ON C-BOCHNER CURVATURE TENSOR OF A CONTACT METRIC MANIFOLD

Title & Authors
ON C-BOCHNER CURVATURE TENSOR OF A CONTACT METRIC MANIFOLD
KIM, JEONG-SIK; TRIPATHI MUKUT MANI; CHOI, JAE-DONG;

Abstract
We prove that a (k, $\small{\mu}$)-manifold with vanishing E­Bochner curvature tensor is a Sasakian manifold. Several interesting corollaries of this result are drawn. Non-Sasakian (k, $\small{\mu}$)­manifolds with C-Bochner curvature tensor B satisfying B $\small{(\xi,\;X)\;\cdot}$ S
Keywords
contact metric manifold;N(K)-contact metric manifold;Sasakian manifold;C-Bochner curvature tensor;Einstein manifold;
Language
English
Cited by
1.
A Study on Conservative C-Bochner Curvature Tensor in K-Contact and Kenmotsu Manifolds Admitting Semisymmetric Metric Connection, ISRN Geometry, 2012, 2012, 1
2.
A Study on Ricci Solitons in Kenmotsu Manifolds, ISRN Geometry, 2013, 2013, 1
3.
E-Bochner curvature tensor on generalized Sasakian space forms, Comptes Rendus Mathematique, 2016, 354, 8, 835
References
1.
D. E. Blair, On the geometric meaning of the Bochner tensor, Geom. Dedicata 4 (1975), 33-38

2.
D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J. 29 (1977), 319-324

3.
, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, 2002

4.
D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), no. 1-3, 189-214

5.
S. Bochner, Curvature and Betti numbers, Ann. of Math. 50 (1949), no. 2, 77-93

6.
W. M. Boothby and H. C.Wang, On contact manifolds, Ann. of Math. 68 (1958), 721-734

7.
H. Endo, On K-contact Riemannian manifolds with vanishing E-contact Bochner curvature tensor, Colloq. Math. 62 (1991), no. 2, 293-297

8.
H. Endo, On the curvature tensor fields of a type of contact metric manifolds and of its certain submanifolds, Publ. Math. Debrecen 48 (1996), no. 3-4, 253-269

9.
T. Koufogiorgos, Contact Riemannian manifolds with constant $\varphi$-sectional curvature, Tokyo J. Math. 20 (1997), no. 1, 13-22

10.
M. Matsumoto and G. Chuman, On the C-Bochner curvature tensor, TRU Math. 5 (1969), 21-30

11.
M. Okumura, Some remarks on space with a certain contact structure, Tohoku Math. J. 14 (1962), 135-145

12.
B. J. Papantoniou, Contact Riemannian manifolds satisfying R($\xi$,X)$\cdot$R = 0 and ${\xi}{\in}(k,\mu)$-nullity distribution, Yokohama Math. J. 40 (1993), no. 2, 149-161

13.
G. Pathak, U. C. De, and Y.-H. Kim, Contact manifolds with C-Bochner curvature tensor, Bull. Calcutta Math. Soc. 96 (2004), no. 1, 45-50

14.
D. Perrone, Contact Riemannian manifolds satisfying R(X, $\xi){\cdot}$R = 0, Yokohama Math. J. 39 (1992), no. 2, 141-149

15.
S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J. 40 (1988), 441-448

16.
Y. Tashiro, On contact structures of tangent sphere bundles, Tohoku Math. J. 21 (1969), 117-143