Indefinite Almost Paracontact Metric Manifolds

In this paper we introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it can not admit an (ε)-para Sasakian structure. We show that, for an (ε)-para Sasakian manifold, the conditions of being symmetric, semisymmetric or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp. timelike) (ε)-para Sasakian manifold M is locally isometric to a pseudohyperbolic space H ν (1) (resp. pseudosphere S n ν (1)). In last, it is proved that for an (ε)-para Sasakian manifold, the conditions of being Ricci-semisymmetric, Ricci-symmetric and Einstein are all identical. Mathematics Subject Classification: 53C25, 53C50.