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Sasakian Finsler Structures on Diffusion of Tangent Bundles Satisfying 푪∗ 푿푯, 풀푯 푺 = ퟎ

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Sasakian Finsler Structures on Diffusion of Tangent Bundles Satisfying 푪∗ 푿푯, 풀푯 푺 = ퟎ International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 3, Issue 10 October 2016 Sasakian Finsler Structures on Diffusion of Tangent Bundles ∗ 푯 푯 Satisfying 푪 푿 , 풀 푺 = ퟎ Nesrin Caliskan1,, A. Funda Yaliniz2 1Faculty of Education, Department of Elementary Mathematics Education, Usak University, 64200, Usak- TURKEY 2Department of Mathematics, Faculty of Art and Sciences, Dumlupinar University, 43100, Kutahya-TURKEY Abstract. Inthis paper,the quasi-conformal curvature tensor 퐶∗ of Sasakian Finsler structures ontangent bundlesis considered. The type of these manifolds satisfying퐶∗ 푋퐻, 푌퐻 푆 = 0 are discussed where푆 denotes the Ricci tensor field 0 0 of type and 푋퐻, 푌퐻휖 푇 푇푀 퐻 . In addition, relations between these structures and 휂-Einstein structures are 2 2 푢 0 examined. Mathematics Subject Classification (2010). 53D15, 53C05, 53C15, 53C60. Keywords.Quasi-conformal curvature tensor; 휼-Einstein;tangent bundle; Sasakian Finsler structure. 1. Introduction and main results There are many different frameworks to study Finsler geometry (like in [1,2]).In this study, we contextualized by using double tangent bundles of Finsler manifolds (such as [3,4,5]). Quasi-conformal curvature tensor is studied in papers for several structures(like[2,6-9,10]). In this study, the notion of “quasi-conformal curvature tensor” for Sasakian Finsler Structures on diffusions of tangent bundles is defined and we found different invariants. In order to study this, essential characteristics of these structures are given; Let 푀 be an 푚 = 2푛 + 1 -dimensional smooth manifold and푇푀 its tangent bundle, 푢 = 푥, 푦 ∈ 푇푀. Ifa function퐹: 푇푀 → 0, ∞ satisfies 1 ∞ i) 퐹 ∈ 퐶 (푇푀) and 퐹 ∈ 퐶 (푇푀0), 푖 푖 푖 푖 ii) 퐹 푥 , 휆푦 = 휆퐹 푥 , 푦 , 휆 > 0, 1 휕2퐹2 iii) The 푚 × 푚 Hessian matrix 푔 푥, 푦 = is positive definite on 푇푀 , 푖푗 2 휕푦 푖휕푦 푗 0 푚 then퐹 = 푀, 퐹 is called a Finsler manifold and푔 is a Finsler metric tensor with 푔푖푗 coefficients [11]. It is well known that, double tangent bundle can be stated by horizontal and vertical complementary 퐻 푉 diffusions; 푇푇푀0 = 푇푇푀0 푇푇푀0 and it is possible to found Sasakian Finsler structures with the help of these subbundle structures. 퐻 퐻 퐻 퐻 푉 푉 푉 푉 퐻 푉 휙 , 휉 , 휂 , 퐺 and 휙 , 휉 , 휂 , 퐺 are 푚-dimensional Sasakian Finsler structures on푇푀0 and 푇푀0 , respectively. 퐻 푉 The following relations hold for Sasakian structures on푇푀0 and푇푀0 [5]: 퐻 퐻 푉 푉 훷. 훷 = −훪푚 + 휂 ⊗ 휉 + 휂 ⊗ 휉 (1.1) 훷휉퐻 = 0 , 훷휉푉 = 0 (1.2) 휂퐻 휉퐻 + 휂푉 휉푉 = 1(1.3) 휂퐻 훷푋퐻 = 0,휂푉 훷푋푉 = 0(1.4) 퐺퐻 훷푋, 훷푌 = 퐺퐻 푋, 푌 − 휂퐻 푋퐻 휂퐻 푌퐻 , 퐺푉 훷푋, 훷푌 = 퐺푉 푋, 푌 − 휂푉 푋푉 휂푉 푌푉 (1.5) Corresponding author. Nesrin CALISKAN (Usak University). E-mail addresses: [email protected]; [email protected] 1 Nesrin Caliskan, A. Funda Yaliniz International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 3, Issue 10 October 2016 퐺퐻 푋, 휉 = 휂퐻 푋퐻 , 퐺푉 푋, 휉 = 휂푉 푋푉 (1.6) 퐻 푉 훻휉 훷 = 0, 훻휉 훷 = 0(1.7) 1 1 훻퐻휉퐻 = − 훷푋퐻, 훻푉휉푉 = − 훷푋푉(1.8) 푋 2 푋 2 1 훻퐻훷 푌퐻 = 퐺퐻 푋퐻, 푌퐻 휉퐻 − 휂퐻 푌퐻 푋퐻 , 푋 2 1 훻푉훷 푌푉 = 퐺푉 푋푉, 푌푉 휉푉 − 휂푉 푌푉 푋푉 (1.9) 푋 2 1 푅 푋퐻, 푌퐻 휉퐻 = 휂퐻 푌퐻 푋퐻 − 휂퐻 푋퐻 푌퐻 , 4 1 푅 푋푉, 푌푉 휉푉 = 휂푉 푌푉 푋푉 − 휂푉 푋푉 푌푉 (1.10) 4 1 푅 푋퐻, 휉퐻 푌퐻 = 휂퐻 푌퐻 푋퐻 − 퐺퐻 푋, 푌 휉퐻 , 4 1 푅 푋푉, 휉푉 푌푉 = 휂푉 푌푉 푋푉 − 퐺푉 푋, 푌 휉푉 (1.11) 4 푛 푛 푆 휉퐻, 휉퐻 = , 푆 휉푉, 휉푉 = (1.12) 2 2 퐻 퐻 푉 푉 푛 퐻 퐻 푉 푉 푆 푋, 휉 = 푆 푋 , 휉 + 푆 푋 , 휉 = 휂 푋 + 휂 푋 (1.13) 2 0 0 Here 퐺 = 퐺퐻 + 퐺푉 is the Sasaki- Finsler metric obtained from the Finsler function 퐹 of type , 훷 2 2 1 1 0 0 denotes a tensor field of type , 휉 is a structure vector field of type , 휂 is an 1-form of type 1 1 1 1 1 1 , 훻 is the Finsler connection with respect to 퐺 on 푇푀, 퐿 is the Lie differential operator, 푅 is the 0 0 1 1 0 0 Riemann curvature tensor field of type of the manifold, 푆 is the Ricci tensor field of type , 3 3 2 2 퐻 퐻 퐻 퐻 푉 푉 푉 푉 푋 , 푌 , 휉 휖 푇푢 푇푀0 and 푋 , 푌 , 휉 휖 푇푢 푇푀0 are tangent vectors. 퐻 For a Sasakian Finsler metric manifold푇푀0 ,quasi-conformal curvature tensor is described as follows: 퐶∗ 푋퐻, 푌퐻 푍퐻 = 푟 푎 푎푅 푋퐻, 푌퐻 푍퐻 + 푏 푆 푌퐻, 푍퐻 푋퐻 − 푆 푋퐻, 푍퐻 푌퐻 + 퐺 푌퐻, 푍퐻 푄푋퐻 − 퐺 푋퐻, 푍퐻 푄푌퐻 − + 2푛+1 2푛 2푏퐺푌퐻,푍퐻푋퐻−퐺푋퐻,푍퐻푌퐻 (1.14) 퐻 퐻 퐻 퐻 for 푋 , 푌 , 푍 휖 푇푢 푇푀0 and where a and b are constants. 1 If푎 = 1and푏 = − , (1.14) takes the following form: 2푛−1 퐶∗ 푋퐻, 푌퐻 푍퐻 = 푅 푋퐻, 푌퐻 푍퐻 1 + 푆 푌퐻, 푍퐻 푋퐻 − 푆 푋퐻, 푍퐻 푌퐻 + 퐺 푌퐻, 푍퐻 푄푋퐻 − 퐺 푋퐻, 푍퐻 푄푌퐻 2푛 − 1 푟 + 퐺 푌퐻, 푍퐻 푋퐻 − 퐺 푋퐻, 푍퐻 푌퐻 = 퐶 푋퐻, 푌퐻 푍퐻 2푛 2푛 − 1 where퐶is conformal curvature tensor. ∗ 퐻 If퐶 = 0then Sasakian Finsler metric manifold 푇푀0 , 퐺 is quasi-conformally flat. If 푆 푋퐻, 푌퐻 = δ퐺 푋퐻, 푌퐻 + ζ휂퐻 푋퐻 휂퐻 푌퐻 (1.15) 퐻 퐻 퐻 퐻 for 푋 , 푌 휖 푇푢 푇푀 and the constantsδ, ζ, then Sasakian Finsler metric manifold 푇푀0 , 퐺 is an휂-Einstein manifold. 2 Nesrin Caliskan, A. Funda Yaliniz International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 3, Issue 10 October 2016 푯 푯 푯 푯 ∗ 푯 푯 2. Sasakian Finsler Structures 흓 , 흃 , 휼 , 푮 on푻푴ퟎ satisfying푪 푿 , 풀 푺 = ퟎ 퐻 The following condition is satisfied for a m-dimensional Sasakian Finsler manifold푇푀0 : 퐶∗ 푋퐻, 푌퐻 푆 = 0. (2.1) By using (1.14), (1.10), (1.11) in (2.1) and putting푋퐻 = 푊퐻 = 휉퐻, we get 푎 푆 푅 휉퐻, 푌퐻 푍퐻, 휉퐻 + 푆 푅 휉퐻, 푌퐻 휉퐻, 휉퐻 + 푏푆 푆 푌퐻, 푍퐻 휉퐻 − 푆 휉퐻, 푍퐻 푌퐻, 휉퐻 + 푏푆 푆 휉퐻, 푌퐻 휉퐻 − 푆 휉퐻, 휉퐻 푌퐻, 푍퐻 + 푏푆 퐺 푌퐻, 푍퐻 푄휉퐻 − 퐺 휉퐻, 푍퐻 푄푌퐻, 휉퐻 + 푏푆 퐺 푌퐻, 휉퐻 푄휉퐻 − 퐺휉퐻,휉퐻푄푌퐻,푍퐻−푟2푛+1푎2푛+2푏푆퐺푌퐻,푍퐻휉퐻−퐺휉퐻,푍퐻푌퐻,휉퐻+ 푆퐺푌퐻,휉퐻휉퐻−퐺휉퐻,휉퐻푌퐻,푍퐻=0 (2.2) With the help of (1.3), (1.6), (1.12) and (1.13), we obtain 푆 푌퐻, 푍퐻 = δ퐺 푌퐻, 푍퐻 + ζ휂퐻 푌퐻 휂퐻 푍퐻 (2.3) where δ and ζ constants are as follows: 푟 푎 푛 − +2푏 2푛+1 2푛 2 δ = 푎 푟 푎 (2.4) − +2푏 4 2푛+1 2푛 푎푛 8 ζ = 푎 푟 푎 (2.5) − +2푏 4 2푛+1 2푛 푛 and it is seen that δ + ζ= , this result enables to write theorem below: 2 퐻 ∗ 퐻 퐻 Theorem 2.1 A Sasakian Finsler manifold 푇푀0 satisfying 퐶 푋 , 푌 푆 = 0 is an 휂-Einstein manifold. 3. Conclusion 퐻 푉 In a Finsler manifold 푀, the complementary tangent bundle structure푇푇푀0 = 푇푇푀0 푇푇푀0 enables to 퐻 퐻 퐻 퐻 study Sasakian Finsler structures 휙 , 휉 , 휂 , 퐺 on diffusion of tangent bundles 푇푀0 , if these structures ∗ 퐻 퐻 퐻 퐻 퐻 satisfy the condition퐶 푋 , 푌 푆 = 0, they are called 휂-Einstein structures for 푋 , 푌 휖 푇푢 푇푀. References [1] A. Bejancu, Tangent bundle and indicatrix bundle of a Finsler manifold, Kodai Math. J., Volume 31, Number 2 (2008), 272-306. [2] F. Massamba and S. J. Mbatakou, Sasakian Finsler structures on the pulled-back bundle, 2016arXiv160405425M, 2016. [3] A. Asanjarani, B. Bidabad, Classification of complete Finsler manifolds through a second order differential equation, Differential Geometry and its Applications 26, (2008), 434-444. [4] B. B. Sinha and R. K. Yadav, On almost Sasakian Finsler manifods, 22 (1991), no 3, 185-192. [5] A.F. Yalınız and N. Caliskan, Sasakian Finsler manifolds, Turkish Journal of Mathematics, doi: 10.3906/mat-1006- 371 (2013) 37: 319-339. [6] M. C. Chaki and M. Tarafdar, On a type of Sasakian manifold, Soochow J. Math.16, No. 1, (1990), 23-28. [7] U.C. De, J. B. Jun and A.K. Gazi, Sasakian manifolds with quasi-conformal curvature tensor, Bull. Korean Math. Soc. 45, No. 2, (2008), 313-319. [8] R. Prasad, On (k,μ)-manifolds with quasi-conformal curvature tensor, Int. J. Contemp. Math. Sciences, Vol. 5, no.34, (2010), 1663-1676. [9] A. A. Shaikh and K. K. Baishya, On weakly quasi-conformally symmetric manifolds, Soochow Journal of Math., Vol. 31, No.4,( 2005), 581-595. [10] M. M. Tripathi, P. Gupta and J. S. Kim, Index of quasi-conformally symmetric semi- Riemannian manifolds, International Journal of Mathematics and Mathematical Sciences, Vol. 2012 , Article ID 461383, doi:10.1155/2012/461383, (2012), 14 p. [11] P. L, Antonelli, Handbook of Finsler geometry, Volume 1, Kluver Academic Publishers, ISBN-10:1402015550, (2003), 1437 p. 3 Nesrin Caliskan, A. Funda Yaliniz .
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