Riemannian manifolds with Einstein-like metrics

Abstract

In this thesis, we investigate properties of manifolds with Riemannian metrics which satisfy conditions more general than those of Einstein metrics, including the latter as special cases. The Einstein condition is well known for being the Euler- Lagrange equation of a variational problem. There is not a great deal of difference between such metrics and metrics with Ricci tensor parallel for the latter are locally Riemannian products of the former. More general classes of metrics considered include Ricci- Codazzi and Ricci cyclic parallel. Both of these are of constant scalar curvature. Our study is divided into three parts. We begin with certain metrics in 4-dimensions and conclude our results with three theorems, the first of which is equivalent to a result of Kasner [Kal] while the second and part of the third is known to Derdzinski [Del.2].Next we construct the metrics mentioned above on spheres of odd dimension. The construction is similar to Jensen's [Jel] but more direct and is due essentially to Gray and Vanhecke [GV]. In this way we obtain .beside the standard metric, the second Einstein metric of Jensen. As for the Ricci- Codazzi metrics, they are essentially Einstein, but the Ricci cyclic parallel metrics seem to form a larger class. Finally, we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute all the corresponding metrics on the quotient spaces associated with G2.