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On quasi Einstein manifolds

for all vector fields X, U being a unit vector field. In such a case a, b are called associated scalars. A is called the associated 1-form and U is called the generator of the manifold. An ndimensional manifold of this kind shall be denoted by the symbol (QE)n. This paper deals with (QE)n (n > 3) which are not conformally flat. The significance of the associated scalars a, b is pointed out by showing that in a (QE)n the Ricci tensor S has only two distinct eigenvalues a + b and a of which the former is simple and the latter is of multiplicity n−1, the generator U being an eigenvector corresponding to the eigenvalue a+b. In a (QE)n the relation R(X, Y ) ·S = 0 does not in general hold. A necessary and sufficient condition is obtained in order that this relation may hold.