IJPAM: Volume 77, No. 4 (2012)

ON $(\varepsilon )$-PARA SASAKIAN $3$-MANIFOLDS

Selcen Yüksel Perktaş$^1$, Erol Kılıç$^2$,
Mukut Mani Tripathi$^3$, Sadık Keleş$^4$
$^1$Department of Mathematics
Faculty of Arts and Sciences
Adıyaman University
02040, Adıyaman, TURKEY
$^{2,4}$Department of Mathematics
Faculty of Arts and Sciences
İnönü University
44280, Malatya, TURKEY
$^3$Department of Mathematics and DST-CIMS
Banaras Hindu University
Varanasi, 221 005, INDIA


Abstract. In this paper we study the $3$-dimensional $\left( \varepsilon
\right) $-para Sasakian manifolds. We obtain a necessary and sufficient condition for an $\left( \varepsilon
\right) $-para Sasakian $3$-manifold to be an indefinite space form. We show that a Ricci-semi-symmetric $\left( \varepsilon
\right) $-para Sasakian $3$-manifold is an indefinite space form. We investigate the necessary and sufficient condition for an $\left( \varepsilon
\right) $-para Sasakian $3$-manifold to be locally $\varphi $-symmetric. It is proved that in an $%
\left( \varepsilon \right) $-para Sasakian $3$-manifold with $\eta $-parallel Ricci tensor the scalar curvature is constant. It is also shown that every $\left( \varepsilon
\right) $-para Sasakian $3$-manifolds is pseudosymmetric in the sense of R. Deszcz.

Received: December 14, 2011

AMS Subject Classification: 53C25, 53C50

Key Words and Phrases: $(\varepsilon )$-para Sasakian $3$-manifold, Ricci-semi-symmetric space, locally $\varphi $-symmetric space, $\eta $-parallel Ricci tensor, pseudosymmetric space

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 77
Issue: 4