SOME PROPERTIES OF $N(k)$-QUASI EINSTEIN MANIFOLDS

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 (M1 1 , g1) and an Einstein manifold (M n2 2 , g2) satisfy a certain condition, then the Riemannian product manifold (M, g) = (M1 1 × M n2 2 , g1 + g2) is a quasi Einstein manifold. In particular, in N(k)-quasi Einstein case, we show that there exists a quasi Einstein product manifold (M, g) = (M1 1 × M n2 2 , g1 + g2) but not an N(k)-quasi Einstein manifold, which consists of an N(k)-quasi Einstein manifold (M1 1 , g1) and an Einstein manifold (M n2 2 , g2) satisfying the certain condition. Finally we study an N(k)-quasi Einstein manifold satisfying the condition R(U,X) ·G = 0. AMS Subject Classification: 53A30, 53A40, 53B20