$\bar{T}-$curvature and $\bar{C}-$Bochner curvature on LP-Sasakian manifolds admitting a general connection
Main Article Content
Abstract
In the present work, it is proved that LP-Sasakian manifolds which are $\bar{T}$-flat, quasi-$\bar{T}$-flat, $\xi$-$\bar{T}$-flat, $\phi$-$\bar{T}$-flat, $\bar{T}$-semi-symmetric, $\phi$-$\bar{T}$-Ricci recurrent, $\bar{C}$-Bochner flat, or satisfy $\bar{T}\cdot\bar{S}=0$, are generalized $\eta$-Einstein manifolds. The $\bar{T}$-curvature and $\bar{C}$-Bochner curvature tensors are defined with respect to a general connection $\bar{\nabla}$ which includes the quarter-symmetric metric, Schouten--van Kampen, Tanaka--Webster, and Zamkovoy connections.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
Supporting Agencies
References
[1] T. Adati and K. Matsumoto, On conformally recurrent and conformally symmetric P-Sasakian manifolds, TRU Math. 13 (1977), 25–32. Google Scholar
[2] A. Biswas and K. K. Baishya, A general connection on Sasakian manifolds and the case of almost pseudo-symmetric Sasakian manifolds, Sci. Stud. Res. Ser. Math. Inform. 29(1) (2019), 59–72. Google Scholar
[3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin–New York, 1976. Google Scholar
[4] S. Bochner, Curvature and Betti numbers II, Ann. of Math. 50 (1949), 77–93. https://doi.org/10.2307/1969353 Google Scholar
[5] W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. 68 (1958), 721–734. Google Scholar
[6] H. Geiges, A brief history of contact geometry and topology, Expo. Math. 19(1) (2001), 25–53. Google Scholar
[7] S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor (N.S.) 29 (1975), 249–254. Google Scholar
[8] R. Kumar, L. Chawngthu, O. Bahadır, and M. A. Khan, Geometry of LP-Sasakian manifolds admitting a general connection, Mathematics 13(6) (2025), 902. https://doi.org/10.3390/math13060902 Google Scholar
[9] K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Natur. Sci. 12(2) (1989), 151–156. Google Scholar
[10] M. Matsumoto and G. Chūman, On the C-Bochner curvature tensor, TRU Math. 5 (1969), 21–30. Google Scholar
[11] H. G. Nagaraja and G. Somashekhara, T-curvature tensor in (k,µ)-contact manifolds, Mathematica Aeterna 2 (2012), 523–532. Google Scholar
[12] S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure. I, Tohoku Math. J. 12 (1960), 459–476. https://doi.org/10.2748/tmj/1178244407 Google Scholar
[13] I. Sato, On a structure similar to the almost contact structure, Tensor (N.S.) 30 (1976), 219–224. Google Scholar
[14] J. A. Schouten and E. R. van Kampen, Zur Einbettungs- und Krümmungstheorie nichtholonomer Gebilde, Math. Ann. 103 (1930), 752–783. Google Scholar
[15] R. J. Shah, On T-curvature tensor in LP-Sasakian manifolds, Kathmandu Univ. J. Sci. Eng. Technol. 9(2) (2013), 69–79. https://doi.org/10.3126/kuset.v9i2.65842 Google Scholar
[16] N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. 2(1) (1976), 131–190. Google Scholar
[17] S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349–379. Google Scholar
[18] M. M. Tripathi and P. Gupta, T-curvature tensor on a semi-Riemannian manifold, J. Adv. Math. Stud. 4(1) (2011), 117–129. Google Scholar
[19] M. M. Tripathi and P. Gupta, On (N(k),ξ)-semi-Riemannian manifolds: pseudosymmetries, Int. Electron. J. Geom. 5(2) (2012), 95–167. Google Scholar
[20] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom. 36(1) (2009), 37–60. https://doi.org/10.1007/s10455-008-9147-3 Google Scholar