International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 846195, 19 pages
Research Article

Indefinite Almost Paracontact Metric Manifolds

1Department of Mathematics, Banaras Hindu University, Varanasi 221 005, India
2Department of Mathematics, Faculty of Arts and Sciences, İnönü University, 44280, Malatya, Turkey

Received 24 September 2009; Accepted 8 March 2010

Academic Editor: Wolfgang Kuehnel

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 (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 Hνn(1) (resp., pseudosphere Sνn(1)). 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.