BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal On contact conformal curvature tensor in LP-Sasakian manifolds Riddhi Jung Shah Department of Mathematics, Janata Campus, Nepal Sanskrit University, Dang, Nepal. *Email: shahrjgeo@gmail.com Article history: Received 01 February, 2017; Accepted 27 October , 2017 DOI: http://dx.doi.org/10.3126/bibechana.v15i0.16507 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ Abstract The purpose of the present paper is to study the contact conformal curvature tensor in LP-Sasakian manifolds. Some properties of contact conformally flat,  -contact conformally flat and contact conformally semi-symmetric LP-Sasakian manifolds are obtained. Keywords: Contact conformal curvature tensor; LP-Sasakian manifold;  -Einstein manifold. 1. Introduction The contact conformal curvature tensor is a curvature like tensor defined on a contact metric manifold which is constructed from the conformal curvature tensor by using the Boothby-Wang's fibration [1]. Jeong, Lee and Pak [2] defined the contact conformal curvature tensor on  12 n -dimensional Sasakian manifolds and proved that it is invariant under D-homothetic deformation. They also proved that a Sasakian manifold  212  nM n with vanishing contact conformal curvature tensor field is of constant  -homothetic sectional curvature    .1/]13[  nnnnr Pak and Shin [3] gave a geometric characterization of a contact metric manifold with vanishing contact conformal curvature tensor by showing that for ,2n every  12 n -dimensional contact metric manifold with vanishing contact conformal curvature tensor is a Sasakian space form. Bang and Kye [4] studied contact conformal curvature tensor on 3-dimensional Sasakian manifolds and gave a partial extension of Pak and Shin's result to 3-dimensional locally  -symmetric contact metric manifold and also showed that the contact conformal curvature tensor on 3-dimensional Sasakian manifold always vanishes. On the other hand, Matsumoto [5] introduced the notion of Lorentzian para-Sasakian manifold. Then Mihai and Rosca [6] introduced the same notion independently and obtained many results on this manifold. Lorentzian para-Sasakian manifolds have also been studied by Matsumoto and Mihai [7], De et al. [8], Shaikh and Biswas [9] and Bagewadi et al. [10]. 2. Preliminaries A differentiable manifold of dimension  12 n is called 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 Riddhi Jung Shah / BIBECHANA 15 (2018) 24-29: RCOST p.24 http://nepjol.info/index.php/BIBECHANA mailto:shahrjgeo@gmail.com http://dx.doi.org/10.3126/bibechana.v15i0.16507 https://creativecommons.org/licenses/by-nc/4.0/ (2.1)       ,,1 2  XXX  (2.2)        ,,, YXYXgYXg   (2.3)    ,, XXg   (2.4) ,X X   (2.5)           ,2,  YXXYYXgY X  where  denotes the covariant differentiation with respect to the Lorentzian metric g [5,7]. In an LP-Sasakian manifold it can be easily seen that; (2.6) .2rank,0,0 n   If we put (2.7)    ,,, YXgYX  for any vector fields X and ,Y then the tensor field  YX , is a symmetric (0, 2) tensor field [5]. Also since the 1-form  is closed in an LP-Sasakian manifold we have (2.8)            ,0,,,,,   XYXgYXgYXY X for any vector fields X and Y [5,9]. An LP-Sasakian manifold M is said to be  -Einstein if its Ricci tensor S of type (0, 2) is of the form (2.9)        ,,, YXbYXagYXS  for any vector fields X and ,Y where ba, are smooth functions on the manifold. In particular, if ,0b then the manifold is said to be an Einstein manifold. In a  12 n -dimensional LP-Sasakian manifold the following relations hold: (2.10)           ,,,, YZXgXZYgZYXR   (2.11)       ,,, XYYXgYXR   (2.12)       ,, YXXYYXR   (2.13)     ,,  XXXR  (2.14)    ,2, XnXS   (2.15)        ,2,, YXnYXSYXS   for any vector fields YX , and ,Z where R and S are the Riemannian curvature tensor and Ricci tensor of the manifold, respectively [8, 9]. In a  12 n -dimensional LP-Sasakian manifold the contact conformal curvature tensor 0 C of type (1, 3) is defined by [2] can be written as (2.16)                                                                                     },,, }{ 2 23 254{ 12 1 }, ,}{ 2 23 2{ 12 1 },2 ,,}{ 2 2 22{ 12 1 },2,2, ,,, ,,, ,,,{ 2 1 ,, 2 2 0        ZXgYZYgXYZX XZY n rn nn nn YZXg XZYg n rn n nn ZYXg YZXgXZYg n rn nn nn ZYXSZQYXgXQZYg YQZXgXZYSYZXSQXZY QYZXZYSXZXSYQYZXg QXZYgYZXSXZYS n ZYXRZYXC                   Riddhi Jung Shah / BIBECHANA 15 (2018) 24-29: RCOST p.25 where QSR ,, and r denote the curvature tensor, the Ricci tensor, the Ricci operator and the scalar curvature, respectively. Definition 2.1 A  12 n -dimensional LP-Sasakian manifold M is said to be contact conformally flat if the condition (2.17)   0, 0 ZYXC holds. Definition 2.2 A  12 n -dimensional LP-Sasakian manifold M is said to be  -contact conformally flat if (2.18)   .0, 0 YXC Definition 2.3 A Riemannian or pseudo-Riemannian manifold is said to be semi-symmetric if the condition (2.19)   0., RYXR holds, where  YXR , denotes the derivation in the tensor algebra at each point of the manifold. Definition 2.4 A  12 n -dimensional LP-Sasakian manifold M is said to be contact conformally semi-symmetric if (2.20)   .0., 0 CYXR 3. Results and Discussion We prove the following results which are related with above definitions Theorem 3.1 A contact conformally flat LP-Sasakian manifold M of dimension  12 n is an  - Einstein manifold. Proof: Let us consider a contact conformally flat LP-Sasakian manifold ,M then (2.17) holds and from (2.16) we have (3.1)                                                                                   }.,, }{ 2 23 254{ 12 1 }, ,}{ 2 23 2{ 12 1 },2 ,,}{ 2 2 22{ 12 1 },2,2, ,,, ,,, ,,,{ 2 1 ,0 2 2        ZXgYZYgXYZX XZY n rn nn nn YZXg XZYg n rn n nn ZYXg YZXgXZYg n rn nn nn ZYXSZQYXgXQZYg YQZXgXZYSYZXSQXZY QYZXZYSXZXSYQYZXg QXZYgYZXSXZYS n ZYXR                   Taking inner product on both sides of (3.1) by ,W we get Riddhi Jung Shah / BIBECHANA 15 (2018) 24-29: RCOST p.26 (3.2)                                                                              WYgZXgWXgZYg n rn nn nn WZgYXSWZSYXgWXSZYg WYSZXgWXgZYSWYgZXSWXSZY WYSZXWZYSXWZXSYWYSZXg WXSZYgWYgZXSWXgZYS n WZYXR ,,,,}{ 2 2 22{ 12 1 },,2,,2,, ,,,,,,, ,,,,, ,,,,,,{ 2 1 ,,, ~ 0 2                                                         },,,, ,}{ 2 23 254{ 12 1 },, ,,}{ 2 23 2{ 12 1 },,2 2 WZXgYWZYgXWYgZX WXgZY n rn nn nn WYgZXg WXgZYg n rn n nn WZgYXg             where     .,,,,, ~ WZYXRgWZYXR  Setting W in (3.2) and using (2.1), (2.3), (2.6), (2.10), (2.14) and then further simplifying yields (3.3)                     .,, ,, 122 234108 0 23 YZXSXZYS YZXgXZYg nn rnnnn              In (3.3) replacing X by  and using (2.1), (2.3) and (2.14), we get (3.4)                    . 12 232762 , 12 232542 , 2 2 ZY nn rnnnn ZYg nn rnnnn ZYS                    Equation (3.4) implies that (3.5)        ,,, ZYZYgZYS   where      12 232542 2    nn rnnnn  and       . 12 232762 2    nn rnnnn  The relation (3.5) implies that the manifold is an  -Einstein manifold. This completes the proof of the theorem. Theorem 3.2 Let M be a  12 n -dimensional LP-Sasakian manifold. If the condition   0, 0 YXC holds in ,M then the manifold is an  -Einstein manifold. Proof: Let us consider a  12 n -dimensional LP-Sasakian manifold M which is  -contact conformally flat, then we have   .0, 0 YXC Now, replacing Z by  in (2.16) and using (2.1), (2.3), (2.6), (2.12), (2.14) and (2.18), we get (3.6)                . 12 232542 0 2 QYXQXY YXXY nn rnnnn              Taking inner product on both sides of (3.6) by ,W we obtain Riddhi Jung Shah / BIBECHANA 15 (2018) 24-29: RCOST p.27 (3.7)                       ., ,, 12 232542 , 2 YWXS XWYgYWXg nn rnnnn XWYS              Putting X in (3.7) and using (2.1), (2.3) and (2.14), we get (3.8)                    . 12 232762 , 12 232542 , 2 2 WY nn rnnnn WYg nn rnnnn WYS                    From (3.8), we have (3.9)        ,,, WYBWYAgWYS  where                12 232542 2 nn rnnnn A and     . 12 )23(2762 2          nn rnnnn B Hence the manifold is an  -Einstein manifold. This completes the proof of the theorem. Theorem 3.3 A contact conformally semi-symmetric LP-Sasakian manifold  gM n ,12  is an Einstein manifold and a manifold of constant curvature  .122  nnr Proof: Let us consider an LP-Sasakian manifold  gM n ,12  satisfying the condition   .0., 0 CYXR Now, we have (3.10)                   .,,,, ,,,,,., 00 000 ZYXRVUCZVYXRUC ZVUYXRCZVUCYXRZVUCYXR   In view of (2.20) and (3.10), we get (3.11)               .,,,, ,,,,0 00 00 ZYXRVUCZVYXRUC ZVUYXRCZVUCYXR   Taking X in (3.11) and using (2.11), we obtain (3.12)                               .,,, ,,,, ,,,,,0 00 000 000 YVUCZVUCZYg ZYUCVZUCVYgZVYCU ZVCUYgYZVUCYZVUCg       Taking inner product on both sides of (3.12) by , we get (3.13)                                      .,,, ,,,, ,,,,,0 00 000 000 YVUCZVUCZYg ZYUCVZUCVYgZVYCU ZVCUYgYZVUCYZVUCg       Putting UY  in (3.13) we obtain (3.14)                            .,,,, ,,,,,,0 000 000 UVUCZVUCZUgZUUCV ZUCVUgZVCUUgUZVUCg     Now, from (2.16) we have (3.15)    ,0, 0  VUC Riddhi Jung Shah / BIBECHANA 15 (2018) 24-29: RCOST p.28 (3.16)                       ZV nn rnnnn ZVg nn rnnnn ZVS n ZVC   12 232762 , 12 232542 , 1 , 2 2 2 2 0       and (3.17)    .0, 0 ZUUC By virtue of (3.15) and (3.17), (3.14) reduces to (3.18)                  .,,, ,,,,0 00 00 UVUCZZUCVUg ZVCUUgUZVUCg     Let  12,..,2,1:  nie i be an orthonormal basis of the tangent space at any point of the manifold. Putting i eU  in (3.18) and taking summation over ,121,  nii we get (3.19)                  12 1 12 1 000 ,0,,2,, n i n i iiii eVeCZZVCneZVeCg  since            12 1 00 .,,, n i ii ZVCZeCVeg  Again, from (2.16) it follows (3.20)                         ZV nn rnnnnnn ZVg nn rnnnnn ZVS n n eZVeCg n i ii                        12 4222524 , 12 22242 , 12 ,, 2 223 2 22 12 1 0 under the condition   0..  Qtrtr  and by the use of (2.2), (2.8) and (2.15). From the definition of contact conformal curvature tensor, we also have (3.21)                      12 1 0 . 1 12212 , n i ii ZV nn nnrn eVeCZ  In view of (3.16), (3.20) and (3.21), (3.21) takes the form (3.22)        ,,, 21 ZVZVgZVS   where      14 234231482 223 1    nn rnnnnnn  and      . 1221 2 n rnnn   Taking an orthonormal frame field at any point of the manifold and contracting over V and Z in (3.22) we get (3.23)  .122  nnr Using (3.23) in (3.22) we obtain (3.24)    .,2, ZVngZVS  In view of (3.23) and (3.24), the theorem is proved. 4. Conclusions In this paper, we have studied on contact conformal curvature tensor in a  12 n -dimensional Lorentzian para-Sasakian manifold (briefly, LP-Sasakian manifold). We have investigated that Riddhi Jung Shah / BIBECHANA 15 (2018) 24-29: RCOST p.28 contact conformally flat and  -contact conformally flat LP-Sasakian manifold is an  -Einstein manifold. 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