Žπ‘—π‘˜ π·β„Žπ‘—π‘˜ PROPN cana-2735 130 28 𝑖 𝑖 SYM cana-2735 130 29 = = NOUN cana-2735 130 30 0 0 PROPN cana-2735 130 31 . . PUNCT cana-2735 131 1 this this DET cana-2735 131 2 discussion discussion NOUN cana-2735 131 3 can can AUX cana-2735 131 4 be be AUX cana-2735 131 5 summarized summarize VERB cana-2735 131 6 as as SCONJ cana-2735 131 7 follows follow VERB cana-2735 131 8 : : PUNCT cana-2735 131 9 proposition proposition NOUN cana-2735 131 10 3.1 3.1 NUM cana-2735 131 11 a a DET cana-2735 131 12 cartan cartan ADJ cana-2735 131 13 space space NOUN cana-2735 131 14 𝐢 𝐢 PROPN cana-2735 131 15 with with ADP cana-2735 131 16 the the DET cana-2735 131 17 ( ( PUNCT cana-2735 131 18 𝛼 𝛼 PROPN cana-2735 131 19 , , PUNCT cana-2735 131 20 𝛽 𝛽 NOUN cana-2735 131 21 ) ) PUNCT cana-2735 131 22 -metric -metric NOUN cana-2735 131 23 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 131 24 , , PUNCT cana-2735 131 25 πœ” πœ” ADJ cana-2735 131 26 ) ) PUNCT cana-2735 131 27 = = SYM cana-2735 131 28 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 131 29 , , PUNCT cana-2735 131 30 πœ” πœ” PRON cana-2735 131 31 ) ) PUNCT cana-2735 131 32 + + NUM cana-2735 131 33 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 131 34 , , PUNCT cana-2735 131 35 πœ” πœ” PRON cana-2735 131 36 ) ) PUNCT cana-2735 131 37 + + CCONJ cana-2735 131 38 2π‘˜ 2π‘˜ NUM cana-2735 131 39 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 131 40 ) ) PUNCT cana-2735 131 41 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 131 42 ) ) PUNCT cana-2735 132 1 βˆ’ βˆ’ PROPN cana-2735 133 1 π‘˜2 π‘˜2 PROPN cana-2735 133 2 3 3 NUM cana-2735 133 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 133 4 ) ) PUNCT cana-2735 133 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 133 6 ) ) PUNCT cana-2735 133 7 admitting admit VERB cana-2735 133 8 a a DET cana-2735 133 9 h h NOUN cana-2735 133 10 - - PUNCT cana-2735 133 11 metrical metrical ADJ cana-2735 133 12 d d NOUN cana-2735 133 13 - - PUNCT cana-2735 133 14 connection connection NOUN cana-2735 133 15 is be AUX cana-2735 133 16 locally locally ADV cana-2735 133 17 flat flat ADJ cana-2735 133 18 if if SCONJ cana-2735 133 19 and and CCONJ cana-2735 133 20 only only ADV cana-2735 133 21 if if SCONJ cana-2735 133 22 the the DET cana-2735 133 23 associated associated ADJ cana-2735 133 24 riemannian riemannian ADJ cana-2735 133 25 space space NOUN cana-2735 133 26 is be AUX cana-2735 133 27 locally locally ADV cana-2735 133 28 flat flat ADJ cana-2735 133 29 . . PUNCT cana-2735 134 1 now now ADV cana-2735 134 2 , , PUNCT cana-2735 134 3 we we PRON cana-2735 134 4 find find VERB cana-2735 134 5 β„Ž-covariant β„Ž-covariant ADJ cana-2735 134 6 derivatives derivative NOUN cana-2735 134 7 of of ADP cana-2735 134 8 the the DET cana-2735 134 9 coefficients coefficient NOUN cana-2735 134 10 of of ADP cana-2735 134 11 cartan cartan ADJ cana-2735 134 12 torsion torsion NOUN cana-2735 134 13 tensor tensor NOUN cana-2735 134 14 πΆπ‘–π‘—π‘˜ πΆπ‘–π‘—π‘˜ PROPN cana-2735 134 15 and and CCONJ cana-2735 134 16 then then ADV cana-2735 134 17 use use VERB cana-2735 134 18 conditions condition NOUN cana-2735 134 19 of of ADP cana-2735 134 20 β„Ž-metrical β„Ž-metrical ADJ cana-2735 134 21 d d NOUN cana-2735 134 22 - - PUNCT cana-2735 134 23 connection connection NOUN cana-2735 134 24 𝐷γ 𝐷γ PROPN cana-2735 134 25 of of ADP cana-2735 134 26 cartan cartan ADJ cana-2735 134 27 space space NOUN cana-2735 134 28 and and CCONJ cana-2735 134 29 equation equation NOUN cana-2735 134 30 ( ( PUNCT cana-2735 134 31 7 7 NUM cana-2735 134 32 ) ) PUNCT cana-2735 134 33 as as SCONJ cana-2735 134 34 follows follow VERB cana-2735 134 35 : : PUNCT cana-2735 134 36 ∡ ∡ NOUN cana-2735 134 37 π‘Ÿβˆ’1 π‘Ÿβˆ’1 PROPN cana-2735 135 1 = = PUNCT cana-2735 135 2 βˆ’4π‘˜2𝛽2 βˆ’4π‘˜2𝛽2 PROPN cana-2735 135 3 𝛼3 𝛼3 ADV cana-2735 135 4 ∴ ∴ PROPN cana-2735 135 5 π‘Ÿβˆ’1|β„Ž π‘Ÿβˆ’1|β„Ž PROPN cana-2735 135 6 = = SYM cana-2735 135 7 0 0 NUM cana-2735 135 8 ( ( PUNCT cana-2735 135 9 14 14 NUM cana-2735 135 10 ) ) PUNCT cana-2735 135 11 ∡ ∡ NOUN cana-2735 135 12 π‘Ÿβˆ’2 π‘Ÿβˆ’2 PROPN cana-2735 135 13 = = PUNCT cana-2735 136 1 6π‘˜2𝛽2βˆ’2π‘˜π›Ό2 6π‘˜2𝛽2βˆ’2π‘˜π›Ό2 NOUN cana-2735 136 2 𝛼5 𝛼5 ADP cana-2735 136 3 ∴ ∴ NOUN cana-2735 136 4 π‘Ÿβˆ’2|β„Ž π‘Ÿβˆ’2|β„Ž NOUN cana-2735 136 5 = = SYM cana-2735 136 6 0 0 NUM cana-2735 136 7 ( ( PUNCT cana-2735 136 8 15 15 NUM cana-2735 136 9 ) ) PUNCT cana-2735 136 10 ∡ ∡ NOUN cana-2735 136 11 π‘Ÿβˆ’3 π‘Ÿβˆ’3 PROPN cana-2735 136 12 = = SYM cana-2735 136 13 6π‘˜π›Ό2π›½βˆ’10π‘˜2𝛽3 6π‘˜π›Ό2π›½βˆ’10π‘˜2𝛽3 NUM cana-2735 137 1 𝛼7 𝛼7 NOUN cana-2735 137 2 ∴ ∴ NOUN cana-2735 137 3 π‘Ÿβˆ’3|β„Ž π‘Ÿβˆ’3|β„Ž NOUN cana-2735 137 4 = = SYM cana-2735 137 5 0 0 NUM cana-2735 137 6 ( ( PUNCT cana-2735 137 7 16 16 NUM cana-2735 137 8 ) ) PUNCT cana-2735 137 9 ∡ ∡ NOUN cana-2735 137 10 π‘Ÿβˆ’4 π‘Ÿβˆ’4 PROPN cana-2735 137 11 = = PUNCT cana-2735 137 12 35π‘˜2𝛽4βˆ’30π‘˜π›Ό2𝛽2 35π‘˜2𝛽4βˆ’30π‘˜π›Ό2𝛽2 NUM cana-2735 137 13 + + NOUN cana-2735 137 14 3𝛼4 3𝛼4 NUM cana-2735 137 15 2𝛼9 2𝛼9 NUM cana-2735 137 16 ∴ ∴ NOUN cana-2735 137 17 π‘Ÿβˆ’4|β„Ž π‘Ÿβˆ’4|β„Ž NOUN cana-2735 137 18 = = SYM cana-2735 137 19 0 0 NUM cana-2735 137 20 ( ( PUNCT cana-2735 137 21 17 17 NUM cana-2735 137 22 ) ) PUNCT cana-2735 137 23 now now ADV cana-2735 137 24 we we PRON cana-2735 137 25 calculate calculate VERB cana-2735 137 26 the the DET cana-2735 137 27 value value NOUN cana-2735 137 28 of of ADP cana-2735 137 29 β„Ž-covariant β„Ž-covariant ADJ cana-2735 137 30 derivative derivative NOUN cana-2735 137 31 of of ADP cana-2735 137 32 𝑑-tensor 𝑑-tensor NOUN cana-2735 137 33 field field NOUN cana-2735 137 34 𝐢𝑖 𝐢𝑖 PROPN cana-2735 137 35 π‘—π‘˜ π‘—π‘˜ ADP cana-2735 137 36 of of ADP cana-2735 137 37 type type NOUN cana-2735 137 38 ( ( PUNCT cana-2735 137 39 2,1 2,1 NUM cana-2735 137 40 ) ) PUNCT cana-2735 137 41 under under ADP cana-2735 137 42 the the DET cana-2735 137 43 assumption assumption NOUN cana-2735 137 44 of of ADP cana-2735 137 45 β„Ž-metrical β„Ž-metrical PROPN cana-2735 137 46 𝑑-connection 𝑑-connection NOUN cana-2735 137 47 as as SCONJ cana-2735 137 48 follows follow VERB cana-2735 137 49 : : PUNCT cana-2735 137 50 ∡ ∡ X cana-2735 137 51 πΆπ‘˜ πΆπ‘˜ VERB cana-2735 137 52 𝑖𝑗 𝑖𝑗 NOUN cana-2735 137 53 = = PUNCT cana-2735 137 54 π‘”π‘˜π‘ŸπΆπ‘Ÿπ‘–π‘— π‘”π‘˜π‘ŸπΆπ‘Ÿπ‘–π‘— PROPN cana-2735 137 55 communications communication NOUN cana-2735 137 56 on on ADP cana-2735 137 57 applied apply VERB cana-2735 137 58 nonlinear nonlinear ADJ cana-2735 137 59 analysis analysis NOUN cana-2735 137 60 issn issn NOUN cana-2735 137 61 : : PUNCT cana-2735 137 62 1074 1074 NUM cana-2735 137 63 - - PUNCT cana-2735 137 64 133x 133x NUM cana-2735 137 65 vol vol NOUN cana-2735 137 66 32 32 NUM cana-2735 137 67 no no NOUN cana-2735 137 68 . . PUNCT cana-2735 138 1 4s 4s NUM cana-2735 138 2 ( ( PUNCT cana-2735 138 3 2025 2025 NUM cana-2735 138 4 ) ) PUNCT cana-2735 138 5 8 8 NUM cana-2735 138 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 138 7 ∴ ∴ PROPN cana-2735 138 8 πΆπ‘˜|β„Ž πΆπ‘˜|β„Ž PROPN cana-2735 138 9 𝑖𝑗 𝑖𝑗 ADP cana-2735 138 10 = = PUNCT cana-2735 138 11 ( ( PUNCT cana-2735 138 12 π‘”π‘˜π‘ŸπΆπ‘Ÿπ‘–π‘— π‘”π‘˜π‘ŸπΆπ‘Ÿπ‘–π‘— PROPN cana-2735 138 13 ) ) PUNCT cana-2735 138 14 |β„Ž |β„Ž X cana-2735 138 15 = = SYM cana-2735 138 16 π‘”π‘˜π‘Ÿ π‘”π‘˜π‘Ÿ PROPN cana-2735 138 17 Γ— Γ— PROPN cana-2735 138 18 𝐢|β„Ž 𝐢|β„Ž PROPN cana-2735 138 19 π‘Ÿπ‘–π‘— π‘Ÿπ‘–π‘— VERB cana-2735 138 20 + + X cana-2735 139 1 πΆπ‘Ÿπ‘–π‘— πΆπ‘Ÿπ‘–π‘— PROPN cana-2735 139 2 Γ— Γ— NOUN cana-2735 139 3 0π‘”π‘˜π‘Ÿ|β„Ž 0π‘”π‘˜π‘Ÿ|β„Ž NOUN cana-2735 139 4 = = SYM cana-2735 140 1 π‘”π‘˜π‘ŸπΆ|β„Ž π‘”π‘˜π‘ŸπΆ|β„Ž PROPN cana-2735 140 2 π‘Ÿπ‘–π‘— π‘Ÿπ‘–π‘— VERB cana-2735 140 3 = = PUNCT cana-2735 141 1 βˆ’π‘”π‘˜π‘Ÿ βˆ’π‘”π‘˜π‘Ÿ PRON cana-2735 141 2 1 1 NUM cana-2735 141 3 2 2 NUM cana-2735 142 1 [ [ X cana-2735 142 2 π‘Ÿβˆ’1π‘π‘Ÿπ‘π‘–π‘π‘— π‘Ÿβˆ’1π‘π‘Ÿπ‘π‘–π‘π‘— NOUN cana-2735 142 3 + + CCONJ cana-2735 142 4 π‘Ÿβˆ’2π‘π‘Ÿπ‘π‘–πœ”π‘— π‘Ÿβˆ’2π‘π‘Ÿπ‘π‘–πœ”π‘— NOUN cana-2735 142 5 + + CCONJ cana-2735 142 6 π‘Ÿβˆ’3π‘π‘Ÿπœ”π‘–πœ”π‘— π‘Ÿβˆ’3π‘π‘Ÿπœ”π‘–πœ”π‘— NOUN cana-2735 142 7 + + CCONJ cana-2735 142 8 π‘Ÿβˆ’4πœ”π‘Ÿπœ”π‘–πœ”π‘— π‘Ÿβˆ’4πœ”π‘Ÿπœ”π‘–πœ”π‘— NOUN cana-2735 142 9 + + CCONJ cana-2735 142 10 πœŒβˆ’1π‘Žπ‘Ÿπ‘–π‘π‘— πœŒβˆ’1π‘Žπ‘Ÿπ‘–π‘π‘— NOUN cana-2735 142 11 + + CCONJ cana-2735 142 12 πœŒβˆ’2π‘Žπ‘Ÿπ‘–πœ”π‘— πœŒβˆ’2π‘Žπ‘Ÿπ‘–πœ”π‘— NOUN cana-2735 142 13 + + CCONJ cana-2735 142 14 π‘Ÿ|𝑖|𝑗]|β„Ž π‘Ÿ|𝑖|𝑗]|β„Ž ADJ cana-2735 142 15 = = PUNCT cana-2735 142 16 βˆ’π‘”π‘˜π‘Ÿ βˆ’π‘”π‘˜π‘Ÿ PRON cana-2735 143 1 1 1 NUM cana-2735 143 2 2 2 NUM cana-2735 144 1 [ [ X cana-2735 144 2 π‘Ÿβˆ’1 π‘Ÿβˆ’1 PROPN cana-2735 144 3 Γ— Γ— NOUN cana-2735 144 4 ( ( PUNCT cana-2735 144 5 π‘π‘Ÿπ‘π‘–π‘π‘—)|β„Ž π‘π‘Ÿπ‘π‘–π‘π‘—)|β„Ž NOUN cana-2735 144 6 + + CCONJ cana-2735 144 7 π‘π‘Ÿπ‘π‘–π‘π‘— π‘π‘Ÿπ‘π‘–π‘π‘— VERB cana-2735 144 8 Γ— Γ— NOUN cana-2735 144 9 0π‘Ÿβˆ’1|β„Ž 0π‘Ÿβˆ’1|β„Ž NOUN cana-2735 145 1 + + CCONJ cana-2735 145 2 π‘Ÿβˆ’2 π‘Ÿβˆ’2 PROPN cana-2735 145 3 Γ— Γ— NOUN cana-2735 145 4 ( ( PUNCT cana-2735 145 5 π‘π‘Ÿπ‘π‘–πœ”π‘—)|β„Ž π‘π‘Ÿπ‘π‘–πœ”π‘—)|β„Ž PROPN cana-2735 145 6 + + NUM cana-2735 145 7 π‘π‘Ÿπ‘π‘–πœ”π‘— π‘π‘Ÿπ‘π‘–πœ”π‘— ADJ cana-2735 145 8 Γ— Γ— NOUN cana-2735 145 9 0π‘Ÿβˆ’2|β„Ž 0π‘Ÿβˆ’2|β„Ž NOUN cana-2735 145 10 + + CCONJ cana-2735 145 11 π‘Ÿβˆ’3 π‘Ÿβˆ’3 INTJ cana-2735 145 12 Γ— Γ— NOUN cana-2735 145 13 ( ( PUNCT cana-2735 145 14 π‘π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž π‘π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž PROPN cana-2735 145 15 + + CCONJ cana-2735 145 16 π‘π‘Ÿπœ”π‘–πœ”π‘— π‘π‘Ÿπœ”π‘–πœ”π‘— ADJ cana-2735 145 17 Γ— Γ— NOUN cana-2735 145 18 0π‘Ÿβˆ’3|β„Ž 0π‘Ÿβˆ’3|β„Ž NOUN cana-2735 145 19 + + CCONJ cana-2735 145 20 π‘Ÿβˆ’4 π‘Ÿβˆ’4 PROPN cana-2735 145 21 Γ— Γ— NOUN cana-2735 145 22 ( ( PUNCT cana-2735 145 23 πœ”π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž πœ”π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž NOUN cana-2735 145 24 + + NUM cana-2735 145 25 πœ”π‘Ÿπœ”π‘–πœ”π‘— πœ”π‘Ÿπœ”π‘–πœ”π‘— NOUN cana-2735 145 26 Γ— Γ— NOUN cana-2735 145 27 0π‘Ÿβˆ’4|β„Ž 0π‘Ÿβˆ’4|β„Ž NOUN cana-2735 145 28 + + CCONJ cana-2735 145 29 πœŒβˆ’1 πœŒβˆ’1 PROPN cana-2735 145 30 Γ— Γ— NOUN cana-2735 145 31 ( ( PUNCT cana-2735 145 32 π‘Žπ‘Ÿπ‘–π‘π‘—)|β„Ž π‘Žπ‘Ÿπ‘–π‘π‘—)|β„Ž NOUN cana-2735 145 33 + + CCONJ cana-2735 145 34 π‘Žπ‘Ÿπ‘–π‘π‘— π‘Žπ‘Ÿπ‘–π‘π‘— VERB cana-2735 145 35 Γ— Γ— NOUN cana-2735 145 36 0πœŒβˆ’1|β„Ž 0πœŒβˆ’1|β„Ž NOUN cana-2735 145 37 + + CCONJ cana-2735 145 38 πœŒβˆ’2 πœŒβˆ’2 PROPN cana-2735 145 39 Γ— Γ— NOUN cana-2735 145 40 ( ( PUNCT cana-2735 145 41 π‘Žπ‘Ÿπ‘–πœ”π‘—)|β„Ž π‘Žπ‘Ÿπ‘–πœ”π‘—)|β„Ž PROPN cana-2735 145 42 + + X cana-2735 145 43 π‘Žπ‘Ÿπ‘–πœ”π‘— π‘Žπ‘Ÿπ‘–πœ”π‘— NOUN cana-2735 145 44 Γ— Γ— NOUN cana-2735 145 45 0πœŒβˆ’2|β„Ž 0πœŒβˆ’2|β„Ž PRON cana-2735 145 46 + + CCONJ cana-2735 145 47 ( ( PUNCT cana-2735 145 48 π‘Ÿ|𝑖|𝑗)|β„Ž π‘Ÿ|𝑖|𝑗)|β„Ž NUM cana-2735 145 49 ] ] X cana-2735 145 50 = = X cana-2735 146 1 βˆ’π‘”π‘˜π‘Ÿ βˆ’π‘”π‘˜π‘Ÿ PRON cana-2735 146 2 1 1 NUM cana-2735 146 3 2 2 NUM cana-2735 146 4 [ [ X cana-2735 146 5 π‘Ÿβˆ’1(π‘π‘Ÿπ‘π‘–π‘π‘—)|β„Ž π‘Ÿβˆ’1(π‘π‘Ÿπ‘π‘–π‘π‘—)|β„Ž NOUN cana-2735 146 6 + + CCONJ cana-2735 146 7 π‘Ÿβˆ’2(π‘π‘Ÿπ‘π‘–πœ”π‘—)|β„Ž π‘Ÿβˆ’2(π‘π‘Ÿπ‘π‘–πœ”π‘—)|β„Ž NOUN cana-2735 146 8 + + CCONJ cana-2735 146 9 π‘Ÿβˆ’3(π‘π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž π‘Ÿβˆ’3(π‘π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž NUM cana-2735 146 10 + + NUM cana-2735 146 11 π‘Ÿβˆ’4(πœ”π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž π‘Ÿβˆ’4(πœ”π‘Ÿπœ”π‘–πœ”π‘—)|β„Ž NOUN cana-2735 146 12 + + CCONJ cana-2735 146 13 πœŒβˆ’1(π‘Žπ‘Ÿπ‘–π‘π‘—)|β„Ž πœŒβˆ’1(π‘Žπ‘Ÿπ‘–π‘π‘—)|β„Ž ADJ cana-2735 146 14 + + CCONJ cana-2735 146 15 πœŒβˆ’2(π‘Žπ‘Ÿπ‘–πœ”π‘—)|β„Ž πœŒβˆ’2(π‘Žπ‘Ÿπ‘–πœ”π‘—)|β„Ž X cana-2735 146 16 + + X cana-2735 146 17 ( ( PUNCT cana-2735 146 18 π‘Ÿ|𝑖|𝑗)|β„Ž π‘Ÿ|𝑖|𝑗)|β„Ž NUM cana-2735 146 19 ] ] X cana-2735 146 20 = = X cana-2735 147 1 βˆ’π‘”π‘˜π‘Ÿ βˆ’π‘”π‘˜π‘Ÿ PRON cana-2735 147 2 1 1 NUM cana-2735 147 3 2 2 NUM cana-2735 148 1 [ [ X cana-2735 148 2 π‘Ÿβˆ’1(π‘π‘Ÿπ‘π‘–0𝑏|β„Ž π‘Ÿβˆ’1(π‘π‘Ÿπ‘π‘–0𝑏|β„Ž ADP cana-2735 148 3 𝑗 𝑗 X cana-2735 148 4 + + PUNCT cana-2735 148 5 π‘π‘Ÿπ‘π‘—0𝑏|β„Ž π‘π‘Ÿπ‘π‘—0𝑏|β„Ž NOUN cana-2735 148 6 𝑖 𝑖 SYM cana-2735 148 7 + + NUM cana-2735 148 8 𝑏𝑖𝑏𝑗0𝑏|β„Ž 𝑏𝑖𝑏𝑗0𝑏|β„Ž ADJ cana-2735 148 9 π‘Ÿ π‘Ÿ NOUN cana-2735 148 10 ) ) PUNCT cana-2735 149 1 + + CCONJ cana-2735 149 2 π‘Ÿβˆ’2(π‘π‘Ÿπ‘π‘–0πœ”|β„Ž π‘Ÿβˆ’2(π‘π‘Ÿπ‘π‘–0πœ”|β„Ž PROPN cana-2735 149 3 𝑗 𝑗 PROPN cana-2735 149 4 + + NOUN cana-2735 149 5 π‘π‘Ÿπœ”π‘—0𝑏|β„Ž π‘π‘Ÿπœ”π‘—0𝑏|β„Ž NOUN cana-2735 149 6 𝑖 𝑖 NOUN cana-2735 149 7 + + NOUN cana-2735 149 8 π‘π‘–πœ”π‘—0𝑏|β„Ž π‘π‘–πœ”π‘—0𝑏|β„Ž X cana-2735 149 9 π‘Ÿ π‘Ÿ NOUN cana-2735 149 10 ) ) PUNCT cana-2735 150 1 + + CCONJ cana-2735 150 2 π‘Ÿβˆ’3(π‘π‘Ÿπœ”π‘–0πœ”|β„Ž π‘Ÿβˆ’3(π‘π‘Ÿπœ”π‘–0πœ”|β„Ž NOUN cana-2735 150 3 𝑗 𝑗 VERB cana-2735 150 4 + + NOUN cana-2735 150 5 π‘π‘Ÿπœ”π‘—0πœ”|β„Ž π‘π‘Ÿπœ”π‘—0πœ”|β„Ž PROPN cana-2735 150 6 𝑖 𝑖 PROPN cana-2735 150 7 + + NOUN cana-2735 150 8 πœ”π‘–πœ”π‘—0𝑏|β„Ž πœ”π‘–πœ”π‘—0𝑏|β„Ž VERB cana-2735 150 9 π‘Ÿ π‘Ÿ NOUN cana-2735 150 10 ) ) PUNCT cana-2735 151 1 + + CCONJ cana-2735 151 2 π‘Ÿβˆ’4(πœ”π‘Ÿπœ”π‘–0πœ”|β„Ž π‘Ÿβˆ’4(πœ”π‘Ÿπœ”π‘–0πœ”|β„Ž X cana-2735 151 3 𝑗 𝑗 X cana-2735 151 4 + + X cana-2735 151 5 πœ”π‘Ÿπœ”π‘—0πœ”|β„Ž πœ”π‘Ÿπœ”π‘—0πœ”|β„Ž PRON cana-2735 151 6 𝑖 𝑖 X cana-2735 151 7 + + NUM cana-2735 151 8 πœ”π‘–πœ”π‘—0πœ”|β„Ž πœ”π‘–πœ”π‘—0πœ”|β„Ž PROPN cana-2735 151 9 π‘Ÿ π‘Ÿ NOUN cana-2735 151 10 ) ) PUNCT cana-2735 152 1 + + CCONJ cana-2735 152 2 πœŒβˆ’1(π‘Žπ‘Ÿπ‘–0𝑏|β„Ž πœŒβˆ’1(π‘Žπ‘Ÿπ‘–0𝑏|β„Ž NUM cana-2735 152 3 𝑗 𝑗 PROPN cana-2735 152 4 + + NUM cana-2735 152 5 𝑏𝑗0π‘Ž|β„Ž 𝑏𝑗0π‘Ž|β„Ž PROPN cana-2735 152 6 π‘Ÿπ‘– π‘Ÿπ‘– PROPN cana-2735 152 7 ) ) PUNCT cana-2735 152 8 + + CCONJ cana-2735 152 9 πœŒβˆ’2(π‘Žπ‘Ÿπ‘–0πœ”|β„Ž πœŒβˆ’2(π‘Žπ‘Ÿπ‘–0πœ”|β„Ž PROPN cana-2735 152 10 𝑗 𝑗 PROPN cana-2735 152 11 + + X cana-2735 152 12 πœ”π‘—0π‘Ž|β„Ž πœ”π‘—0π‘Ž|β„Ž X cana-2735 152 13 π‘Ÿπ‘– π‘Ÿπ‘– X cana-2735 152 14 ) ) PUNCT cana-2735 152 15 + + CCONJ cana-2735 152 16 0(π‘Ÿ|𝑖|𝑗)|β„Ž 0(π‘Ÿ|𝑖|𝑗)|β„Ž NOUN cana-2735 152 17 ] ] X cana-2735 152 18 πΆπ‘˜|β„Ž πΆπ‘˜|β„Ž PROPN cana-2735 152 19 𝑖𝑗 𝑖𝑗 ADP cana-2735 152 20 = = SYM cana-2735 152 21 0 0 NUM cana-2735 152 22 ( ( PUNCT cana-2735 152 23 18 18 NUM cana-2735 152 24 ) ) PUNCT cana-2735 152 25 thus thus ADV cana-2735 152 26 we we PRON cana-2735 152 27 shown show VERB cana-2735 152 28 that that SCONJ cana-2735 152 29 cartan cartan PROPN cana-2735 152 30 torsion torsion NOUN cana-2735 152 31 tensor tensor NOUN cana-2735 152 32 vanishes vanish VERB cana-2735 152 33 under under ADP cana-2735 152 34 the the DET cana-2735 152 35 assumption assumption NOUN cana-2735 152 36 of of ADP cana-2735 152 37 β„Ž-metrical β„Ž-metrical PROPN cana-2735 152 38 𝑑-connection 𝑑-connection PROPN cana-2735 152 39 . . PUNCT cana-2735 153 1 one one NUM cana-2735 153 2 knows know VERB cana-2735 153 3 that that SCONJ cana-2735 153 4 a a DET cana-2735 153 5 cartan cartan ADJ cana-2735 153 6 space space NOUN cana-2735 153 7 𝐢 𝐢 PROPN cana-2735 153 8 is be AUX cana-2735 153 9 berwald berwald NOUN cana-2735 153 10 space space NOUN cana-2735 153 11 if if SCONJ cana-2735 153 12 and and CCONJ cana-2735 153 13 only only ADV cana-2735 153 14 if if SCONJ cana-2735 153 15 πΆπ‘˜|β„Ž πΆπ‘˜|β„Ž PROPN cana-2735 153 16 𝑖𝑗 𝑖𝑗 ADP cana-2735 153 17 = = NOUN cana-2735 153 18 0 0 PUNCT cana-2735 154 1 [ [ X cana-2735 154 2 12 12 NUM cana-2735 154 3 ] ] PUNCT cana-2735 154 4 . . PUNCT cana-2735 155 1 hence hence ADV cana-2735 155 2 from from ADP cana-2735 155 3 equation equation NOUN cana-2735 155 4 ( ( PUNCT cana-2735 155 5 18 18 NUM cana-2735 155 6 ) ) PUNCT cana-2735 155 7 , , PUNCT cana-2735 155 8 we we PRON cana-2735 155 9 have have VERB cana-2735 155 10 the the DET cana-2735 155 11 following follow VERB cana-2735 155 12 proposition proposition NOUN cana-2735 155 13 : : PUNCT cana-2735 155 14 proposition proposition NOUN cana-2735 155 15 3.2 3.2 NUM cana-2735 155 16 a a DET cana-2735 155 17 cartan cartan ADJ cana-2735 155 18 space space NOUN cana-2735 155 19 𝐢 𝐢 PROPN cana-2735 155 20 with with ADP cana-2735 155 21 the the DET cana-2735 155 22 ( ( PUNCT cana-2735 155 23 𝛼 𝛼 PROPN cana-2735 155 24 , , PUNCT cana-2735 155 25 𝛽 𝛽 NOUN cana-2735 155 26 ) ) PUNCT cana-2735 155 27 -metric -metric NOUN cana-2735 155 28 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 155 29 , , PUNCT cana-2735 155 30 πœ” πœ” ADJ cana-2735 155 31 ) ) PUNCT cana-2735 155 32 = = SYM cana-2735 156 1 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 156 2 , , PUNCT cana-2735 156 3 πœ” πœ” PRON cana-2735 156 4 ) ) PUNCT cana-2735 156 5 + + NUM cana-2735 156 6 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 156 7 , , PUNCT cana-2735 156 8 πœ” πœ” PRON cana-2735 156 9 ) ) PUNCT cana-2735 156 10 + + CCONJ cana-2735 156 11 2π‘˜ 2π‘˜ NUM cana-2735 156 12 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 156 13 ) ) PUNCT cana-2735 156 14 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 156 15 ) ) PUNCT cana-2735 156 16 βˆ’ βˆ’ PROPN cana-2735 157 1 π‘˜2 π‘˜2 PROPN cana-2735 157 2 3 3 NUM cana-2735 157 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 157 4 ) ) PUNCT cana-2735 157 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 157 6 ) ) PUNCT cana-2735 157 7 admitting admit VERB cana-2735 157 8 h h NOUN cana-2735 157 9 - - ADJ cana-2735 157 10 metrical metrical ADJ cana-2735 157 11 d d NOUN cana-2735 157 12 - - PUNCT cana-2735 157 13 connection connection NOUN cana-2735 157 14 is be AUX cana-2735 157 15 a a DET cana-2735 157 16 berwald berwald ADJ cana-2735 157 17 space space NOUN cana-2735 157 18 in in ADP cana-2735 157 19 [ [ X cana-2735 157 20 12 12 NUM cana-2735 157 21 ] ] PUNCT cana-2735 157 22 , , PUNCT cana-2735 157 23 it it PRON cana-2735 157 24 is be AUX cana-2735 157 25 deduced deduce VERB cana-2735 157 26 that that SCONJ cana-2735 157 27 a a DET cana-2735 157 28 locally locally ADV cana-2735 157 29 minkowski minkowski ADJ cana-2735 157 30 space space NOUN cana-2735 157 31 is be AUX cana-2735 157 32 a a DET cana-2735 157 33 berwald berwald ADJ cana-2735 157 34 space space NOUN cana-2735 157 35 in in ADP cana-2735 157 36 which which PRON cana-2735 157 37 curvature curvature NOUN cana-2735 157 38 tensor tensor NOUN cana-2735 157 39 vanishes vanish VERB cana-2735 157 40 . . PUNCT cana-2735 158 1 hence hence ADV cana-2735 158 2 , , PUNCT cana-2735 158 3 from from ADP cana-2735 158 4 the the DET cana-2735 158 5 propositions proposition NOUN cana-2735 158 6 3.1 3.1 NUM cana-2735 158 7 and and CCONJ cana-2735 158 8 3.2 3.2 NUM cana-2735 158 9 , , PUNCT cana-2735 158 10 we we PRON cana-2735 158 11 have have AUX cana-2735 158 12 following follow VERB cana-2735 158 13 theorem theorem VERB cana-2735 158 14 : : PUNCT cana-2735 158 15 theorem theorem VERB cana-2735 158 16 3.3 3.3 NUM cana-2735 158 17 a a DET cana-2735 158 18 cartan cartan ADJ cana-2735 158 19 space space NOUN cana-2735 158 20 𝐢 𝐢 PROPN cana-2735 158 21 with with ADP cana-2735 158 22 the the DET cana-2735 158 23 ( ( PUNCT cana-2735 158 24 𝛼 𝛼 PROPN cana-2735 158 25 , , PUNCT cana-2735 158 26 𝛽 𝛽 NOUN cana-2735 158 27 ) ) PUNCT cana-2735 158 28 -metric -metric NOUN cana-2735 158 29 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 158 30 , , PUNCT cana-2735 158 31 πœ” πœ” ADJ cana-2735 158 32 ) ) PUNCT cana-2735 158 33 = = SYM cana-2735 158 34 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 158 35 , , PUNCT cana-2735 158 36 πœ” πœ” PRON cana-2735 158 37 ) ) PUNCT cana-2735 158 38 + + NUM cana-2735 158 39 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 158 40 , , PUNCT cana-2735 158 41 πœ” πœ” PRON cana-2735 158 42 ) ) PUNCT cana-2735 158 43 + + CCONJ cana-2735 158 44 2π‘˜ 2π‘˜ NUM cana-2735 158 45 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 158 46 ) ) PUNCT cana-2735 158 47 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 158 48 ) ) PUNCT cana-2735 158 49 βˆ’ βˆ’ PROPN cana-2735 159 1 π‘˜2 π‘˜2 PROPN cana-2735 159 2 3 3 NUM cana-2735 159 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 159 4 ) ) PUNCT cana-2735 159 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 159 6 ) ) PUNCT cana-2735 159 7 admitting admit VERB cana-2735 159 8 β„Ž-metrical β„Ž-metrical ADJ cana-2735 159 9 𝑑-connection 𝑑-connection NOUN cana-2735 159 10 is be AUX cana-2735 159 11 locally locally ADV cana-2735 159 12 minkowski minkowski ADJ cana-2735 159 13 space space NOUN cana-2735 159 14 if if SCONJ cana-2735 159 15 and and CCONJ cana-2735 159 16 only only ADV cana-2735 159 17 if if SCONJ cana-2735 159 18 the the DET cana-2735 159 19 associated associated ADJ cana-2735 159 20 riemannian riemannian ADJ cana-2735 159 21 space space NOUN cana-2735 159 22 is be AUX cana-2735 159 23 locally locally ADV cana-2735 159 24 flat flat ADJ cana-2735 159 25 . . PUNCT cana-2735 160 1 communications communication NOUN cana-2735 160 2 on on ADP cana-2735 160 3 applied apply VERB cana-2735 160 4 nonlinear nonlinear ADJ cana-2735 160 5 analysis analysis NOUN cana-2735 160 6 issn issn NOUN cana-2735 160 7 : : PUNCT cana-2735 160 8 1074 1074 NUM cana-2735 160 9 - - PUNCT cana-2735 160 10 133x 133x NUM cana-2735 160 11 vol vol NOUN cana-2735 160 12 32 32 NUM cana-2735 160 13 no no NOUN cana-2735 160 14 . . PUNCT cana-2735 161 1 4s 4s NUM cana-2735 161 2 ( ( PUNCT cana-2735 161 3 2025 2025 NUM cana-2735 161 4 ) ) PUNCT cana-2735 161 5 9 9 NUM cana-2735 161 6 https://internationalpubls.com https://internationalpubls.com SYM cana-2735 161 7 4 4 NUM cana-2735 161 8 conformal conformal NOUN cana-2735 161 9 change change NOUN cana-2735 161 10 of of ADP cana-2735 161 11 cartan cartan ADJ cana-2735 161 12 space space NOUN cana-2735 161 13 with with ADP cana-2735 161 14 an an DET cana-2735 161 15 ( ( PUNCT cana-2735 161 16 𝜢 𝜢 NOUN cana-2735 161 17 , , PUNCT cana-2735 161 18 𝜷)-metric 𝜷)-metric VERB cana-2735 161 19 in in ADP cana-2735 161 20 this this DET cana-2735 161 21 section section NOUN cana-2735 161 22 our our PRON cana-2735 161 23 aim aim NOUN cana-2735 161 24 is be AUX cana-2735 161 25 to to PART cana-2735 161 26 conformally conformally ADV cana-2735 161 27 transform transform VERB cana-2735 161 28 a a DET cana-2735 161 29 cartan cartan ADJ cana-2735 161 30 space space NOUN cana-2735 161 31 ( ( PUNCT cana-2735 161 32 𝑀 𝑀 PROPN cana-2735 161 33 , , PUNCT cana-2735 161 34 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 161 35 , , PUNCT cana-2735 161 36 πœ” πœ” NOUN cana-2735 161 37 ) ) PUNCT cana-2735 161 38 ) ) PUNCT cana-2735 161 39 to to ADP cana-2735 161 40 another another DET cana-2735 161 41 cartan cartan ADJ cana-2735 161 42 space space NOUN cana-2735 161 43 ( ( PUNCT cana-2735 161 44 𝑀 𝑀 PROPN cana-2735 161 45 , , PUNCT cana-2735 161 46 οΏ½ οΏ½ PROPN cana-2735 161 47 Μƒ Μƒ PROPN cana-2735 161 48 οΏ½ οΏ½ PROPN cana-2735 161 49 (π‘₯ (π‘₯ NOUN cana-2735 161 50 , , PUNCT cana-2735 161 51 πœ” πœ” NOUN cana-2735 161 52 ) ) PUNCT cana-2735 161 53 ) ) PUNCT cana-2735 161 54 and and CCONJ cana-2735 161 55 then then ADV cana-2735 161 56 to to PART cana-2735 161 57 determine determine VERB cana-2735 161 58 the the DET cana-2735 161 59 nature nature NOUN cana-2735 161 60 of of ADP cana-2735 161 61 curvature curvature NOUN cana-2735 161 62 tensor tensor NOUN cana-2735 161 63 οΏ½ οΏ½ PROPN cana-2735 161 64 Μƒ Μƒ PROPN cana-2735 161 65 οΏ½ οΏ½ NOUN cana-2735 161 66 β„Žπ‘—π‘˜ β„Žπ‘—π‘˜ NOUN cana-2735 161 67 𝑖 𝑖 SYM cana-2735 161 68 in in ADP cana-2735 161 69 the the DET cana-2735 161 70 conformally conformally ADV cana-2735 161 71 transformed transform VERB cana-2735 161 72 space space NOUN cana-2735 161 73 ( ( PUNCT cana-2735 161 74 𝑀 𝑀 PROPN cana-2735 161 75 , , PUNCT cana-2735 161 76 οΏ½ οΏ½ PROPN cana-2735 161 77 Μƒ Μƒ PROPN cana-2735 161 78 οΏ½ οΏ½ PROPN cana-2735 161 79 (π‘₯ (π‘₯ NOUN cana-2735 161 80 , , PUNCT cana-2735 161 81 πœ” πœ” NOUN cana-2735 161 82 ) ) PUNCT cana-2735 161 83 ) ) PUNCT cana-2735 161 84 under under ADP cana-2735 161 85 the the DET cana-2735 161 86 influence influence NOUN cana-2735 161 87 of of ADP cana-2735 161 88 β„Ž β„Ž PROPN cana-2735 161 89 -metrical -metrical ADJ cana-2735 161 90 𝑑 𝑑 NOUN cana-2735 161 91 -connection -connection NOUN cana-2735 161 92 on on ADP cana-2735 161 93 the the DET cana-2735 161 94 original original ADJ cana-2735 161 95 cartan cartan ADJ cana-2735 161 96 space space NOUN cana-2735 161 97 ( ( PUNCT cana-2735 161 98 𝑀 𝑀 PROPN cana-2735 161 99 , , PUNCT cana-2735 161 100 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 161 101 , , PUNCT cana-2735 161 102 πœ” πœ” NOUN cana-2735 161 103 ) ) PUNCT cana-2735 161 104 ) ) PUNCT cana-2735 161 105 . . PUNCT cana-2735 162 1 that that PRON cana-2735 162 2 is be AUX cana-2735 162 3 , , PUNCT cana-2735 162 4 we we PRON cana-2735 162 5 are be AUX cana-2735 162 6 going go VERB cana-2735 162 7 to to PART cana-2735 162 8 determine determine VERB cana-2735 162 9 the the DET cana-2735 162 10 shape shape NOUN cana-2735 162 11 of of ADP cana-2735 162 12 conformally conformally ADV cana-2735 162 13 transformed transform VERB cana-2735 162 14 space space NOUN cana-2735 162 15 ( ( PUNCT cana-2735 162 16 𝑀 𝑀 PROPN cana-2735 162 17 , , PUNCT cana-2735 162 18 οΏ½ οΏ½ PROPN cana-2735 162 19 Μƒ Μƒ PROPN cana-2735 162 20 οΏ½ οΏ½ PROPN cana-2735 162 21 (π‘₯ (π‘₯ NOUN cana-2735 162 22 , , PUNCT cana-2735 162 23 πœ” πœ” NOUN cana-2735 162 24 ) ) PUNCT cana-2735 162 25 ) ) PUNCT cana-2735 162 26 under under ADP cana-2735 162 27 the the DET cana-2735 162 28 stipulation stipulation NOUN cana-2735 162 29 of of ADP cana-2735 162 30 β„Ž-metrical β„Ž-metrical ADJ cana-2735 162 31 𝑑-connection 𝑑-connection NOUN cana-2735 162 32 on on ADP cana-2735 162 33 ( ( PUNCT cana-2735 162 34 𝑀 𝑀 PROPN cana-2735 162 35 , , PUNCT cana-2735 162 36 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 162 37 , , PUNCT cana-2735 162 38 πœ” πœ” NOUN cana-2735 162 39 ) ) PUNCT cana-2735 162 40 ) ) PUNCT cana-2735 162 41 . . PUNCT cana-2735 163 1 for for ADP cana-2735 163 2 that that PRON cana-2735 163 3 , , PUNCT cana-2735 163 4 consider consider VERB cana-2735 163 5 an an DET cana-2735 163 6 𝑛-dimensional 𝑛-dimensional ADJ cana-2735 163 7 cartan cartan PROPN cana-2735 163 8 space space NOUN cana-2735 163 9 𝐢 𝐢 PROPN cana-2735 163 10 = = SYM cana-2735 163 11 ( ( PUNCT cana-2735 163 12 𝑀 𝑀 PROPN cana-2735 163 13 , , PUNCT cana-2735 163 14 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 163 15 , , PUNCT cana-2735 163 16 πœ” πœ” NOUN cana-2735 163 17 ) ) PUNCT cana-2735 163 18 ) ) PUNCT cana-2735 163 19 equipped equip VERB cana-2735 163 20 with with ADP cana-2735 163 21 a a DET cana-2735 163 22 real real ADJ cana-2735 163 23 smooth smooth ADJ cana-2735 163 24 𝑛manifold 𝑛manifold ADJ cana-2735 163 25 𝑀 𝑀 PROPN cana-2735 163 26 and and CCONJ cana-2735 163 27 the the DET cana-2735 163 28 ( ( PUNCT cana-2735 163 29 𝛼 𝛼 PROPN cana-2735 163 30 , , PUNCT cana-2735 163 31 𝛽 𝛽 NOUN cana-2735 163 32 ) ) PUNCT cana-2735 163 33 -metric -metric NOUN cana-2735 163 34 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 163 35 , , PUNCT cana-2735 163 36 πœ” πœ” ADJ cana-2735 163 37 ) ) PUNCT cana-2735 163 38 = = SYM cana-2735 164 1 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 164 2 , , PUNCT cana-2735 164 3 πœ” πœ” PRON cana-2735 164 4 ) ) PUNCT cana-2735 164 5 + + NUM cana-2735 164 6 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 164 7 , , PUNCT cana-2735 164 8 πœ” πœ” PRON cana-2735 164 9 ) ) PUNCT cana-2735 164 10 + + CCONJ cana-2735 164 11 2π‘˜ 2π‘˜ NUM cana-2735 164 12 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 164 13 ) ) PUNCT cana-2735 164 14 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 164 15 ) ) PUNCT cana-2735 164 16 βˆ’ βˆ’ PROPN cana-2735 165 1 π‘˜2 π‘˜2 PROPN cana-2735 165 2 3 3 NUM cana-2735 165 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 165 4 ) ) PUNCT cana-2735 165 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 165 6 ) ) PUNCT cana-2735 165 7 , , PUNCT cana-2735 165 8 where where SCONJ cana-2735 165 9 𝛼 𝛼 X cana-2735 165 10 = = SYM cana-2735 165 11 ( ( PUNCT cana-2735 165 12 π‘Žπ‘–π‘—(π‘₯ π‘Žπ‘–π‘—(π‘₯ PROPN cana-2735 165 13 , , PUNCT cana-2735 165 14 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— PRON cana-2735 165 15 ) ) PUNCT cana-2735 165 16 1 1 NUM cana-2735 165 17 2 2 NUM cana-2735 165 18 and and CCONJ cana-2735 165 19 𝛽 𝛽 NOUN cana-2735 165 20 = = PUNCT cana-2735 165 21 πœ”π‘–π‘ πœ”π‘–π‘ ADJ cana-2735 165 22 𝑖(π‘₯ 𝑖(π‘₯ PROPN cana-2735 165 23 ) ) PUNCT cana-2735 165 24 . . PUNCT cana-2735 166 1 by by ADP cana-2735 166 2 a a DET cana-2735 166 3 conformal conformal ADJ cana-2735 166 4 change change NOUN cana-2735 166 5 𝜎 𝜎 NOUN cana-2735 166 6 : : PUNCT cana-2735 166 7 𝐾 𝐾 PROPN cana-2735 166 8 β†’ β†’ SYM cana-2735 166 9 οΏ½ οΏ½ PROPN cana-2735 166 10 Μƒ Μƒ PROPN cana-2735 166 11 οΏ½ οΏ½ PROPN cana-2735 166 12 such such ADJ cana-2735 166 13 that that PRON cana-2735 166 14 οΏ½ οΏ½ PROPN cana-2735 166 15 Μƒ Μƒ PROPN cana-2735 166 16 οΏ½ οΏ½ PROPN cana-2735 166 17 ( ( PUNCT cana-2735 166 18 οΏ½ οΏ½ PROPN cana-2735 166 19 Μƒ Μƒ PROPN cana-2735 166 20 οΏ½ οΏ½ PROPN cana-2735 166 21 , , PUNCT cana-2735 166 22 𝛽 𝛽 NOUN cana-2735 166 23 ) ) PUNCT cana-2735 166 24 = = SYM cana-2735 166 25 π‘’πœŽπΎ(𝛼 π‘’πœŽπΎ(𝛼 PROPN cana-2735 166 26 , , PUNCT cana-2735 166 27 𝛽 𝛽 NOUN cana-2735 166 28 ) ) PUNCT cana-2735 166 29 , , PUNCT cana-2735 166 30 we we PRON cana-2735 166 31 have have VERB cana-2735 166 32 the the DET cana-2735 166 33 another another DET cana-2735 166 34 cartan cartan ADJ cana-2735 166 35 space space NOUN cana-2735 166 36 οΏ½ οΏ½ PROPN cana-2735 166 37 Μƒ Μƒ PROPN cana-2735 166 38 οΏ½ οΏ½ NOUN cana-2735 166 39 𝑛 𝑛 PRON cana-2735 166 40 = = SYM cana-2735 166 41 ( ( PUNCT cana-2735 166 42 𝑀 𝑀 PROPN cana-2735 166 43 , , PUNCT cana-2735 166 44 οΏ½ οΏ½ PROPN cana-2735 166 45 Μƒ Μƒ PROPN cana-2735 166 46 οΏ½ οΏ½ PROPN cana-2735 166 47 ( ( PUNCT cana-2735 166 48 οΏ½ οΏ½ PROPN cana-2735 166 49 Μƒ Μƒ PROPN cana-2735 166 50 οΏ½ οΏ½ PROPN cana-2735 166 51 , , PUNCT cana-2735 166 52 𝛽 𝛽 NOUN cana-2735 166 53 ) ) PUNCT cana-2735 166 54 ) ) PUNCT cana-2735 166 55 , , PUNCT cana-2735 166 56 where where SCONJ cana-2735 166 57 οΏ½ οΏ½ PROPN cana-2735 166 58 Μƒ Μƒ PROPN cana-2735 166 59 οΏ½ οΏ½ PROPN cana-2735 166 60 = = SYM cana-2735 166 61 π‘’πœŽπ›Ό π‘’πœŽπ›Ό PROPN cana-2735 166 62 and and CCONJ cana-2735 166 63 𝛽 𝛽 NOUN cana-2735 166 64 = = PRON cana-2735 166 65 π‘’πœŽπ›½. π‘’πœŽπ›½. NOUN cana-2735 166 66 putting put VERB cana-2735 166 67 𝛼 𝛼 NOUN cana-2735 166 68 = = SYM cana-2735 166 69 ( ( PUNCT cana-2735 166 70 π‘Žπ‘–π‘—(π‘₯ π‘Žπ‘–π‘—(π‘₯ PROPN cana-2735 166 71 , , PUNCT cana-2735 166 72 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— PRON cana-2735 166 73 ) ) PUNCT cana-2735 166 74 1 1 NUM cana-2735 166 75 2 2 NUM cana-2735 166 76 and and CCONJ cana-2735 166 77 𝛽 𝛽 NOUN cana-2735 166 78 = = PUNCT cana-2735 166 79 πœ”π‘–π‘ πœ”π‘–π‘ ADJ cana-2735 166 80 𝑖(π‘₯ 𝑖(π‘₯ PROPN cana-2735 166 81 ) ) PUNCT cana-2735 166 82 in in ADP cana-2735 166 83 the the DET cana-2735 166 84 above above ADJ cana-2735 166 85 relations relation NOUN cana-2735 166 86 , , PUNCT cana-2735 166 87 we we PRON cana-2735 166 88 get get VERB cana-2735 166 89 οΏ½ οΏ½ PROPN cana-2735 166 90 Μƒ Μƒ NOUN cana-2735 166 91 οΏ½ οΏ½ NOUN cana-2735 166 92 = = SYM cana-2735 166 93 π‘’πœŽπ›Ό π‘’πœŽπ›Ό PROPN cana-2735 166 94 οΏ½ οΏ½ PROPN cana-2735 166 95 Μƒ Μƒ PROPN cana-2735 166 96 οΏ½ οΏ½ PROPN cana-2735 166 97 = = SYM cana-2735 166 98 π‘’πœŽ(π‘Žπ‘–π‘—(π‘₯ π‘’πœŽ(π‘Žπ‘–π‘—(π‘₯ PROPN cana-2735 166 99 , , PUNCT cana-2735 166 100 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— PRON cana-2735 166 101 ) ) PUNCT cana-2735 166 102 1 1 NUM cana-2735 166 103 2 2 NUM cana-2735 166 104 οΏ½ οΏ½ PROPN cana-2735 166 105 Μƒ Μƒ NOUN cana-2735 166 106 οΏ½ οΏ½ NOUN cana-2735 166 107 = = SYM cana-2735 166 108 ( ( PUNCT cana-2735 166 109 𝑒2πœŽπ‘Žπ‘–π‘—(π‘₯ 𝑒2πœŽπ‘Žπ‘–π‘—(π‘₯ ADP cana-2735 166 110 , , PUNCT cana-2735 166 111 πœ”)πœ”π‘–πœ”π‘— πœ”)πœ”π‘–πœ”π‘— PRON cana-2735 166 112 ) ) PUNCT cana-2735 166 113 1 1 NUM cana-2735 166 114 2 2 NUM cana-2735 166 115 οΏ½ οΏ½ PROPN cana-2735 166 116 Μƒ Μƒ NOUN cana-2735 166 117 οΏ½ οΏ½ NOUN cana-2735 166 118 = = SYM cana-2735 166 119 ( ( PUNCT cana-2735 166 120 οΏ½ οΏ½ PROPN cana-2735 166 121 Μƒ Μƒ NOUN cana-2735 166 122 οΏ½ οΏ½ NOUN cana-2735 166 123 π‘–π‘—πœ”π‘–πœ”π‘— π‘–π‘—πœ”π‘–πœ”π‘— NOUN cana-2735 166 124 ) ) PUNCT cana-2735 166 125 1 1 NUM cana-2735 166 126 2 2 NUM cana-2735 166 127 οΏ½ οΏ½ PROPN cana-2735 166 128 Μƒ Μƒ NOUN cana-2735 166 129 οΏ½ οΏ½ NOUN cana-2735 166 130 𝑖𝑗 𝑖𝑗 NOUN cana-2735 166 131 = = SYM cana-2735 166 132 𝑒2πœŽπ‘Žπ‘–π‘—(π‘₯ 𝑒2πœŽπ‘Žπ‘–π‘—(π‘₯ NOUN cana-2735 166 133 , , PUNCT cana-2735 166 134 πœ” πœ” NOUN cana-2735 166 135 ) ) PUNCT cana-2735 166 136 and and CCONJ cana-2735 166 137 𝛽 𝛽 NOUN cana-2735 166 138 = = SYM cana-2735 166 139 π‘’πœŽπ›½ π‘’πœŽπ›½ PROPN cana-2735 166 140 𝛽 𝛽 NOUN cana-2735 166 141 = = PUNCT cana-2735 166 142 π‘’πœŽπœ”π‘–π‘ π‘’πœŽπœ”π‘–π‘ VERB cana-2735 166 143 𝑖(π‘₯ 𝑖(π‘₯ NUM cana-2735 166 144 ) ) PUNCT cana-2735 166 145 𝛽 𝛽 NOUN cana-2735 166 146 = = SYM cana-2735 166 147 πœ”π‘–π‘’ πœ”π‘–π‘’ NOUN cana-2735 166 148 πœŽπ‘π‘–(π‘₯ πœŽπ‘π‘–(π‘₯ X cana-2735 166 149 ) ) PUNCT cana-2735 166 150 𝛽 𝛽 NOUN cana-2735 166 151 = = SYM cana-2735 166 152 πœ”π‘–π‘ πœ”π‘–π‘ NOUN cana-2735 166 153 𝑖 𝑖 SYM cana-2735 166 154 οΏ½ οΏ½ PROPN cana-2735 166 155 Μƒ Μƒ NOUN cana-2735 166 156 οΏ½ οΏ½ NOUN cana-2735 166 157 𝑖 𝑖 NOUN cana-2735 166 158 = = SYM cana-2735 166 159 π‘’πœŽπ‘π‘–(π‘₯ π‘’πœŽπ‘π‘–(π‘₯ PROPN cana-2735 166 160 ) ) PUNCT cana-2735 166 161 now now ADV cana-2735 166 162 we we PRON cana-2735 166 163 calculate calculate VERB cana-2735 166 164 the the DET cana-2735 166 165 christoffel christoffel ADJ cana-2735 166 166 symbols symbol NOUN cana-2735 166 167 οΏ½ οΏ½ PROPN cana-2735 166 168 Μƒ Μƒ PROPN cana-2735 166 169 οΏ½ οΏ½ PROPN cana-2735 166 170 π‘Ÿπ‘˜ π‘Ÿπ‘˜ NOUN cana-2735 166 171 𝑝 𝑝 PROPN cana-2735 166 172 in in ADP cana-2735 166 173 conformally conformally ADV cana-2735 166 174 transformed transform VERB cana-2735 166 175 space space NOUN cana-2735 166 176 ( ( PUNCT cana-2735 166 177 𝑀 𝑀 PROPN cana-2735 166 178 , , PUNCT cana-2735 166 179 οΏ½ οΏ½ PROPN cana-2735 166 180 Μƒ Μƒ PROPN cana-2735 166 181 οΏ½ οΏ½ PROPN cana-2735 166 182 (π‘₯ (π‘₯ NOUN cana-2735 166 183 , , PUNCT cana-2735 166 184 πœ” πœ” NOUN cana-2735 166 185 ) ) PUNCT cana-2735 166 186 ) ) PUNCT cana-2735 166 187 as as SCONJ cana-2735 166 188 follows follow VERB cana-2735 166 189 : : PUNCT cana-2735 166 190 we we PRON cana-2735 166 191 know know VERB cana-2735 166 192 from from ADP cana-2735 166 193 riemannian riemannian ADJ cana-2735 166 194 geometry geometry NOUN cana-2735 166 195 christoffel christoffel NOUN cana-2735 166 196 symbols symbol NOUN cana-2735 166 197 of of ADP cana-2735 166 198 second second ADJ cana-2735 166 199 kind kind NOUN cana-2735 166 200 π›Ύπ‘Ÿπ‘˜ π›Ύπ‘Ÿπ‘˜ VERB cana-2735 166 201 𝑝 𝑝 NOUN cana-2735 166 202 from from ADP cana-2735 166 203 fundamental fundamental ADJ cana-2735 166 204 metric metric ADJ cana-2735 166 205 tensor tensor NOUN cana-2735 166 206 π‘Žπ‘π‘ž(π‘₯ π‘Žπ‘π‘ž(π‘₯ PROPN cana-2735 166 207 , , PUNCT cana-2735 166 208 πœ” πœ” NOUN cana-2735 166 209 ) ) PUNCT cana-2735 166 210 can can AUX cana-2735 166 211 be be AUX cana-2735 166 212 defined define VERB cana-2735 166 213 as as ADP cana-2735 166 214 π›Ύπ‘žπ‘˜ π›Ύπ‘žπ‘˜ PROPN cana-2735 166 215 𝑝 𝑝 NOUN cana-2735 166 216 = = SYM cana-2735 166 217 1 1 NUM cana-2735 166 218 2 2 NUM cana-2735 166 219 π‘Žπ‘™π‘ π‘Žπ‘™π‘ VERB cana-2735 166 220 ( ( PUNCT cana-2735 166 221 πœ•π‘Žπ‘˜π‘™ πœ•π‘Žπ‘˜π‘™ NOUN cana-2735 166 222 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 167 1 + + CCONJ cana-2735 167 2 πœ•π‘Žπ‘™π‘ž πœ•π‘Žπ‘™π‘ž PROPN cana-2735 167 3 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 167 4 βˆ’ βˆ’ PROPN cana-2735 167 5 πœ•π‘Žπ‘žπ‘˜ πœ•π‘Žπ‘žπ‘˜ PROPN cana-2735 167 6 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 167 7 ) ) PUNCT cana-2735 167 8 similarly similarly ADV cana-2735 167 9 , , PUNCT cana-2735 167 10 we we PRON cana-2735 167 11 can can AUX cana-2735 167 12 also also ADV cana-2735 167 13 define define VERB cana-2735 167 14 the the DET cana-2735 167 15 christoffel christoffel ADJ cana-2735 167 16 symbols symbol NOUN cana-2735 167 17 οΏ½ οΏ½ PROPN cana-2735 167 18 Μƒ Μƒ PROPN cana-2735 167 19 οΏ½ οΏ½ PROPN cana-2735 167 20 π‘Ÿπ‘˜ π‘Ÿπ‘˜ NOUN cana-2735 167 21 𝑝 𝑝 PROPN cana-2735 167 22 in in ADP cana-2735 167 23 conformally conformally ADV cana-2735 167 24 transformed transform VERB cana-2735 167 25 space space NOUN cana-2735 167 26 ( ( PUNCT cana-2735 167 27 𝑀 𝑀 PROPN cana-2735 167 28 , , PUNCT cana-2735 167 29 οΏ½ οΏ½ PROPN cana-2735 167 30 Μƒ Μƒ PROPN cana-2735 167 31 οΏ½ οΏ½ PROPN cana-2735 167 32 (π‘₯ (π‘₯ NOUN cana-2735 167 33 , , PUNCT cana-2735 167 34 πœ” πœ” NOUN cana-2735 167 35 ) ) PUNCT cana-2735 167 36 ) ) PUNCT cana-2735 167 37 as as SCONJ cana-2735 167 38 οΏ½ οΏ½ PROPN cana-2735 167 39 Μƒ Μƒ PROPN cana-2735 167 40 οΏ½ οΏ½ PROPN cana-2735 167 41 π‘žπ‘˜ π‘žπ‘˜ NOUN cana-2735 167 42 𝑝 𝑝 NOUN cana-2735 167 43 = = SYM cana-2735 167 44 1 1 NUM cana-2735 167 45 2 2 NUM cana-2735 167 46 οΏ½ οΏ½ PROPN cana-2735 167 47 Μƒ Μƒ PROPN cana-2735 167 48 οΏ½ οΏ½ PROPN cana-2735 167 49 𝑙𝑝 𝑙𝑝 PROPN cana-2735 167 50 ( ( PUNCT cana-2735 167 51 πœ• πœ• PROPN cana-2735 167 52 οΏ½ οΏ½ PROPN cana-2735 167 53 Μƒ Μƒ PROPN cana-2735 167 54 οΏ½ οΏ½ PROPN cana-2735 167 55 π‘˜π‘™ π‘˜π‘™ PRON cana-2735 167 56 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 167 57 + + PUNCT cana-2735 167 58 πœ• πœ• PROPN cana-2735 167 59 οΏ½ οΏ½ PROPN cana-2735 167 60 Μƒ Μƒ PROPN cana-2735 167 61 οΏ½ οΏ½ PROPN cana-2735 167 62 π‘™π‘ž π‘™π‘ž PROPN cana-2735 167 63 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 167 64 βˆ’ βˆ’ PROPN cana-2735 167 65 πœ• πœ• PROPN cana-2735 167 66 οΏ½ οΏ½ PROPN cana-2735 167 67 Μƒ Μƒ PROPN cana-2735 167 68 οΏ½ οΏ½ PROPN cana-2735 167 69 π‘žπ‘˜ π‘žπ‘˜ NOUN cana-2735 167 70 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 167 71 ) ) PUNCT cana-2735 167 72 communications communication NOUN cana-2735 167 73 on on ADP cana-2735 167 74 applied apply VERB cana-2735 167 75 nonlinear nonlinear ADJ cana-2735 167 76 analysis analysis NOUN cana-2735 167 77 issn issn NOUN cana-2735 167 78 : : PUNCT cana-2735 167 79 1074 1074 NUM cana-2735 167 80 - - PUNCT cana-2735 167 81 133x 133x NUM cana-2735 167 82 vol vol NOUN cana-2735 167 83 32 32 NUM cana-2735 167 84 no no NOUN cana-2735 167 85 . . PUNCT cana-2735 168 1 4s 4s NUM cana-2735 168 2 ( ( PUNCT cana-2735 168 3 2025 2025 NUM cana-2735 168 4 ) ) PUNCT cana-2735 168 5 10 10 NUM cana-2735 168 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 168 7 = = SYM cana-2735 168 8 1 1 NUM cana-2735 168 9 2 2 NUM cana-2735 168 10 𝑒2πœŽπ‘Žπ‘™π‘(π‘₯ 𝑒2πœŽπ‘Žπ‘™π‘(π‘₯ ADP cana-2735 168 11 , , PUNCT cana-2735 168 12 πœ” πœ” PRON cana-2735 168 13 ) ) PUNCT cana-2735 168 14 ( ( PUNCT cana-2735 168 15 πœ•π‘’2πœŽπ‘Žπ‘˜π‘™(π‘₯,πœ” πœ•π‘’2πœŽπ‘Žπ‘˜π‘™(π‘₯,πœ” NOUN cana-2735 168 16 ) ) PUNCT cana-2735 168 17 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 169 1 + + CCONJ cana-2735 169 2 πœ•π‘’2πœŽπ‘Žπ‘™π‘ž(π‘₯,πœ” πœ•π‘’2πœŽπ‘Žπ‘™π‘ž(π‘₯,πœ” ADJ cana-2735 169 3 ) ) PUNCT cana-2735 169 4 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ NOUN cana-2735 169 5 βˆ’ βˆ’ PROPN cana-2735 169 6 πœ•π‘’2πœŽπ‘Žπ‘žπ‘˜(π‘₯,πœ” πœ•π‘’2πœŽπ‘Žπ‘žπ‘˜(π‘₯,πœ” NOUN cana-2735 169 7 ) ) PUNCT cana-2735 169 8 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 169 9 ) ) PUNCT cana-2735 170 1 = = SYM cana-2735 170 2 1 1 NUM cana-2735 170 3 2 2 NUM cana-2735 170 4 𝑒2πœŽπ‘Žπ‘™π‘ 𝑒2πœŽπ‘Žπ‘™π‘ VERB cana-2735 170 5 [ [ X cana-2735 170 6 ( ( PUNCT cana-2735 170 7 𝑒2𝜎 𝑒2𝜎 NOUN cana-2735 170 8 πœ•π‘Žπ‘˜π‘™ πœ•π‘Žπ‘˜π‘™ NOUN cana-2735 170 9 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 171 1 + + PUNCT cana-2735 171 2 π‘Žπ‘˜π‘™ π‘Žπ‘˜π‘™ PROPN cana-2735 171 3 πœ•π‘’2𝜎 πœ•π‘’2𝜎 NOUN cana-2735 171 4 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 171 5 ) ) PUNCT cana-2735 172 1 + + CCONJ cana-2735 172 2 ( ( PUNCT cana-2735 172 3 𝑒2𝜎 𝑒2𝜎 X cana-2735 172 4 πœ•π‘Žπ‘™π‘ž πœ•π‘Žπ‘™π‘ž PROPN cana-2735 172 5 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 172 6 + + CCONJ cana-2735 172 7 π‘Žπ‘™π‘ž π‘Žπ‘™π‘ž PROPN cana-2735 172 8 πœ•π‘’2𝜎 πœ•π‘’2𝜎 PROPN cana-2735 172 9 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 172 10 ) ) PUNCT cana-2735 172 11 βˆ’ βˆ’ PROPN cana-2735 172 12 ( ( PUNCT cana-2735 172 13 𝑒2𝜎 𝑒2𝜎 VERB cana-2735 172 14 πœ•π‘Žπ‘—π‘ž πœ•π‘Žπ‘—π‘ž VERB cana-2735 172 15 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PRON cana-2735 172 16 + + CCONJ cana-2735 172 17 π‘Žπ‘žπ‘˜ π‘Žπ‘žπ‘˜ NOUN cana-2735 172 18 πœ•π‘’2𝜎 πœ•π‘’2𝜎 PROPN cana-2735 172 19 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 172 20 ) ) PUNCT cana-2735 172 21 ] ] PUNCT cana-2735 173 1 = = SYM cana-2735 173 2 1 1 NUM cana-2735 173 3 2 2 NUM cana-2735 173 4 𝑒2πœŽπ‘Žπ‘™π‘ 𝑒2πœŽπ‘Žπ‘™π‘ VERB cana-2735 173 5 [ [ X cana-2735 173 6 ( ( PUNCT cana-2735 173 7 𝑒2𝜎 𝑒2𝜎 NOUN cana-2735 173 8 πœ•π‘Žπ‘˜π‘™ πœ•π‘Žπ‘˜π‘™ NOUN cana-2735 173 9 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 174 1 + + PUNCT cana-2735 174 2 2𝑒2πœŽπ‘Žπ‘˜π‘™ 2𝑒2πœŽπ‘Žπ‘˜π‘™ NUM cana-2735 174 3 πœ•πœŽ πœ•πœŽ NOUN cana-2735 174 4 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 174 5 ) ) PUNCT cana-2735 174 6 + + CCONJ cana-2735 174 7 ( ( PUNCT cana-2735 174 8 𝑒2𝜎 𝑒2𝜎 X cana-2735 174 9 πœ•π‘Žπ‘™π‘ž πœ•π‘Žπ‘™π‘ž PROPN cana-2735 174 10 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 174 11 + + CCONJ cana-2735 174 12 2𝑒2πœŽπ‘Žπ‘™π‘ž 2𝑒2πœŽπ‘Žπ‘™π‘ž PROPN cana-2735 174 13 πœ•πœŽ πœ•πœŽ NOUN cana-2735 174 14 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ NOUN cana-2735 174 15 ) ) PUNCT cana-2735 174 16 βˆ’ βˆ’ PROPN cana-2735 174 17 ( ( PUNCT cana-2735 174 18 𝑒2𝜎 𝑒2𝜎 VERB cana-2735 174 19 πœ•π‘Žπ‘žπ‘˜ πœ•π‘Žπ‘žπ‘˜ VERB cana-2735 174 20 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 174 21 + + NUM cana-2735 174 22 2𝑒2πœŽπ‘Žπ‘žπ‘˜ 2𝑒2πœŽπ‘Žπ‘žπ‘˜ NUM cana-2735 174 23 πœ•πœŽ πœ•πœŽ NOUN cana-2735 174 24 πœ•π‘₯𝑙 πœ•π‘₯𝑙 NOUN cana-2735 174 25 ) ) PUNCT cana-2735 174 26 ] ] PUNCT cana-2735 175 1 = = SYM cana-2735 175 2 1 1 NUM cana-2735 175 3 2 2 NUM cana-2735 175 4 𝑒4πœŽπ‘Žπ‘™π‘ 𝑒4πœŽπ‘Žπ‘™π‘ VERB cana-2735 175 5 [ [ X cana-2735 175 6 ( ( PUNCT cana-2735 175 7 πœ•π‘Žπ‘˜π‘™ πœ•π‘Žπ‘˜π‘™ PROPN cana-2735 175 8 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 176 1 + + CCONJ cana-2735 176 2 πœ•π‘Žπ‘™π‘ž πœ•π‘Žπ‘™π‘ž PROPN cana-2735 176 3 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 176 4 βˆ’ βˆ’ PROPN cana-2735 176 5 πœ•π‘Žπ‘žπ‘˜ πœ•π‘Žπ‘žπ‘˜ PROPN cana-2735 176 6 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 176 7 ) ) PUNCT cana-2735 177 1 + + CCONJ cana-2735 177 2 ( ( PUNCT cana-2735 177 3 2π‘Žπ‘˜π‘™ 2π‘Žπ‘˜π‘™ PROPN cana-2735 177 4 πœ•πœŽ πœ•πœŽ NOUN cana-2735 177 5 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 178 1 + + CCONJ cana-2735 178 2 2π‘Žπ‘™π‘ž 2π‘Žπ‘™π‘ž PROPN cana-2735 178 3 πœ•πœŽ πœ•πœŽ NOUN cana-2735 178 4 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ NOUN cana-2735 178 5 βˆ’ βˆ’ PROPN cana-2735 178 6 2π‘Žπ‘žπ‘˜ 2π‘Žπ‘žπ‘˜ NUM cana-2735 178 7 πœ•πœŽ πœ•πœŽ NOUN cana-2735 178 8 πœ•π‘₯𝑙 πœ•π‘₯𝑙 NOUN cana-2735 178 9 ) ) PUNCT cana-2735 178 10 ] ] PUNCT cana-2735 178 11 = = SYM cana-2735 178 12 𝑒4𝜎 𝑒4𝜎 PROPN cana-2735 178 13 [ [ PUNCT cana-2735 178 14 1 1 NUM cana-2735 178 15 2 2 NUM cana-2735 178 16 π‘Žπ‘™π‘ π‘Žπ‘™π‘ VERB cana-2735 178 17 ( ( PUNCT cana-2735 178 18 πœ•π‘Žπ‘˜π‘™ πœ•π‘Žπ‘˜π‘™ NOUN cana-2735 178 19 πœ•π‘₯π‘ž πœ•π‘₯π‘ž PUNCT cana-2735 179 1 + + CCONJ cana-2735 179 2 πœ•π‘Žπ‘™π‘ž πœ•π‘Žπ‘™π‘ž PROPN cana-2735 179 3 πœ•π‘₯π‘˜ πœ•π‘₯π‘˜ PROPN cana-2735 179 4 βˆ’ βˆ’ PROPN cana-2735 179 5 πœ•π‘Žπ‘žπ‘˜ πœ•π‘Žπ‘žπ‘˜ PROPN cana-2735 179 6 πœ•π‘₯𝑙 πœ•π‘₯𝑙 PUNCT cana-2735 179 7 ) ) PUNCT cana-2735 180 1 + + CCONJ cana-2735 180 2 ( ( PUNCT cana-2735 180 3 π‘Žπ‘™π‘π‘Žπ‘˜π‘™πœŽπ‘ž π‘Žπ‘™π‘π‘Žπ‘˜π‘™πœŽπ‘ž NOUN cana-2735 180 4 + + CCONJ cana-2735 180 5 π‘Žπ‘™π‘π‘Žπ‘™π‘žπœŽπ‘˜ π‘Žπ‘™π‘π‘Žπ‘™π‘žπœŽπ‘˜ NOUN cana-2735 180 6 βˆ’ βˆ’ PROPN cana-2735 180 7 π‘Žπ‘™π‘π‘Žπ‘žπ‘˜πœŽπ‘™ π‘Žπ‘™π‘π‘Žπ‘žπ‘˜πœŽπ‘™ PROPN cana-2735 180 8 ) ) PUNCT cana-2735 180 9 ] ] PUNCT cana-2735 181 1 = = PUNCT cana-2735 181 2 𝑒4𝜎[π›Ύπ‘žπ‘˜ 𝑒4𝜎[π›Ύπ‘žπ‘˜ PROPN cana-2735 181 3 𝑝 𝑝 PROPN cana-2735 181 4 + + CCONJ cana-2735 181 5 ( ( PUNCT cana-2735 181 6 π›Ώπ‘˜ π›Ώπ‘˜ ADP cana-2735 181 7 π‘πœŽπ‘ž π‘πœŽπ‘ž PROPN cana-2735 181 8 + + CCONJ cana-2735 181 9 π›Ώπ‘ž π›Ώπ‘ž PROPN cana-2735 181 10 π‘πœŽπ‘˜ π‘πœŽπ‘˜ NOUN cana-2735 181 11 βˆ’ βˆ’ NOUN cana-2735 181 12 π‘Žπ‘žπ‘˜πœŽπ‘– π‘Žπ‘žπ‘˜πœŽπ‘– PROPN cana-2735 181 13 ) ) PUNCT cana-2735 181 14 ] ] PUNCT cana-2735 182 1 hence hence ADV cana-2735 182 2 , , PUNCT cana-2735 182 3 the the DET cana-2735 182 4 components component NOUN cana-2735 182 5 of of ADP cana-2735 182 6 christoffel christoffel ADJ cana-2735 182 7 symbols symbol NOUN cana-2735 182 8 οΏ½ οΏ½ PROPN cana-2735 182 9 Μƒ Μƒ PROPN cana-2735 182 10 οΏ½ οΏ½ PROPN cana-2735 182 11 π‘žπ‘˜ π‘žπ‘˜ ADP cana-2735 182 12 𝑝 𝑝 NOUN cana-2735 182 13 , , PUNCT cana-2735 182 14 constructed construct VERB cana-2735 182 15 from from ADP cana-2735 182 16 οΏ½ οΏ½ PROPN cana-2735 182 17 Μƒ Μƒ PROPN cana-2735 182 18 οΏ½ οΏ½ NOUN cana-2735 182 19 π‘π‘ž π‘π‘ž NOUN cana-2735 182 20 , , PUNCT cana-2735 182 21 in in ADP cana-2735 182 22 conformally conformally ADV cana-2735 182 23 transformed transform VERB cana-2735 182 24 space space NOUN cana-2735 182 25 are be AUX cana-2735 182 26 given give VERB cana-2735 182 27 by by ADP cana-2735 182 28 οΏ½ οΏ½ PROPN cana-2735 182 29 Μƒ Μƒ PROPN cana-2735 182 30 οΏ½ οΏ½ PROPN cana-2735 182 31 π‘žπ‘˜ π‘žπ‘˜ NOUN cana-2735 182 32 𝑝 𝑝 NOUN cana-2735 182 33 = = PRON cana-2735 182 34 π›Ύπ‘žπ‘˜ π›Ύπ‘žπ‘˜ PROPN cana-2735 182 35 𝑝 𝑝 NOUN cana-2735 182 36 + + CCONJ cana-2735 182 37 π΅π‘žπ‘˜ π΅π‘žπ‘˜ PROPN cana-2735 182 38 𝑝 𝑝 PROPN cana-2735 182 39 , , PUNCT cana-2735 182 40 ( ( PUNCT cana-2735 182 41 19 19 NUM cana-2735 182 42 ) ) PUNCT cana-2735 182 43 where where SCONJ cana-2735 182 44 π΅π‘žπ‘˜ π΅π‘žπ‘˜ PROPN cana-2735 182 45 𝑝 𝑝 PROPN cana-2735 182 46 = = PUNCT cana-2735 182 47 πœŽπ‘˜π›Ώπ‘ž πœŽπ‘˜π›Ώπ‘ž NOUN cana-2735 182 48 𝑝 𝑝 NOUN cana-2735 182 49 + + CCONJ cana-2735 182 50 πœŽπ‘žπ›Ώπ‘˜ πœŽπ‘žπ›Ώπ‘˜ PROPN cana-2735 182 51 𝑝 𝑝 PROPN cana-2735 182 52 βˆ’ βˆ’ PROPN cana-2735 182 53 π‘Žπ‘˜π‘žπœŽπ‘ π‘Žπ‘˜π‘žπœŽπ‘ ADJ cana-2735 182 54 , , PUNCT cana-2735 182 55 πœŽπ‘ πœŽπ‘ NOUN cana-2735 182 56 = = PUNCT cana-2735 182 57 πœŽπ‘žπ‘Žπ‘π‘ž πœŽπ‘žπ‘Žπ‘π‘ž PROPN cana-2735 182 58 . . PUNCT cana-2735 183 1 the the DET cana-2735 183 2 covariant covariant ADJ cana-2735 183 3 derivative derivative NOUN cana-2735 183 4 of of ADP cana-2735 183 5 οΏ½ οΏ½ PROPN cana-2735 183 6 Μƒ Μƒ PROPN cana-2735 183 7 οΏ½ οΏ½ PROPN cana-2735 183 8 𝑝 𝑝 NOUN cana-2735 183 9 with with ADP cana-2735 183 10 respect respect NOUN cana-2735 183 11 to to ADP cana-2735 183 12 οΏ½ οΏ½ PROPN cana-2735 183 13 Μƒ Μƒ NOUN cana-2735 183 14 οΏ½ οΏ½ PROPN cana-2735 183 15 π‘Ÿπ‘˜ π‘Ÿπ‘˜ NOUN cana-2735 183 16 𝑝 𝑝 PROPN cana-2735 183 17 , , PUNCT cana-2735 183 18 yields yield NOUN cana-2735 183 19 οΏ½ οΏ½ PROPN cana-2735 183 20 Μƒ Μƒ PROPN cana-2735 183 21 οΏ½ οΏ½ NOUN cana-2735 183 22 :π‘˜ :π‘˜ ADJ cana-2735 183 23 𝑝 𝑝 NOUN cana-2735 183 24 = = SYM cana-2735 183 25 π‘’πœŽ(𝑏:π‘˜ π‘’πœŽ(𝑏:π‘˜ PROPN cana-2735 183 26 𝑝 𝑝 PROPN cana-2735 183 27 + + NUM cana-2735 183 28 2πœŽπ‘˜π‘π‘ 2πœŽπ‘˜π‘π‘ NUM cana-2735 184 1 + + NUM cana-2735 184 2 π‘π‘ŸπœŽπ‘Ÿπ›Ώπ‘˜ π‘π‘ŸπœŽπ‘Ÿπ›Ώπ‘˜ PROPN cana-2735 184 3 𝑝 𝑝 ADP cana-2735 184 4 βˆ’ βˆ’ PROPN cana-2735 184 5 πœŽπ‘π‘π‘Ÿπ‘Žπ‘Ÿπ‘˜ πœŽπ‘π‘π‘Ÿπ‘Žπ‘Ÿπ‘˜ ADJ cana-2735 184 6 ) ) PUNCT cana-2735 184 7 . . PUNCT cana-2735 185 1 ( ( PUNCT cana-2735 185 2 20 20 X cana-2735 185 3 ) ) PUNCT cana-2735 185 4 transvecting transvecte VERB cana-2735 185 5 the the DET cana-2735 185 6 equation equation NOUN cana-2735 185 7 ( ( PUNCT cana-2735 185 8 20 20 NUM cana-2735 185 9 ) ) PUNCT cana-2735 185 10 by by ADP cana-2735 185 11 οΏ½ οΏ½ PROPN cana-2735 185 12 Μƒ Μƒ PROPN cana-2735 185 13 οΏ½ οΏ½ NOUN cana-2735 185 14 π‘˜ π‘˜ NOUN cana-2735 185 15 , , PUNCT cana-2735 185 16 and and CCONJ cana-2735 185 17 putting put VERB cana-2735 185 18 𝑀𝑝 𝑀𝑝 NOUN cana-2735 185 19 = = SYM cana-2735 185 20 1 1 NUM cana-2735 185 21 𝐡2 𝐡2 NOUN cana-2735 185 22 ( ( PUNCT cana-2735 185 23 π‘π‘˜π‘:π‘˜ π‘π‘˜π‘:π‘˜ PROPN cana-2735 185 24 𝑝 𝑝 PROPN cana-2735 185 25 βˆ’ βˆ’ PROPN cana-2735 185 26 𝑏:π‘Ÿ 𝑏:π‘Ÿ PROPN cana-2735 185 27 π‘Ÿ π‘Ÿ PRON cana-2735 185 28 𝑏𝑝 𝑏𝑝 INTJ cana-2735 185 29 𝑛+4 𝑛+4 NUM cana-2735 185 30 ) ) PUNCT cana-2735 185 31 , , PUNCT cana-2735 185 32 ( ( PUNCT cana-2735 185 33 21 21 NUM cana-2735 185 34 ) ) PUNCT cana-2735 185 35 we we PRON cana-2735 185 36 have have VERB cana-2735 185 37 πœŽπ‘ πœŽπ‘ NUM cana-2735 185 38 = = PUNCT cana-2735 185 39 οΏ½ οΏ½ PROPN cana-2735 185 40 Μƒ Μƒ PROPN cana-2735 185 41 οΏ½ οΏ½ PROPN cana-2735 185 42 𝑝 𝑝 PROPN cana-2735 185 43 βˆ’ βˆ’ PROPN cana-2735 185 44 𝑀𝑝 𝑀𝑝 PROPN cana-2735 185 45 , , PUNCT cana-2735 185 46 from from ADP cana-2735 185 47 which which PRON cana-2735 185 48 we we PRON cana-2735 185 49 get get VERB cana-2735 185 50 πœŽπ‘ πœŽπ‘ NOUN cana-2735 185 51 = = PUNCT cana-2735 185 52 οΏ½ οΏ½ PROPN cana-2735 185 53 Μƒ Μƒ PROPN cana-2735 185 54 οΏ½ οΏ½ PROPN cana-2735 185 55 𝑝 𝑝 NOUN cana-2735 185 56 βˆ’ βˆ’ PROPN cana-2735 185 57 𝑀𝑝. 𝑀𝑝. NOUN cana-2735 185 58 substituting substitute VERB cana-2735 185 59 the the DET cana-2735 185 60 values value NOUN cana-2735 185 61 of of ADP cana-2735 185 62 πœŽπ‘ πœŽπ‘ NOUN cana-2735 185 63 and and CCONJ cana-2735 185 64 πœŽπ‘ πœŽπ‘ NOUN cana-2735 185 65 in in ADP cana-2735 185 66 equation equation NOUN cana-2735 185 67 ( ( PUNCT cana-2735 185 68 19 19 NUM cana-2735 185 69 ) ) PUNCT cana-2735 185 70 and and CCONJ cana-2735 185 71 using use VERB cana-2735 185 72 π·β„Žπ‘ž π·β„Žπ‘ž PROPN cana-2735 185 73 𝑝 𝑝 NOUN cana-2735 185 74 = = NOUN cana-2735 185 75 π›Ύβ„Žπ‘ž π›Ύβ„Žπ‘ž NOUN cana-2735 185 76 𝑝 𝑝 NOUN cana-2735 186 1 + + CCONJ cana-2735 186 2 π›Ώβ„Ž π›Ώβ„Ž NOUN cana-2735 186 3 π‘π‘€π‘ž π‘π‘€π‘ž VERB cana-2735 186 4 + + CCONJ cana-2735 186 5 π›Ώβ„Ž π›Ώβ„Ž NOUN cana-2735 186 6 π‘π‘€π‘ž π‘π‘€π‘ž NOUN cana-2735 186 7 + + CCONJ cana-2735 186 8 π›Ώπ‘ž π›Ώπ‘ž PROPN cana-2735 186 9 π‘π‘€β„Ž π‘π‘€β„Ž NOUN cana-2735 186 10 βˆ’ βˆ’ PROPN cana-2735 186 11 π‘€π‘π‘Žβ„Žπ‘ž π‘€π‘π‘Žβ„Žπ‘ž PROPN cana-2735 186 12 , , PUNCT cana-2735 186 13 we we PRON cana-2735 186 14 find find VERB cana-2735 186 15 οΏ½ οΏ½ PROPN cana-2735 186 16 Μƒ Μƒ NOUN cana-2735 186 17 οΏ½ οΏ½ PROPN cana-2735 186 18 β„Žπ‘ž β„Žπ‘ž PROPN cana-2735 186 19 𝑝 𝑝 NOUN cana-2735 186 20 = = PUNCT cana-2735 186 21 π·β„Žπ‘ž π·β„Žπ‘ž PROPN cana-2735 186 22 𝑝 𝑝 PROPN cana-2735 186 23 . . PUNCT cana-2735 187 1 ( ( PUNCT cana-2735 187 2 22 22 NUM cana-2735 187 3 ) ) PUNCT cana-2735 187 4 here here ADV cana-2735 187 5 π·β„Žπ‘ž π·β„Žπ‘ž PROPN cana-2735 187 6 𝑝 𝑝 PROPN cana-2735 187 7 is be AUX cana-2735 187 8 a a DET cana-2735 187 9 symmetric symmetric ADJ cana-2735 187 10 and and CCONJ cana-2735 187 11 conformally conformally ADV cana-2735 187 12 invariant invariant ADJ cana-2735 187 13 linear linear ADJ cana-2735 187 14 connection connection NOUN cana-2735 187 15 on on ADP cana-2735 187 16 𝑀. 𝑀. PROPN cana-2735 187 17 the the DET cana-2735 187 18 whole whole ADJ cana-2735 187 19 discussion discussion NOUN cana-2735 187 20 can can AUX cana-2735 187 21 be be AUX cana-2735 187 22 summarized summarize VERB cana-2735 187 23 in in ADP cana-2735 187 24 the the DET cana-2735 187 25 following follow VERB cana-2735 187 26 proposition proposition NOUN cana-2735 187 27 . . PUNCT cana-2735 188 1 proposition proposition NOUN cana-2735 188 2 4.1 4.1 NUM cana-2735 188 3 let let VERB cana-2735 188 4 𝐢 𝐢 PROPN cana-2735 188 5 = = SYM cana-2735 188 6 ( ( PUNCT cana-2735 188 7 𝑀 𝑀 PROPN cana-2735 188 8 , , PUNCT cana-2735 188 9 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 188 10 , , PUNCT cana-2735 188 11 πœ” πœ” NOUN cana-2735 188 12 ) ) PUNCT cana-2735 188 13 ) ) PUNCT cana-2735 188 14 be be AUX cana-2735 188 15 a a DET cana-2735 188 16 cartan cartan ADJ cana-2735 188 17 space space NOUN cana-2735 188 18 the the DET cana-2735 188 19 ( ( PUNCT cana-2735 188 20 𝛼 𝛼 PROPN cana-2735 188 21 , , PUNCT cana-2735 188 22 𝛽)-metric 𝛽)-metric ADJ cana-2735 188 23 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 188 24 , , PUNCT cana-2735 188 25 πœ” πœ” ADJ cana-2735 188 26 ) ) PUNCT cana-2735 188 27 = = SYM cana-2735 189 1 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 189 2 , , PUNCT cana-2735 189 3 πœ” πœ” PRON cana-2735 189 4 ) ) PUNCT cana-2735 189 5 + + NUM cana-2735 189 6 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 189 7 , , PUNCT cana-2735 189 8 πœ” πœ” PRON cana-2735 189 9 ) ) PUNCT cana-2735 189 10 + + CCONJ cana-2735 189 11 2π‘˜ 2π‘˜ NUM cana-2735 189 12 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 189 13 ) ) PUNCT cana-2735 189 14 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 189 15 ) ) PUNCT cana-2735 189 16 βˆ’ βˆ’ PROPN cana-2735 190 1 π‘˜2 π‘˜2 PROPN cana-2735 190 2 3 3 NUM cana-2735 190 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 190 4 ) ) PUNCT cana-2735 190 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 190 6 ) ) PUNCT cana-2735 190 7 . . PUNCT cana-2735 191 1 hhen hhen NOUN cana-2735 191 2 , , PUNCT cana-2735 191 3 there there PRON cana-2735 191 4 eiists eiist VERB cana-2735 191 5 a a DET cana-2735 191 6 conformally conformally ADV cana-2735 191 7 invariant invariant ADJ cana-2735 191 8 symmetric symmetric ADJ cana-2735 191 9 linear linear PROPN cana-2735 191 10 connection connection NOUN cana-2735 191 11 π·π‘žπ‘˜ π·π‘žπ‘˜ PROPN cana-2735 191 12 𝑝 𝑝 PROPN cana-2735 191 13 on on ADP cana-2735 191 14 𝑀. 𝑀. PROPN cana-2735 191 15 next next ADV cana-2735 191 16 , , PUNCT cana-2735 191 17 if if SCONJ cana-2735 191 18 we we PRON cana-2735 191 19 denote denote VERB cana-2735 191 20 the the DET cana-2735 191 21 curvature curvature NOUN cana-2735 191 22 tensor tensor NOUN cana-2735 191 23 of of ADP cana-2735 191 24 π·π‘žπ‘˜ π·π‘žπ‘˜ PROPN cana-2735 191 25 𝑝 𝑝 PROPN cana-2735 191 26 by by ADP cana-2735 191 27 π·β„Žπ‘žπ‘˜ π·β„Žπ‘žπ‘˜ PROPN cana-2735 191 28 𝑝 𝑝 PROPN cana-2735 191 29 , , PUNCT cana-2735 191 30 then then ADV cana-2735 191 31 from from ADP cana-2735 191 32 the the DET cana-2735 191 33 equation equation NOUN cana-2735 191 34 ( ( PUNCT cana-2735 191 35 22 22 NUM cana-2735 191 36 ) ) PUNCT cana-2735 191 37 , , PUNCT cana-2735 191 38 we we PRON cana-2735 191 39 get get VERB cana-2735 191 40 οΏ½ οΏ½ PROPN cana-2735 191 41 Μƒ Μƒ PROPN cana-2735 191 42 οΏ½ οΏ½ NOUN cana-2735 191 43 β„Žπ‘žπ‘˜ β„Žπ‘žπ‘˜ NOUN cana-2735 191 44 𝑝 𝑝 PROPN cana-2735 191 45 = = SYM cana-2735 191 46 π·β„Žπ‘žπ‘˜ π·β„Žπ‘žπ‘˜ PROPN cana-2735 191 47 𝑝 𝑝 PROPN cana-2735 191 48 . . PUNCT cana-2735 192 1 ( ( PUNCT cana-2735 192 2 23 23 NUM cana-2735 192 3 ) ) PUNCT cana-2735 192 4 since since SCONJ cana-2735 192 5 𝑏:π‘˜ 𝑏:π‘˜ NOUN cana-2735 192 6 𝑝 𝑝 NOUN cana-2735 192 7 = = SYM cana-2735 192 8 0 0 NUM cana-2735 192 9 , , PUNCT cana-2735 192 10 from from ADP cana-2735 192 11 equation equation NOUN cana-2735 192 12 ( ( PUNCT cana-2735 192 13 21 21 NUM cana-2735 192 14 ) ) PUNCT cana-2735 192 15 , , PUNCT cana-2735 192 16 we we PRON cana-2735 192 17 get get VERB cana-2735 192 18 𝑀𝑖 𝑀𝑖 PROPN cana-2735 192 19 = = NOUN cana-2735 192 20 0 0 NUM cana-2735 192 21 . . PUNCT cana-2735 193 1 hence hence ADV cana-2735 193 2 , , PUNCT cana-2735 193 3 we we PRON cana-2735 193 4 deduce deduce VERB cana-2735 193 5 that that SCONJ cana-2735 193 6 π·π‘žπ‘˜ π·π‘žπ‘˜ PROPN cana-2735 193 7 𝑝 𝑝 PROPN cana-2735 193 8 = = PRON cana-2735 193 9 π›Ύπ‘žπ‘˜ π›Ύπ‘žπ‘˜ PROPN cana-2735 193 10 𝑝 𝑝 PROPN cana-2735 193 11 and and CCONJ cana-2735 193 12 π·β„Žπ‘žπ‘˜ π·β„Žπ‘žπ‘˜ PROPN cana-2735 193 13 𝑝 𝑝 PROPN cana-2735 193 14 = = NOUN cana-2735 193 15 communications communication NOUN cana-2735 193 16 on on ADP cana-2735 193 17 applied apply VERB cana-2735 193 18 nonlinear nonlinear ADJ cana-2735 193 19 analysis analysis NOUN cana-2735 193 20 issn issn NOUN cana-2735 193 21 : : PUNCT cana-2735 193 22 1074 1074 NUM cana-2735 193 23 - - PUNCT cana-2735 193 24 133x 133x NUM cana-2735 193 25 vol vol NOUN cana-2735 193 26 32 32 NUM cana-2735 194 1 no no NOUN cana-2735 194 2 . . PUNCT cana-2735 195 1 4s 4s NUM cana-2735 195 2 ( ( PUNCT cana-2735 195 3 2025 2025 NUM cana-2735 195 4 ) ) PUNCT cana-2735 195 5 11 11 NUM cana-2735 195 6 https://internationalpubls.com https://internationalpubls.com X cana-2735 195 7 π‘…β„Žπ‘žπ‘˜ π‘…β„Žπ‘žπ‘˜ PROPN cana-2735 195 8 𝑝 𝑝 PROPN cana-2735 195 9 . . PUNCT cana-2735 196 1 thus thus ADV cana-2735 196 2 we we PRON cana-2735 196 3 have have VERB cana-2735 196 4 the the DET cana-2735 196 5 following follow VERB cana-2735 196 6 proposition proposition NOUN cana-2735 196 7 . . PUNCT cana-2735 197 1 proposition proposition NOUN cana-2735 197 2 4.2 4.2 NUM cana-2735 197 3 let let VERB cana-2735 197 4 𝐢 𝐢 PROPN cana-2735 197 5 = = SYM cana-2735 197 6 ( ( PUNCT cana-2735 197 7 𝑀 𝑀 PROPN cana-2735 197 8 , , PUNCT cana-2735 197 9 𝐾 𝐾 PROPN cana-2735 197 10 ) ) PUNCT cana-2735 197 11 be be AUX cana-2735 197 12 a a DET cana-2735 197 13 cartan cartan ADJ cana-2735 197 14 space space NOUN cana-2735 197 15 the the DET cana-2735 197 16 ( ( PUNCT cana-2735 197 17 𝛼 𝛼 PROPN cana-2735 197 18 , , PUNCT cana-2735 197 19 𝛽 𝛽 NOUN cana-2735 197 20 ) ) PUNCT cana-2735 197 21 -metric -metric NOUN cana-2735 197 22 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 197 23 , , PUNCT cana-2735 197 24 πœ” πœ” ADJ cana-2735 197 25 ) ) PUNCT cana-2735 197 26 = = SYM cana-2735 198 1 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 198 2 , , PUNCT cana-2735 198 3 πœ” πœ” PRON cana-2735 198 4 ) ) PUNCT cana-2735 198 5 + + NUM cana-2735 198 6 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 198 7 , , PUNCT cana-2735 198 8 πœ” πœ” PRON cana-2735 198 9 ) ) PUNCT cana-2735 198 10 + + CCONJ cana-2735 198 11 2π‘˜ 2π‘˜ NUM cana-2735 198 12 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 198 13 ) ) PUNCT cana-2735 198 14 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 198 15 ) ) PUNCT cana-2735 198 16 βˆ’ βˆ’ PROPN cana-2735 199 1 π‘˜2 π‘˜2 PROPN cana-2735 199 2 3 3 NUM cana-2735 199 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 199 4 ) ) PUNCT cana-2735 199 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 199 6 ) ) PUNCT cana-2735 199 7 admitting admit VERB cana-2735 199 8 β„Ž β„Ž ADP cana-2735 199 9 -metrical -metrical ADJ cana-2735 199 10 𝑑 𝑑 NOUN cana-2735 199 11 -connection -connection NOUN cana-2735 199 12 . . PUNCT cana-2735 200 1 hhen hhen NOUN cana-2735 201 1 , , PUNCT cana-2735 201 2 there there PRON cana-2735 201 3 eiists eiist VERB cana-2735 201 4 a a DET cana-2735 201 5 conformally conformally ADV cana-2735 201 6 invariant invariant ADJ cana-2735 201 7 symmetric symmetric ADJ cana-2735 201 8 linear linear PROPN cana-2735 201 9 connection connection NOUN cana-2735 201 10 π·π‘žβ„Ž π·π‘žβ„Ž PROPN cana-2735 201 11 𝑝 𝑝 PROPN cana-2735 201 12 such such ADJ cana-2735 201 13 that that SCONJ cana-2735 201 14 π·π‘žπ‘˜ π·π‘žπ‘˜ PROPN cana-2735 201 15 𝑝 𝑝 PROPN cana-2735 201 16 = = PRON cana-2735 201 17 π›Ύπ‘žπ‘˜ π›Ύπ‘žπ‘˜ PROPN cana-2735 201 18 𝑝 𝑝 PROPN cana-2735 202 1 and and CCONJ cana-2735 202 2 it it PRON cana-2735 202 3 ’s ’ VERB cana-2735 202 4 curvature curvature NOUN cana-2735 202 5 tensor tensor NOUN cana-2735 202 6 π·β„Žπ‘žπ‘˜ π·β„Žπ‘žπ‘˜ PROPN cana-2735 202 7 𝑝 𝑝 PROPN cana-2735 202 8 = = SYM cana-2735 202 9 π‘…β„Žπ‘žπ‘˜ π‘…β„Žπ‘žπ‘˜ PROPN cana-2735 202 10 𝑝 𝑝 PROPN cana-2735 202 11 . . PUNCT cana-2735 203 1 next next ADV cana-2735 203 2 , , PUNCT cana-2735 203 3 if if SCONJ cana-2735 203 4 the the DET cana-2735 203 5 associated associated ADJ cana-2735 203 6 riemannian riemannian ADJ cana-2735 203 7 space space NOUN cana-2735 203 8 ( ( PUNCT cana-2735 203 9 𝑀 𝑀 PROPN cana-2735 203 10 , , PUNCT cana-2735 203 11 𝛼 𝛼 NOUN cana-2735 203 12 ) ) PUNCT cana-2735 203 13 is be AUX cana-2735 203 14 locally locally ADV cana-2735 203 15 flat flat ADJ cana-2735 203 16 , , PUNCT cana-2735 203 17 that that ADV cana-2735 203 18 is is ADV cana-2735 203 19 , , PUNCT cana-2735 203 20 π‘…β„Žπ‘žπ‘˜ π‘…β„Žπ‘žπ‘˜ PROPN cana-2735 203 21 𝑝 𝑝 NOUN cana-2735 203 22 = = SYM cana-2735 203 23 0 0 PROPN cana-2735 203 24 , , PUNCT cana-2735 203 25 then then ADV cana-2735 203 26 from from ADP cana-2735 203 27 proposition proposition NOUN cana-2735 203 28 4.2 4.2 NUM cana-2735 203 29 and and CCONJ cana-2735 203 30 equation equation NOUN cana-2735 203 31 ( ( PUNCT cana-2735 203 32 23 23 NUM cana-2735 203 33 ) ) PUNCT cana-2735 203 34 , , PUNCT cana-2735 203 35 we we PRON cana-2735 203 36 deduce deduce VERB cana-2735 203 37 that that DET cana-2735 203 38 οΏ½ οΏ½ PROPN cana-2735 203 39 Μƒ Μƒ PROPN cana-2735 203 40 οΏ½ οΏ½ NOUN cana-2735 203 41 β„Žπ‘žπ‘˜ β„Žπ‘žπ‘˜ NOUN cana-2735 203 42 𝑝 𝑝 NOUN cana-2735 203 43 = = SYM cana-2735 203 44 0 0 NUM cana-2735 203 45 , , PUNCT cana-2735 203 46 that that ADV cana-2735 203 47 is is ADV cana-2735 203 48 , , PUNCT cana-2735 203 49 the the DET cana-2735 203 50 space space NOUN cana-2735 203 51 𝐢 𝐢 PROPN cana-2735 203 52 is be AUX cana-2735 203 53 conformally conformally ADV cana-2735 203 54 flat flat ADJ cana-2735 203 55 . . PUNCT cana-2735 204 1 thus thus ADV cana-2735 204 2 we we PRON cana-2735 204 3 have have VERB cana-2735 204 4 the the DET cana-2735 204 5 following follow VERB cana-2735 204 6 theorem theorem PROPN cana-2735 204 7 . . PUNCT cana-2735 204 8 theorem theorem VERB cana-2735 204 9 4.3 4.3 NUM cana-2735 204 10 let let VERB cana-2735 204 11 𝐢 𝐢 PROPN cana-2735 204 12 = = SYM cana-2735 204 13 ( ( PUNCT cana-2735 204 14 𝑀 𝑀 PROPN cana-2735 204 15 , , PUNCT cana-2735 204 16 𝐾 𝐾 PROPN cana-2735 204 17 ) ) PUNCT cana-2735 204 18 be be AUX cana-2735 204 19 a a DET cana-2735 204 20 cartan cartan ADJ cana-2735 204 21 space space NOUN cana-2735 204 22 the the DET cana-2735 204 23 ( ( PUNCT cana-2735 204 24 𝛼 𝛼 PROPN cana-2735 204 25 , , PUNCT cana-2735 204 26 𝛽)-metric 𝛽)-metric ADJ cana-2735 204 27 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 204 28 , , PUNCT cana-2735 204 29 πœ” πœ” ADJ cana-2735 204 30 ) ) PUNCT cana-2735 204 31 = = SYM cana-2735 204 32 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 204 33 , , PUNCT cana-2735 204 34 πœ” πœ” PRON cana-2735 204 35 ) ) PUNCT cana-2735 204 36 + + NUM cana-2735 204 37 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 204 38 , , PUNCT cana-2735 204 39 πœ” πœ” PRON cana-2735 204 40 ) ) PUNCT cana-2735 204 41 + + CCONJ cana-2735 204 42 2π‘˜ 2π‘˜ NUM cana-2735 204 43 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 204 44 ) ) PUNCT cana-2735 204 45 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 204 46 ) ) PUNCT cana-2735 204 47 βˆ’ βˆ’ PROPN cana-2735 205 1 π‘˜2 π‘˜2 PROPN cana-2735 205 2 3 3 NUM cana-2735 205 3 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 205 4 ) ) PUNCT cana-2735 205 5 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 205 6 ) ) PUNCT cana-2735 205 7 admitting admit VERB cana-2735 205 8 β„Ž-metrical β„Ž-metrical PROPN cana-2735 205 9 𝑑-connection 𝑑-connection PROPN cana-2735 205 10 . . PUNCT cana-2735 206 1 hhen hhen AUX cana-2735 206 2 the the DET cana-2735 206 3 space space NOUN cana-2735 206 4 𝐢 𝐢 PROPN cana-2735 206 5 is be AUX cana-2735 206 6 conformally conformally ADV cana-2735 206 7 flat flat ADJ cana-2735 206 8 if if SCONJ cana-2735 206 9 and and CCONJ cana-2735 206 10 only only ADV cana-2735 206 11 if if SCONJ cana-2735 206 12 the the DET cana-2735 206 13 associated associated ADJ cana-2735 206 14 riemannian riemannian ADJ cana-2735 206 15 space space NOUN cana-2735 206 16 is be AUX cana-2735 206 17 locally locally ADV cana-2735 206 18 flat flat ADJ cana-2735 206 19 . . PUNCT cana-2735 207 1 conclusion conclusion NOUN cana-2735 207 2 : : PUNCT cana-2735 207 3 the the DET cana-2735 207 4 conditions condition NOUN cana-2735 207 5 derived derive VERB cana-2735 207 6 clarify clarify VERB cana-2735 207 7 how how SCONJ cana-2735 207 8 the the DET cana-2735 207 9 ( ( PUNCT cana-2735 207 10 Ξ± Ξ± NOUN cana-2735 207 11 , , PUNCT cana-2735 207 12 Ξ²)-metric Ξ²)-metric PUNCT cana-2735 207 13 𝐾(π‘₯ 𝐾(π‘₯ NOUN cana-2735 207 14 , , PUNCT cana-2735 207 15 πœ” πœ” ADJ cana-2735 207 16 ) ) PUNCT cana-2735 207 17 = = SYM cana-2735 207 18 𝛼(π‘₯ 𝛼(π‘₯ PROPN cana-2735 207 19 , , PUNCT cana-2735 207 20 πœ” πœ” PRON cana-2735 207 21 ) ) PUNCT cana-2735 207 22 + + NUM cana-2735 207 23 πœ–π›½(π‘₯ πœ–π›½(π‘₯ NOUN cana-2735 207 24 , , PUNCT cana-2735 207 25 πœ” πœ” PRON cana-2735 207 26 ) ) PUNCT cana-2735 207 27 + + CCONJ cana-2735 207 28 2π‘˜ 2π‘˜ NUM cana-2735 207 29 𝛽2(π‘₯,πœ” 𝛽2(π‘₯,πœ” NOUN cana-2735 207 30 ) ) PUNCT cana-2735 207 31 𝛼(π‘₯,πœ” 𝛼(π‘₯,πœ” NUM cana-2735 207 32 ) ) PUNCT cana-2735 208 1 βˆ’ βˆ’ PROPN cana-2735 208 2 π‘˜2 π‘˜2 PROPN cana-2735 208 3 3 3 NUM cana-2735 208 4 𝛽4(π‘₯,πœ” 𝛽4(π‘₯,πœ” NOUN cana-2735 208 5 ) ) PUNCT cana-2735 208 6 𝛼3(π‘₯,πœ” 𝛼3(π‘₯,πœ” NOUN cana-2735 208 7 ) ) PUNCT cana-2735 208 8 influences influence VERB cana-2735 208 9 the the DET cana-2735 208 10 geometric geometric ADJ cana-2735 208 11 structure structure NOUN cana-2735 208 12 of of ADP cana-2735 208 13 the the DET cana-2735 208 14 cartan cartan ADJ cana-2735 208 15 space space NOUN cana-2735 208 16 , , PUNCT cana-2735 208 17 particularly particularly ADV cana-2735 208 18 regarding regard VERB cana-2735 208 19 its its PRON cana-2735 208 20 transformation transformation NOUN cana-2735 208 21 into into ADP cana-2735 208 22 a a DET cana-2735 208 23 locally locally ADV cana-2735 208 24 minkowski minkowski ADJ cana-2735 208 25 space space NOUN cana-2735 208 26 . . PUNCT cana-2735 209 1 moreover moreover ADV cana-2735 209 2 , , PUNCT cana-2735 209 3 it it PRON cana-2735 209 4 identifies identify VERB cana-2735 209 5 the the DET cana-2735 209 6 criteria criterion NOUN cana-2735 209 7 under under ADP cana-2735 209 8 which which PRON cana-2735 209 9 the the DET cana-2735 209 10 space space NOUN cana-2735 209 11 becomes become VERB cana-2735 209 12 conformally conformally ADV cana-2735 209 13 flat flat ADJ cana-2735 209 14 , , PUNCT cana-2735 209 15 suggesting suggest VERB cana-2735 209 16 that that SCONJ cana-2735 209 17 the the DET cana-2735 209 18 curvature curvature NOUN cana-2735 209 19 can can AUX cana-2735 209 20 be be AUX cana-2735 209 21 transformed transform VERB cana-2735 209 22 to to ADP cana-2735 209 23 zero zero NUM cana-2735 209 24 by by ADP cana-2735 209 25 a a DET cana-2735 209 26 conformal conformal ADJ cana-2735 209 27 transformation transformation NOUN cana-2735 209 28 under under ADP cana-2735 209 29 h h NOUN cana-2735 209 30 - - ADJ cana-2735 209 31 metrical metrical ADJ cana-2735 209 32 d d NOUN cana-2735 209 33 - - NOUN cana-2735 209 34 connection connection NOUN cana-2735 209 35 . . PUNCT cana-2735 210 1 these these DET cana-2735 210 2 resuluts resulut NOUN cana-2735 210 3 may may AUX cana-2735 210 4 have have VERB cana-2735 210 5 implications implication NOUN cana-2735 210 6 for for ADP cana-2735 210 7 theoretical theoretical ADJ cana-2735 210 8 physics physic NOUN cana-2735 210 9 , , PUNCT cana-2735 210 10 particularly particularly ADV cana-2735 210 11 in in ADP cana-2735 210 12 contexts context NOUN cana-2735 210 13 like like ADP cana-2735 210 14 general general ADJ cana-2735 210 15 relativity relativity NOUN cana-2735 210 16 , , PUNCT cana-2735 210 17 where where SCONJ cana-2735 210 18 understanding understand VERB cana-2735 210 19 the the DET cana-2735 210 20 structure structure NOUN cana-2735 210 21 of of ADP cana-2735 210 22 spacetime spacetime NOUN cana-2735 210 23 is be AUX cana-2735 210 24 crucial crucial ADJ cana-2735 210 25 . . PUNCT cana-2735 211 1 references reference NOUN cana-2735 211 2 : : PUNCT cana-2735 212 1 [ [ X cana-2735 212 2 1 1 X cana-2735 212 3 ] ] X cana-2735 212 4 f. f. PROPN cana-2735 212 5 brickell brickell PROPN cana-2735 212 6 . . PUNCT cana-2735 213 1 β€œ " PUNCT cana-2735 213 2 a a DET cana-2735 213 3 relation relation NOUN cana-2735 213 4 between between ADP cana-2735 213 5 finsler finsler NOUN cana-2735 213 6 and and CCONJ cana-2735 213 7 cartan cartan PROPN cana-2735 213 8 structures structure NOUN cana-2735 213 9 ” " PUNCT cana-2735 213 10 . . PUNCT cana-2735 214 1 in in ADP cana-2735 214 2 : : PUNCT cana-2735 214 3 tensors tensor NOUN cana-2735 214 4 , , PUNCT cana-2735 214 5 n.s n.s PROPN cana-2735 214 6 . . PROPN cana-2735 214 7 25(1972 25(1972 NUM cana-2735 214 8 ) ) PUNCT cana-2735 214 9 , , PUNCT cana-2735 214 10 pp pp ADP cana-2735 214 11 . . PUNCT cana-2735 215 1 360–364 360–364 NUM cana-2735 215 2 . . PUNCT cana-2735 216 1 [ [ X cana-2735 216 2 2 2 X cana-2735 216 3 ] ] PUNCT cana-2735 216 4 e. e. PROPN cana-2735 216 5 cartan cartan PROPN cana-2735 216 6 . . PUNCT cana-2735 217 1 les les PROPN cana-2735 217 2 espaces espaces X cana-2735 217 3 de de X cana-2735 217 4 finsler finsler NOUN cana-2735 217 5 . . PUNCT cana-2735 218 1 actualites actualite NOUN cana-2735 218 2 79 79 NUM cana-2735 218 3 , , PUNCT cana-2735 218 4 herman herman NOUN cana-2735 218 5 , , PUNCT cana-2735 218 6 peris peris PROPN cana-2735 218 7 , , PUNCT cana-2735 218 8 1934 1934 NUM cana-2735 218 9 . . PUNCT cana-2735 219 1 [ [ X cana-2735 219 2 3 3 X cana-2735 219 3 ] ] PUNCT cana-2735 219 4 t. t. NOUN cana-2735 219 5 igrashi igrashi PROPN cana-2735 219 6 . . PUNCT cana-2735 220 1 β€œ " PUNCT cana-2735 220 2 ( ( PUNCT cana-2735 220 3 Ξ± Ξ± NOUN cana-2735 220 4 , , PUNCT cana-2735 220 5 Ξ²)metric Ξ²)metric ADJ cana-2735 220 6 in in ADP cana-2735 220 7 cartan cartan PROPN cana-2735 220 8 spaces space NOUN cana-2735 220 9 ” " PUNCT cana-2735 220 10 . . PUNCT cana-2735 221 1 in in ADP cana-2735 221 2 : : PUNCT cana-2735 221 3 tensors tensor NOUN cana-2735 221 4 , , PUNCT cana-2735 221 5 n.s n.s PROPN cana-2735 221 6 . . PROPN cana-2735 221 7 57 57 NUM cana-2735 221 8 ( ( PUNCT cana-2735 221 9 1994 1994 NUM cana-2735 221 10 ) ) PUNCT cana-2735 221 11 , , PUNCT cana-2735 221 12 pp pp ADP cana-2735 221 13 . . PUNCT cana-2735 222 1 78–82 78–82 X cana-2735 222 2 . . PUNCT cana-2735 223 1 [ [ X cana-2735 223 2 4 4 X cana-2735 223 3 ] ] PUNCT cana-2735 223 4 t. t. NOUN cana-2735 223 5 igrashi igrashi PROPN cana-2735 223 6 . . PUNCT cana-2735 224 1 β€œ " PUNCT cana-2735 224 2 remarkable remarkable ADJ cana-2735 224 3 connections connection NOUN cana-2735 224 4 in in ADP cana-2735 224 5 hamilton hamilton PROPN cana-2735 224 6 spaces space NOUN cana-2735 224 7 ” " PUNCT cana-2735 224 8 . . PUNCT cana-2735 225 1 in in ADP cana-2735 225 2 : : PUNCT cana-2735 225 3 tensors tensor NOUN cana-2735 225 4 , , PUNCT cana-2735 225 5 n.s n.s PROPN cana-2735 225 6 . . PROPN cana-2735 225 7 55 55 NUM cana-2735 225 8 ( ( PUNCT cana-2735 225 9 1992 1992 NUM cana-2735 225 10 ) ) PUNCT cana-2735 225 11 , , PUNCT cana-2735 225 12 pp pp ADP cana-2735 225 13 . . PUNCT cana-2735 226 1 151–161 151–161 NUM cana-2735 226 2 . . PUNCT cana-2735 227 1 [ [ X cana-2735 227 2 5 5 X cana-2735 227 3 ] ] PUNCT cana-2735 227 4 manoj manoj PROPN cana-2735 227 5 kumar kumar PROPN cana-2735 227 6 et et PROPN cana-2735 227 7 al al PROPN cana-2735 227 8 . . PUNCT cana-2735 228 1 β€œ " PUNCT cana-2735 228 2 on on ADP cana-2735 228 3 general general ADJ cana-2735 228 4 ( ( PUNCT cana-2735 228 5 Ξ± Ξ± X cana-2735 228 6 , , PUNCT cana-2735 228 7 Ξ²)-metrics Ξ²)-metrics PUNCT cana-2735 228 8 with with ADP cana-2735 228 9 cartan cartan ADJ cana-2735 228 10 torsion torsion NOUN cana-2735 228 11 , , PUNCT cana-2735 228 12 mean mean VERB cana-2735 228 13 cartan cartan PROPN cana-2735 228 14 torsion torsion NOUN cana-2735 228 15 and and CCONJ cana-2735 228 16 landsberg landsberg PROPN cana-2735 228 17 curvature curvature NOUN cana-2735 228 18 ” " PUNCT cana-2735 228 19 . . PUNCT cana-2735 229 1 in in ADP cana-2735 229 2 : : PUNCT cana-2735 229 3 aut aut PROPN cana-2735 229 4 journal journal NOUN cana-2735 229 5 of of ADP cana-2735 229 6 mathematics mathematics PROPN cana-2735 229 7 and and CCONJ cana-2735 229 8 computing compute VERB cana-2735 229 9 4.2 4.2 NUM cana-2735 229 10 ( ( PUNCT cana-2735 229 11 2023 2023 NUM cana-2735 229 12 ) ) PUNCT cana-2735 229 13 , , PUNCT cana-2735 229 14 pp pp ADP cana-2735 229 15 . . PUNCT cana-2735 230 1 105–112 105–112 NUM cana-2735 230 2 . . PUNCT cana-2735 231 1 [ [ X cana-2735 231 2 6 6 NUM cana-2735 231 3 ] ] PUNCT cana-2735 231 4 r. r. PROPN cana-2735 231 5 miron miron PROPN cana-2735 231 6 . . PUNCT cana-2735 232 1 β€œ " PUNCT cana-2735 232 2 cartan cartan PROPN cana-2735 232 3 spaces space NOUN cana-2735 232 4 in in ADP cana-2735 232 5 a a DET cana-2735 232 6 new new ADJ cana-2735 232 7 point point NOUN cana-2735 232 8 of of ADP cana-2735 232 9 view view NOUN cana-2735 232 10 by by ADP cana-2735 232 11 considering consider VERB cana-2735 232 12 them they PRON cana-2735 232 13 as as ADP cana-2735 232 14 dual dual ADJ cana-2735 232 15 of of ADP cana-2735 232 16 finsler finsler NOUN cana-2735 232 17 spaces space NOUN cana-2735 232 18 ” " PUNCT cana-2735 232 19 . . PUNCT cana-2735 233 1 in in ADP cana-2735 233 2 : : PUNCT cana-2735 233 3 tensors tensor NOUN cana-2735 233 4 , , PUNCT cana-2735 233 5 n.s n.s PROPN cana-2735 233 6 . . PROPN cana-2735 233 7 46 46 NUM cana-2735 233 8 ( ( PUNCT cana-2735 233 9 1987 1987 NUM cana-2735 233 10 ) ) PUNCT cana-2735 233 11 , , PUNCT cana-2735 233 12 pp pp ADP cana-2735 233 13 . . PUNCT cana-2735 234 1 329–334 329–334 NUM cana-2735 234 2 . . PUNCT cana-2735 235 1 [ [ X cana-2735 235 2 7 7 X cana-2735 235 3 ] ] X cana-2735 235 4 r. r. PROPN cana-2735 235 5 miron miron PROPN cana-2735 235 6 . . PUNCT cana-2735 236 1 β€œ " PUNCT cana-2735 236 2 the the DET cana-2735 236 3 geometry geometry NOUN cana-2735 236 4 of of ADP cana-2735 236 5 cartan cartan PROPN cana-2735 236 6 spaces space NOUN cana-2735 236 7 ” " PUNCT cana-2735 236 8 . . PUNCT cana-2735 237 1 in in ADP cana-2735 237 2 : : PUNCT cana-2735 237 3 progress progress NOUN cana-2735 237 4 of of ADP cana-2735 237 5 math math NOUN cana-2735 237 6 . . PUNCT cana-2735 238 1 22 22 NUM cana-2735 238 2 ( ( PUNCT cana-2735 238 3 1998 1998 NUM cana-2735 238 4 ) ) PUNCT cana-2735 238 5 , , PUNCT cana-2735 238 6 pp pp ADP cana-2735 238 7 . . PUNCT cana-2735 239 1 1–38 1–38 INTJ cana-2735 239 2 . . PUNCT cana-2735 240 1 [ [ X cana-2735 240 2 8 8 NUM cana-2735 240 3 ] ] X cana-2735 240 4 h.g h.g PROPN cana-2735 240 5 . . PROPN cana-2735 240 6 nagaraja nagaraja PROPN cana-2735 240 7 . . PUNCT cana-2735 241 1 β€œ " PUNCT cana-2735 241 2 on on ADP cana-2735 241 3 cartan cartan ADJ cana-2735 241 4 spaces space NOUN cana-2735 241 5 with with ADP cana-2735 241 6 ( ( PUNCT cana-2735 241 7 Ξ± Ξ± NOUN cana-2735 241 8 , , PUNCT cana-2735 241 9 Ξ²)metric Ξ²)metric ADJ cana-2735 241 10 ” " PUNCT cana-2735 241 11 . . PUNCT cana-2735 242 1 in in ADP cana-2735 242 2 : : PUNCT cana-2735 242 3 turk turk PROPN cana-2735 242 4 . . PUNCT cana-2735 243 1 j. j. PROPN cana-2735 243 2 math math PROPN cana-2735 243 3 . . PUNCT cana-2735 244 1 31 31 NUM cana-2735 244 2 ( ( PUNCT cana-2735 244 3 2007 2007 NUM cana-2735 244 4 ) ) PUNCT cana-2735 244 5 , , PUNCT cana-2735 245 1 pp pp ADP cana-2735 245 2 . . PUNCT cana-2735 246 1 363–369 363–369 NUM cana-2735 246 2 . . PUNCT cana-2735 247 1 [ [ X cana-2735 247 2 9 9 NUM cana-2735 247 3 ] ] PUNCT cana-2735 247 4 m. m. NOUN cana-2735 247 5 rafee rafee NOUN cana-2735 247 6 and and CCONJ cana-2735 247 7 gc gc PROPN cana-2735 247 8 chaubey chaubey NOUN cana-2735 247 9 . . PUNCT cana-2735 248 1 β€œ " PUNCT cana-2735 248 2 cartan cartan PROPN cana-2735 248 3 spaces space VERB cana-2735 248 4 with with ADP cana-2735 248 5 z. z. PROPN cana-2735 248 6 shen shen PROPN cana-2735 248 7 ’s ’s PROPN cana-2735 248 8 square square PROPN cana-2735 248 9 metric metric NOUN cana-2735 248 10 ” " PUNCT cana-2735 248 11 . . PUNCT cana-2735 249 1 in in ADP cana-2735 249 2 : : PUNCT cana-2735 249 3 far far PROPN cana-2735 249 4 east east PROPN cana-2735 249 5 journal journal PROPN cana-2735 249 6 of of ADP cana-2735 249 7 mathematical mathematical ADJ cana-2735 249 8 sciences sciences PROPN cana-2735 249 9 ( ( PUNCT cana-2735 249 10 fjms fjms NOUN cana-2735 249 11 ) ) PUNCT cana-2735 249 12 108 108 NUM cana-2735 249 13 ( ( PUNCT cana-2735 249 14 1 1 NUM cana-2735 249 15 ) ) PUNCT cana-2735 249 16 ( ( PUNCT cana-2735 249 17 2018 2018 NUM cana-2735 249 18 ) ) PUNCT cana-2735 249 19 , , PUNCT cana-2735 249 20 pp pp ADP cana-2735 249 21 . . PUNCT cana-2735 250 1 123–131 123–131 NUM cana-2735 250 2 . . PUNCT cana-2735 251 1 [ [ X cana-2735 251 2 10 10 NUM cana-2735 251 3 ] ] X cana-2735 251 4 h. h. PROPN cana-2735 251 5 rund rund PROPN cana-2735 251 6 . . PUNCT cana-2735 252 1 the the DET cana-2735 252 2 hamilton hamilton PROPN cana-2735 252 3 - - PUNCT cana-2735 252 4 jacobi jacobi PROPN cana-2735 252 5 theory theory NOUN cana-2735 252 6 in in ADP cana-2735 252 7 the the DET cana-2735 252 8 calculus calculus NOUN cana-2735 252 9 of of ADP cana-2735 252 10 variations variation NOUN cana-2735 252 11 . . PUNCT cana-2735 253 1 d. d. PROPN cana-2735 253 2 van van PROPN cana-2735 253 3 nostrand nostrand PROPN cana-2735 253 4 co. co. PROPN cana-2735 253 5 , , PUNCT cana-2735 253 6 london london PROPN cana-2735 253 7 , , PUNCT cana-2735 253 8 1966 1966 NUM cana-2735 253 9 . . PUNCT cana-2735 254 1 [ [ X cana-2735 254 2 11 11 NUM cana-2735 254 3 ] ] X cana-2735 254 4 g. g. PROPN cana-2735 254 5 shanker shanker PROPN cana-2735 254 6 . . PUNCT cana-2735 255 1 β€œ " PUNCT cana-2735 255 2 on on ADP cana-2735 255 3 the the DET cana-2735 255 4 cartan cartan ADJ cana-2735 255 5 spaces space NOUN cana-2735 255 6 with with ADP cana-2735 255 7 generalized generalized ADJ cana-2735 255 8 ( ( PUNCT cana-2735 255 9 Ξ± Ξ± NOUN cana-2735 255 10 , , PUNCT cana-2735 255 11 Ξ²)metric Ξ²)metric ADJ cana-2735 255 12 ” " PUNCT cana-2735 255 13 . . PUNCT cana-2735 256 1 in in ADP cana-2735 256 2 : : PUNCT cana-2735 256 3 j.t.s j.t.s VERB cana-2735 256 4 . . PUNCT cana-2735 256 5 4 4 NUM cana-2735 256 6 ( ( PUNCT cana-2735 256 7 2010 2010 NUM cana-2735 256 8 ) ) PUNCT cana-2735 256 9 , , PUNCT cana-2735 256 10 pp pp ADP cana-2735 256 11 . . PUNCT cana-2735 257 1 41–48 41–48 NUM cana-2735 257 2 . . PUNCT cana-2735 258 1 [ [ X cana-2735 258 2 12 12 NUM cana-2735 258 3 ] ] X cana-2735 258 4 s.k s.k PROPN cana-2735 258 5 . . PROPN cana-2735 258 6 singh singh PROPN cana-2735 258 7 . . PUNCT cana-2735 259 1 β€œ " PUNCT cana-2735 259 2 conformally conformally ADV cana-2735 259 3 minkowski minkowski ADJ cana-2735 259 4 type type NOUN cana-2735 259 5 spaces space NOUN cana-2735 259 6 and and CCONJ cana-2735 259 7 certain certain ADJ cana-2735 259 8 d d NOUN cana-2735 259 9 - - PUNCT cana-2735 259 10 connections connection NOUN cana-2735 259 11 in in ADP cana-2735 259 12 miron miron PROPN cana-2735 259 13 space space NOUN cana-2735 259 14 ” " PUNCT cana-2735 259 15 . . PUNCT cana-2735 260 1 in in ADP cana-2735 260 2 : : PUNCT cana-2735 260 3 indian indian ADJ cana-2735 260 4 j. j. PROPN cana-2735 260 5 of of ADP cana-2735 260 6 pure pure ADJ cana-2735 260 7 and and CCONJ cana-2735 260 8 applied applied ADJ cana-2735 260 9 math math NOUN cana-2735 260 10 . . PUNCT cana-2735 261 1 26.4 26.4 NUM cana-2735 261 2 ( ( PUNCT cana-2735 261 3 1993 1993 NUM cana-2735 261 4 ) ) PUNCT cana-2735 261 5 , , PUNCT cana-2735 261 6 pp pp ADP cana-2735 261 7 . . PUNCT cana-2735 262 1 339–346 339–346 NUM cana-2735 262 2 . . PUNCT cana-2735 263 1 [ [ X cana-2735 263 2 13 13 NUM cana-2735 263 3 ] ] X cana-2735 263 4 brijesh brijesh NOUN cana-2735 263 5 kumar kumar PROPN cana-2735 263 6 tripathi tripathi PROPN cana-2735 263 7 and and CCONJ cana-2735 263 8 vk vk PROPN cana-2735 263 9 chaubey chaubey NOUN cana-2735 263 10 . . PUNCT cana-2735 264 1 β€œ " PUNCT cana-2735 264 2 deformed deform VERB cana-2735 264 3 infinite infinite ADJ cana-2735 264 4 series series NOUN cana-2735 264 5 metric metric ADJ cana-2735 264 6 in in ADP cana-2735 264 7 cartan cartan PROPN cana-2735 264 8 spaces space NOUN cana-2735 264 9 ” " PUNCT cana-2735 264 10 . . PUNCT cana-2735 265 1 in in ADP cana-2735 265 2 : : PUNCT cana-2735 265 3 arxiv arxiv PROPN cana-2735 265 4 preprint preprint NOUN cana-2735 265 5 arxiv:1912.12176 arxiv:1912.12176 NOUN cana-2735 265 6 ( ( PUNCT cana-2735 265 7 2019 2019 NUM cana-2735 265 8 ) ) PUNCT cana-2735 265 9 . . PUNCT cana-2735 266 1 [ [ X cana-2735 266 2 14 14 NUM cana-2735 266 3 ] ] X cana-2735 266 4 a.a a.a PROPN cana-2735 266 5 . . PROPN cana-2735 266 6 shaikh shaikh PROPN cana-2735 266 7 u.c u.c PROPN cana-2735 266 8 . . PROPN cana-2735 266 9 de de PROPN cana-2735 266 10 and and CCONJ cana-2735 266 11 j. j. PROPN cana-2735 266 12 sengupta sengupta PROPN cana-2735 266 13 . . PUNCT cana-2735 266 14 tensor tensor NOUN cana-2735 266 15 calculus calculus NOUN cana-2735 266 16 . . PUNCT cana-2735 267 1 narosa narosa PROPN cana-2735 267 2 publishing publishing PROPN cana-2735 267 3 house house PROPN cana-2735 267 4 , , PUNCT cana-2735 267 5 new new PROPN cana-2735 267 6 delhi delhi PROPN cana-2735 267 7 , , PUNCT cana-2735 267 8 2008 2008 NUM cana-2735 267 9 . . PUNCT cana-2735 268 1 [ [ X cana-2735 268 2 15 15 NUM cana-2735 268 3 ] ] X cana-2735 268 4 sonia sonia NOUN cana-2735 268 5 rani rani PROPN cana-2735 268 6 , , PUNCT cana-2735 268 7 vinod vinod PROPN cana-2735 268 8 kumar kumar PROPN cana-2735 268 9 and and CCONJ cana-2735 268 10 mohammad mohammad PROPN cana-2735 268 11 rafee rafee PROPN cana-2735 268 12 . . PUNCT cana-2735 269 1 β€œ " PUNCT cana-2735 269 2 theory theory NOUN cana-2735 269 3 of of ADP cana-2735 269 4 cartan cartan ADJ cana-2735 269 5 space space NOUN cana-2735 269 6 with with ADP cana-2735 269 7 the the DET cana-2735 269 8 generalized generalized ADJ cana-2735 269 9 square square ADJ cana-2735 269 10 metric metric NOUN cana-2735 269 11 ” " PUNCT cana-2735 269 12 . . PUNCT cana-2735 270 1 in in ADP cana-2735 270 2 : : PUNCT cana-2735 270 3 advances advance NOUN cana-2735 270 4 in in ADP cana-2735 270 5 nonlinear nonlinear ADJ cana-2735 270 6 variational variational ADJ cana-2735 270 7 inequalities inequality NOUN cana-2735 270 8 ( ( PUNCT cana-2735 270 9 anvi anvi NOUN cana-2735 270 10 ) ) PUNCT cana-2735 270 11 27(2 27(2 PROPN cana-2735 270 12 ) ) PUNCT cana-2735 270 13 ( ( PUNCT cana-2735 270 14 2024 2024 NUM cana-2735 270 15 ) ) PUNCT cana-2735 270 16 , , PUNCT cana-2735 270 17 pp pp ADP cana-2735 270 18 . . PUNCT cana-2735 271 1 405–415 405–415 NUM cana-2735 271 2 . . PUNCT