Indefinite Almost Paracontact Metric Manifolds

Abstract

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 cannot admit an (ε)‐para Sasakian structure. We show that, for an (ε)‐para Sasakian manifold, the conditions of being symmetric, semi‐symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) (ε)‐para Sasakian manifold Mn is locally isometric to a pseudohyperbolic space (resp., pseudosphere ). At last, it is proved that for an (ε)‐para Sasakian manifold the conditions of being Ricci‐semi‐symmetric, Ricci‐symmetric, and Einstein are all identical.