Devendra Sir Cover Page copy.jpg BIBECHANA 17 (2020) 110-116 110 BIBECHANA ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, Biratnagar, Nepal Ricci solitons on Lorentzian para-Sasakian manifolds Riddhi Jung Shah Department of Mathematics, Janata Campus, Nepal Sanskrit University, Dang, Nepal Email: shahrjgeo@gmail.com Article Information Received: June 5, 2019 Accepted: December 5, 2019 Keywords: Ricci soliton LP-Sasakian manifold 2 W -curvature tensor 4 W -curvature tensor ABSTRACT In this paper we study Ricci solitons in Lorentzian para-Sasakian manifolds. It is proved that the Ricci soliton in a  12 n -dimensinal LP-Sasakian manifold is shrinking. It is also shown that Ricci solitons in an LP-Sasakian manifold satisfying the derivation conditions     0.,,0., 422  WXWWXR  and   0., 24 WXW  are shrinking but are steady for the condition   .0., 2 SXW  Finally, we give an example of 3-dimensional LP-Sasakian manifold and prove that the Ricci soliton is expanding and shrinking in this manifold. 1. Introduction A Ricci soliton is a natural generalization of an Einstein metric and is defined on a Riemannian manifold  ., gM A Ricci soliton is a triple  ,,Vg with g a Riemannian metric, V a vector field and  a real scalar such that ,022  gSgL V  (1.1) where S is the Ricci tensor and gL V denotes the Lie derivative of g along a vector field V [1]. The Ricci soliton is said to be shrinking, steady and expanding according as 0,0   and 0 respectively. Compact Ricci solitons are the fixed points of the Ricci flow S t g 2   projected from the space of metrics onto its quotient modulo diffeomorphism and scalings and often arise as blow-up limits for the Ricci flow on compact manifolds. Metrics satisfying (1.1) are interesting and useful in physics and often are referred as quasi-Einstein (eg, see [2], [3]). The Ricci flow was used by Perelman to prove the Poincare's conjecture theorem and the Thurston's geometrization conjecture theorem in topology [4]. Ricci solitons have also been studied by [5], [6], [7], [8] and others. On the other hand, the notion of a Lorentzian para-Sasakian manifold was introduced by https://creativecommons.org/licenses/by- nc/4.0/ This work is licensed under the Creative Commons CCBY-NC License. DOI: https://doi.org/10.3126/bibechana.v17i0.24341 http://nepjol.info/index.php/BIBECHANA mailto:shahrjgeo@gmail.com https://creativecommons.org/licenses/by-nc/4.0/ https://doi.org/10.3126/bibechana.v17i0.24341 Riddhi Jung Shah / BIBECHANA 17 (2020) 110-116 111 Matsumoto [9]. Mihai and Rosca defined the same notion independently and obtain several results on this manifold [10]. LP-Sasakian manifolds have also been studied by [11], [12], [13] and others. In this paper, we prove some derivation conditions for Ricci solitons in LP- Sasakian manifolds. We investigate shrinking property of Ricci soliton in a LP-Sasakin manifold when a vector field V is collinear with . We obtain some results of Ricci solitons on LP-Sasakian manifolds satisfying the conditions   ,0., 2 WXR    ,0., 2 SXW    0., 42 WXW  and   0., 24 WXW  respectively. Finally, we give an example of 3- dimensional LP-Sasakian manifold which is expanding and shrinking Ricci soliton. 2. Preliminaries Let M be a  12 n -dimensional differentiable manifold. Then M is said to be a Lorentzian para-Sasakian manifold (briefly LP-Sasakian manifold), if it admits a (1, 1) tensor field , a contravariant vector field , a 1-form  and a Lorentzian metric g which satisfy         ,0 ,0 , ,1 2   X XXX   (2.1)        ,,, YXYXgYXg   (2.2)    ,, XXg   (2.3) ,X X   (2.4)           ,2,  YXXYYXgY X  (2.5) for all ,, TMYX  where  denotes the covariant differentiation with respect to the Lorentzian metric g [9, 10]. If we put,    ,,, YXgYX  then  is a symmetric (0,2) tensor field [9]. Since the 1-form  is closed in an LP-Sasakian manifold we have [12], [9]        .,,, YXgYXgYXY X   (2.6) In a  12 n -dimensional LP-Sasakian manifold the following relations hold           ,,, , YZXgXZYg ZYXR    (2.7)       ,,, XYYXgYXR   (2.8)       ,, YXXYYXR   (2.9)    ,2, XnXS   (2.10)        ,2,, YXnYXSYXS   (2.11) for any vector fields ,,, ZYX where R and S are the Riemannian curvature tensor and the Ricci tensor of the manifold, respectively [11]. Let  ,,Vg be a Ricci soliton in a  12 n - dimensional LP-Sasakian manifold .M Then we have       .,,, XgYgYXgL YX    Using (2.4) and (2.6) in this equation we get     .,2, YXgYXgL    (2.12) From (1.1) and (2.12) we obtain      },,,{, YXgYXgYXS   (2.13)   ,12  nr provided .0. tr (2.14) In view of (2.1), (2.3) and (2.13) we get    ., XXS   (2.15) 3. Results and Discussion Now, we have the following results and their proofs Theorem 3.1: If in a  12 n -dimensional LP- Sasakian manifold the metric g is a Ricci soliton and V is pointwise collinear with , then V is a constant multiple of  and g is shrinking. Proof: Let M be a  12 n -dimensional LP- Sasakian manifold with Lorentzian metric .g A Ricci soliton is a generalization of an Einstein metric and defined on a Riemannian manifold  gM , by (1.1). Let V be pointwise collinear with  i.e., cV  where c is a function on a  12 n -dimensional LP-Sasakian mani-fold. Then from (1.1), we have Riddhi Jung Shah / BIBECHANA 17 (2020) 110-116 112        .0,2,2,  YXgYXSYXgL c   (3.1) Further simplification and use of (2.4) in (3.1) yields                 .0,2,2 ,,   YXgYXSXYc XYcgYXcYXcg   (3.2) By virtue of (2.6) and (3.2) we obtain               .0,2,2 ,2   YXgYXS XYcYXcYXcg   (3.3) Putting Y in (3.3) and using (2.1), (2.3) and (2.10) we get        .024  XcXnc  (3.4) Taking X in (3.4) gives  .2   nc (3.5) In view of (3.4) and (3.5) we obtain    ,2 XnXc  which implies   .2  ndc (3.6) Taking exterior derivative on both sides of (3.6) we get   ,02   dn (3.7) since ,0d we have .02  n Hence c is constant from (3.6). Consequently, the equation (3.3) reduces to       .,,, YXcgYXgYXS   (3.8) Comparing (2.13) and (3.8) we get .1c Again, 02  n implies that 02  n for .1n Thus the Ricci soliton is shrinking. This proves the theorem. Theorem 3.2: A Ricci soliton in a 2 W -semi- symmetric LP-Sasakian manifold of dimension  12 n is shrinking. Proof: Let M be a  12 n -dimensional LP- Sasakian manifold admitting a Ricci soliton  .,, Vg The 2 W -curvature tensor in M is defined by [14]            ],,, ,,[ 2 1 ,,,,,, 2 TXRicZYg TYRicZXg n TZYXRTZYXW   this can be written as         ]., ,[ 2 1 ,, 2 QXZYg QYZXg n ZYXRZYXW   (3.9) Putting X in (3.9) and using (2.3) and (2.8) we get           ].,[ 2 1 ,, 2   QZYgQYZ n YZZYgZYW   (3.10) Taking inner product on both sides of (3.9) with  and using (2.7) and (2.15) we obtain           ]. [ 2 1, , , 2 YZXg XZYg n ZYXW             (3.11) Suppose that the condition     0,., 2 UZYWXR  holds in .M Then by definition we have               0 ,,,, ,,,, 22 22    UXRZYWUZXRYW UZYXRWUZYWXR   (3.12) for all vector fields UZYX ,,, on .M In view of (2.8) and (3.12) we get                               .0 ,,, ,,, ,,, ,,, 22 22 22 22      XZYWUZYWUXg UXYWZUYWZXg UZXWYUZWYXg XUZYWUZYWXg     (3.13) Taking inner product on both sides of (3.13) with  and using (2.1) we obtain                                       .0 ,,, ,,, ,,, ,,, 22 22 22 22      XZYWUZYWUXg UXYWZUYWZXg UZXWYUZWYXg XUZYWXUZYWg     (314) In view of (3.9), (3.11) and (3.14), we get                                                                    .0}],,{ },,{ },{, },,{ },{,}, ,{[ 2 1],, ,,[ 2 1 ,,              ZYXgYZXgU XUYgYUXgZ UYUYgZXg ZUXgXUZgY ZUUZgYXgZUYg YZUgX n YXSUZg ZXSUYg n XUZYRg        (3.15) Let  12,...,2,1:  nie i be an orthonormal basis of the tangent space at any point of the manifold. Putting i eYX  in (3.15) and taking summation over ,121 ,  nii we get Riddhi Jung Shah / BIBECHANA 17 (2020) 110-116 113      ., 12 22 , UZg n nnr UZS     (3.16) Again taking an orthonormal frame field at any point of the manifold and contracting over Z and U in (3.16) we have ,02  n for .1n Hence the Ricci soliton is shrinking. This completes the proof of the theorem. Theorem 3.3: Let M be a  12 n -dimensional LP-Sasakian manifold and  ,,Vg be a Ricci soliton satisfying the condition   0., 2 SXW  in ,M then the Ricci soliton is steady. Proof: Let M be a  12 n -dimensional LP- Sasakian manifold and  ,,Vg be a Ricci soliton in .M Suppose that the condition     0,., 2 ZYSXW  holds in ,M then we have       .0,,,, 22  ZXWYSZYXWS  (3.17) In view of (3.10), (2.15) and (3.17) we obtain                                  .0,,,, ,,} { 2 1 ,, , ,     YQSZXgZQSYXg ZYXSYZXS n YZXgZYXg ZYQXS YZQXS      (3.18) Putting Z in (3.18) and using (2.1), (2.3) and (2.15) we get                 .],, } 2 ,1{[2 , YQSXYXS YX n YXgn YQXS       (3.19) Again taking Y in (3.19) and using (2.1), (2.3) and (2.15) we obtain     ,012 2  Xn  (3.20) since   ,0X (3.20) implies that .0 Thus the Ricci soliton is steady. This proves the theorem. Theorem 3.4: A Ricci soliton in a  12 n - dimensional LP-Sasakian manifold satisfying the condition   0., 42 WXW  is shrinking under the condition .0. tr Proof: Let M be a  12 n -dimensional LP- Sasakian manifold and  ,,Vg be a Ricci soliton in .M The 4 W -curvature tensor in M is defined by [15]            ] [ 2 1 ,,, ,,, ,, ,, 4 TZRicYXg TYRicZXg n TZYXR TZYXW   which can be written as         ]. [ 2 1 , , , , 4 QZYXg QYZXg n ZYXR ZYXW   (3.21) Putting X in (3.21) and using (2.3) and (2.8) we obtain           . 2 1 ,, 4 QZYQYZ n YZZYgZYW     (3.22) Taking inner product on both sides of (3.21) with  and using (2.7) and (2.15) we get               ., 2 1 , 2 , , 4 YZXg n ZYXg n XZYg ZYXW               (3.23) Now, we assume that the condition     0,., 42 UZYWXW  holds in ,M then we have               .0,, ,, ,, ,, 24 24 24 42    UXWZYW UZXWYW UZYXWW UZYWXW     (3.24) In view of (3.10) and (3.24) we get                                                             .0, ,,, ,,, ,,],, ,,, ,,, ,,, ,[ 2 1 ,,, 4 44 44 44 44 44 44 4 44          XZYWU ZYWUXgUXYWZ UYWZXgUZXWY UZWYXgQZYWUXg QXZYWUUQYWZXg UQXYWZUZQWYXg UZQXWYQXUZYWg QXUZYW n XUZYWXUZYWg          (3.25) Taking inner product on both sides of (3.25) with  and using (2.1), (2.3) and (2.15) we obtain Riddhi Jung Shah / BIBECHANA 17 (2020) 110-116 114                                                              .0,,, ,,, ,,, },,, ,,, )),((),()),(({ 2 1 ,,, 2 1 1 44 44 44 44 44 44 44              XZYWUZYWUXg UXYWZUYWZXg UZXWYUZWYXg QZYWUXgQXZYWU UQYWZXgUQXYWZ UZQWYXgUZQXWY n UZYWXXUZYWg n        (3.26) In view of (3.21), (3.23), (3.26) and (2.15) we get                                                 .0)()(,, 2 1 },,{ 4 ],, 2 1},, ,, ,,,,{ 2 1 ,,,,[ 2 1 2                   ZUYXSYXg n ZYUXgUXS n UYgZXg n UXgZYg YUZXgYUZXS UXSZYgZXSUYg n UZgYXgXUZYRg n        (3.27) Let  12,...,2,1: ne i be an orthonormal basis of the tangent space at any point of the manifold. Putting i eYX  in (3.27) and summing over ,121 ,  nii we get          . 2 12 ,2, ZU n rn ZUngZUS              (3.28) Again taking  ZU and using (2.1), (2.3), (2.14) and (2.15) we get .02  n Thus  is negative. This concludes that the Ricci soliton is shrinking. This completes the proof of the theorem. Theorem 3.5: Let M be a  12 n -dimensional LP-Sasakian manifold and  ,,Vg be a Ricci soliton in .M If g satisfies the condition   ,0., 24 WXW  then g is shrinking under the condition .0. tr Proof: Let M be a  12 n -dimensional LP- Sasakian manifold and  ,,Vg be a Ricci soliton in .M Suppose that the condition     0,., 24 UZYWXW  holds in ,M then by definition we have               .,,,, ,,,,0 4242 4224 UXWZYWUZXWYW UZYXWWUZYWXW     (3.29) By virtue of (3.22) and (3.29) we have                                                              .0 ,,, ,,, ,,, ],, ,, ,, ,,[ 2 1 ,,, 22 22 22 22 22 22 22 22          XZYWUZYWUXg UXYWZUYWZXg UZXWYUZWYXg QUZYWXQXZYWU UQZYWXUQXYWZ UZQYWXUZQXWY UZYQWXQXUZYW n XUZYWXUZYWg         (3.30) Taking inner product on both sides of (3.30) with  and using (2.1), (2.3) and (2.15) we obtain                                                                               .0,,, ,,, ,,, ],, ,, ,, ,,,[ 2 1 ,,, 22 22 22 22 22 22 22 22         XZYWUZYWUXg UXYWZUYWZXg UZXWYUZWYXg QUZYWXQXZYWU UQZYWXUQXYWZ UZQYWXUZQXWY UZYQWgXUZYWX n UZYWXXUZYWg         (3.31) In view of (3.9) and (3.11), (3.31) yields                          .0)}]()(),( 2 1 )()(),( 2 1 )()(),( ,{ 1 ,, ,,[ 2 1} { 2 1 ,, ,, ,,             ZUYXS UYZXSYXUZS ZXUYS n UZgYXg ZXgUYg n n XUZYRg YXSUZg ZXSUYg     (3.32) Let  12,...,2,1:  nie i be an orthonormal basis of the tangent space at any point of the manifold. Putting i eYX  in (3.32) and summing over ,121 ,  nii we get        . 264 326 , 264 248 , 2 2 2 23 ZU nn rnrn ZUg nn nrnn UZS                        (3.33) Again, putting UZ in (3.33) and using (2.1), (2.3) and (2.15) we get .044 22  nn (3.34) Riddhi Jung Shah / BIBECHANA 17 (2020) 110-116 115 This equation gives nn 2,2  . Hence  is negative. This concludes that the Ricci soliton is shrinking. Thus the theorem is proved. From theorem 3.4 and theorem 3.5 we can state next theorem Theorem 3.6: Ricci solitons in a  12 n - dimensional LP-Sasakian manifold satisfying the derivation conditions   0., 42 WXW  and   0., 24 WXW  are equivalent. Now we give an example of LP-Sasakian manifold. 4. Example for 3-dimensional LP-Sasakian Manifold Let us consider a 3-dimensional manifold     ,,,:,, 3RzyxzyxM  where  zyx ,, are standard coordinates in . 3 R We choose the vector fields , , , 321 x E yz eE y eE xx                   (4.1 which are linearly independent at each point of .M Let g be the Lorentzian metric defined by       ,0,,, 313221  EEgEEgEEg (4.2)       .1,,, 332211  EEgEEgEEg (4.3) Let  be a 1-form defined by     3 , EZgZ  for any vector field Z on .M Let  be a (1, 1) tensor field defined by       0., , 32211  EEEEE  (4.4) The linearity property of  and g yields that       , ,1 3 2 3 EUUUE   (4.5)        ,,, UZUZgUZg   (4.6) for any vector fields UZ , on .M Thus for  gE ,,, , 3  defines a Lorentzian paracontact structure on .M By the definition of Lie bracket and (4.1) we have       ., ,0, ,, 23221131 EEEEEEEE  (4.7) Let  be the Levi-Civita connection with respect to the Lorentzian metric ,g the Koszul formula is defined as                 .,, ,,,, ,,, ,2 YXZg XZYgZYXg YXZgXZYgZYXg ZYg X     (4.8) In view of (4.2), (4.3), (4.7) and (4.8) we get                   .,2 ,,,,,, ,,, ,2 11 311113131 311113131 131 EEg EEEgEEEgEEEg EEgEEEgEEEgE EEg E     Similarly, we can obtain     21231 ,20,2 EEgEEg E  and    .,20,2 31331 EEgEEg E  From above we can write    XEgXEg E ,2,2 131  for all  .MX  Thus . 131 EE E  Proceeding same way we obtain .0 ,0 , ,, ,0 , 132333 12322 232311E 21131          EEE EEE EEEE EEE EEE EE E EE (4.9) Now, we have     . 3 311 111111 E EE EEE E EEE      Again, from definition and by the use of (2.5) we obtain           . 2, 3 31111311 11 E EEEEEEEEg E E      Similarly, we obtain other relations. Thus we have                              .0 ,0,0 ,, ,0, ,0, 33 2313 232322 12131 21311 E EE EEEE EEE EEE E EE EE EE EE      (4.10) From (4.5), (4.6), (4.9) and (4.10), we see that the equations (2.1) - (2.5) are satisfied by the manifold ,M for . 3 E Hence  g,,,  is an LP-Sasakian structure in .M Consequently  gM ,,, 3  is an LP-Sasakian manifold. Now, the Riemannian curvature tensor is defined by     ., , ZZZZYXR YX XYYX  (4.11) By virtue of (4.7), (4.9) and (4.11) we obtain Riddhi Jung Shah / BIBECHANA 17 (2020) 110-116 116     .0 , 1221 32,1312321321   EE EEEEEER EE EEEEEE Similarly, we obtain                                     .0,, 0,,, ,0,,, ,0,,, ,,,, ,,,0, 333222 1113131 1322121 2313232 12211331 2332321 EEEREEER EEEREEEER EEEREEEER EEEREEEER EEEEREEEER EEEEREEER (4.12) By the use of (4.12) we get                .0 ,, ,,,, , 1111 1331 3 1 122111 11 ,,        EEgEEg EEEERgEEEERg EES EEEERg i ii Similarly, we obtain   0, 22 EES and   .2, 33 EES Thus we have           .2, ,0,, 33 2211 EES EESEES (4.13) From (2.13) we have       .,,, iiiiii EEgEEgEES   This equation yields      ,1,, 2211  EESEES by the use of (4.3), (4.4) and (4.13) for .2 ,1i This implies ,01  for .2,1i And   ,, 33 EES for .3i This yields .02  Since 01 for 2,1i and 02  for ,3i this is an example of expanding and shrinking Ricci soliton in 3-dimensional LP-Sasakian manifold. 5. 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