Copyright © 2010 Mukut Mani Tripathi et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.