IJPAM: Volume 88, No. 3 (2013)


Jaeman Kim
Department of Mathematics Education
Kangwon National University
Chunchon, 200-701, KOREA

Abstract. In this paper, an example of an $N(k)$-quasi Einstein manifold with closed associated 1-form is given. Also we show that if a quasi Einstein manifold $(M_{1}^{n_{1}},g_{1})$ and an Einstein manifold $(M_{2}^{n_{2}},g_{2})$ satisfy a certain condition, then the Riemannian product manifold $(M^{n},g)=(M_{1}^{n_{1}}\times
M_{2}^{n_{2}},g_{1}+g_{2})$ is a quasi Einstein manifold. In particular, in $N(k)$-quasi Einstein case, we show that there exists a quasi Einstein product manifold $(M^{n},g)=(M_{1}^{n_{1}}\times
M_{2}^{n_{2}},g_{1}+g_{2})$ but not an $N(k)$-quasi Einstein manifold, which consists of an $N(k)$-quasi Einstein manifold $(M_{1}^{n_{1}},g_{1})$ and an Einstein manifold $(M_{2}^{n_{2}},g_{2})$ satisfying the certain condition. Finally we study an $N(k)$-quasi Einstein manifold satisfying the condition ${R(U,X)\cdot {G}}=0$.

Received: May 16, 2013

AMS Subject Classification: 53A30, 53A40, 53B20

Key Words and Phrases: quasi Einstein manifold, associated 1-form, $N(k)$-quasi Einstein manifold, Riemannian product manifolds, quasi Einstein product manifold, G-curvature tensor, Killing vector field, Ricci-semisymmetric manifold

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DOI: 10.12732/ijpam.v88i3.3 How to cite this paper?
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 88
Issue: 3