On Four Dimensional Einstein Manifolds

Abstract

In this paper we provide a study of 4-dimensional Einstein manifolds, in particular we classify ali 4-dimensional Einstein manifolds (up to local isometry) which have tbe property that tbe metric depends on only one coordinate. Kasner (1923) considered the Solutions of the Einstein equations involving functions cf cniy one variahh.. His calculations are rather long and purely algebraic in nature. In this paper, by using a theorem of Singer and Thorpe (1969), we provide an alternative approach to find ali 4-dimensional Einstein manifolds (up to local isometry) which have the property that the metric depends on only one this we consider the metric 3 coordinate. To do