Classifications of $K$-Contact Semi-Riemannian Manifolds

Year 2019, Volume: 7 Issue: 2, 312 - 318, 15.10.2019

Chiranjib Dey , Uday Chand De

Abstract

The object of the present paper is to characterize a K-contact semi-Riemannian manifold satisfying certain curvature conditions. We study Ricci semi-symmetric K-contact semi-Riemannian manifolds and obtain an equivalent condition. Next we prove that a K-contact semi-Riemannian manifold is of harmonic conformal curvature tensor if and only if the manifold is an Einstein manifold. Also we study $\xi$-conformally flat K-contact semi-Riemannian manifolds. Finally, we charecterize conformally semisymmetric Lorentzian $K$-contact manifolds.

Keywords

Conformally semisymmetric manifolds

References

• [1] Blair, , D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics.509, Springer-Verlag, Berlin-New York, 1976.
• [2] Blair, D. E., Riemannian Geometry of Contact and Sympletic Manifolds, Progress in Math-ematics. 203, Birkhauser Boston, Inc., Boston, MA, 2001.
• [3] Calvaruso, G., and Perrone, D., Contact pseudo-metric manifolds, Dierential Geom. Appl.,28(2010), 615-634.
• [4] Calvaruso, G., and Perrone, D., H-contact semi-Riemannian manifolds, J. Geom. Phys.71(2013),11-21.
• [5] Derdzinski, A., and Roter, W., Some theorems on conformally symmetric manifolds, Tensor(N.S.)32(1978),11-23.
• [6] Derdzinski, A., and Roter, W., On conformally symmetric manifolds with metrics of indices0 and 1, Tensor (N.S.) 31(1977), 255-259.
• [7] Perrone, D., Curvature of K-contact semi-Riemannian manifolds, Can. Math. Bull57(2014),401-412.
• [8] Perrone, D., Contact pseudo-metric manifolds of constant curvature and CR geometry,Results. Math. 66(2014), 213-225.
• [9] Tanno, S., Locally symmetric K-conatct Riemannian manifolds, Proc. Japan Acad.43(1967),581-543.
• [10] Takahasi, T., Sasakian manifold with pseudo-Riemannian metrics, Tohoku Math. J.21(1969),271-290.
• [11] Yildiz, A., and Ata, E., On a type of K-contact manifolds, Hacettepe J. Math. and Stat.41(2012),567-571.
• [12] Zhen, G., Cabrerizo, J. l., Fenandez, L. M., and Fenandez, M., On -conformally at K-contact metric manifolds, Indian J. Pure Appl. Math, 28(1997), 725-734.
• [13] Zhen, G., The conformally symmetric K-contact manifolds, Chinese Quarterly Journal ofMath., 7(1992),5-10.