Indefinite Almost Paracontact Metric Manifolds

We introduce the concept of ( )-almost paracontact manifolds, and in particular, of ( )-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of ( )-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 ( )-para Sasakian structure. We show that, for an ( )-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) ( )-para Sasakian manifold is locally isometric to a pseudohyperbolic space (resp., pseudosphere ). At last, it is proved that for an ( )-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.