Indefinite almost paracontact metric manifolds

Abstract

In this paper we introduce the concept of $(\epsilon )$-almost paracontact manifolds, and in particular, of $(\epsilon )$-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of $(\epsilon )$-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 $(\epsilon )$-para Sasakian structure. We show that, for an $(\epsilon )$-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) $(\epsilon )$-para Sasakian manifold $M^n$ is locally isometric to a pseudohyperbolic space $H_{\nu }^n(1)$ (resp. pseudosphere $S_{\nu }^n(1)$). In last, it is proved that for an $(\epsilon )$-para Sasakian manifold, the conditions of being Ricci-semisymmetric, Ricci-symmetric and Einstein are all identical.