Einstein like -para Sasakian manifolds
Keywords:Einstein like -para Sasakian manifold, indefinite locally Riemannian product manifold.
AbstractEinstein like -para Sasakian manifolds are introduced. For an -para Sasakian manifold to be Einstein like, a necessary and sufficient condition in terms of its curvature tensor is obtained. The scalar curvature of an Einstein like -para Sasakian manifold is obtained and it is shown that the scalar curvature in this case must satisfy certain differential equation. A necessary and sufficient condition for an -almost paracontact metric hypersurface of an indefinite locally Riemannian product manifold to be -para Sasakian is obtained and it is proved that the -para Sasakian hypersurface of an indefinite locally Riemannian product manifold of almost constant curvature is always Einstein like.
Beem, J. K. & Ehrlich, P. E. 1981 . Global Lorentzian geometry. Marcel Dekker, New York.
Bejancu, A. & Duggal, K. L. 1993 . Real hypersurfaces of indefinite Kaehler manifolds<$>. International Journal of Mathematics and Mathematical Sciences 16 (3): 545-556.
Blair, D. E. 2002 . Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics (203). Birkhauser Boston, Inc., Boston, MA.
Chen, B.-Y. 1973 . Geometry of submanifolds. Pure and Applied Mathematics (22). Marcel Dekker, New York.
Duggal, K. L. 1990 . Space time manifolds and contact structures. International Journal of Mathematics and Mathematical Sciences 13 : 545-554.
Matsumoto, K . 1989. On Lorentzian paracontact manifolds. Bulletin of Yamagata University Natural Science 12 (2): 151-156.
O'Neill, B. 1983 . Semi-Riemannian geometry with applications to relativity<$>,<$> Academic Press.
Sasaki, S. 1960 . On differentiable manifolds with certain structures which are closely related to almost contact structure I. Tohoku Mathematical Journal 12 (2): 459-476.
Sato, I. 1976 . On a structure similar to the almost contact structure. Tensor (NS) 30 (3): 219-224.
Sharma, R. 1982 . On Einstein-like P-Sasakian manifold, Matematicki Vesnik 6 (19)(34) (2): 177-184.
Takahashi, T. 1969 . Sasakian manifold with pseudo-Riemannian metric<$>,<$> T^‡‡‡‡‡‡‡‡‡‡‡ohoku Mathematical Journal 21 (2): 644-653.
Tripathi, M. M., Kilõõiµ, E., Perkta<?tlig="lig">_s, S. Y. & Kele<?tlig="lig">_s, S. 2010 . Indefinite almost paracontact metric manifolds, International Journal of Mathematics and Mathematical Sciences. pp. 19.
Yano, K. 1965 . Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics (49) A Pergamon Press Book. The Macmillan Coorporation, New York.