A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifo...A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10431040, No.10501025) and the Liu Hui Center for Applied Mathematics.
文摘A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.
