Riemannian manifolds with Einstein-like metrics (original) (raw)

1 n this thesis, we investigate propaties of manifolds with Riemannian metrics which satisfy conditions more general than those of J:'instein metrics, including the Lauer as special cases. Fhe /:"instein condition is weU known for being the l:"uln-l.agrange equation of a vw iational problem. '/'here is twt a great deal of difference between such metrics and melrics with Ricci tensor parallel for the Latter are locaUy 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 thr•ee 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 pan of the third is ktwwn to Derdzinski / Del,2]. Next we construct the metrics mentioned above on spheres of odd dimension.'J'he construction is similar to Jensen's /Jell but more direct and is due essentiaUy to Gray and Vanhecke /GV}. In this way we obtain ,beside the standard metric, the second l:.'instein metric of Jensen. As for the Ricci-Codazzi metrics, they are essentiaUy Jiinstein, but the Ricci cyclic parallel mell ics seem to form a larger class. FinaUy,we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute aU the corresponding metrics on the quotient spaces associated with G2.