关于爱因斯坦流形的一些注记

SOME NOTES OF EINSTEIN MANIFOLD

爱因斯坦流形是特殊的一种黎曼流形,它有很好的特征,其定义弱于常曲率黎曼流形.本文对其有关性质进行了讨论,得到了2维和n(n≥3)维爱因斯坦流形的数曲率的一些结果:ρ可能为常数和ρ为常数,以及爱因斯坦流形与常曲率黎曼流形之间的关系;3维连通的爱因斯坦流形(M,g)必为常曲率黎曼流形,它的截面曲率的几个结论;最后得到了一个关于其上非零的平行向量场的存在性定理,并且对爱因斯坦流形作了几点总结.

Einstein manifold is a particular kind of Riemannian Manifold, it has good characters, its definition is weaker than Riemannian Manifold with constant sectional curvature. In this paper some properties about it are discussed. We get some results about the scalar curvature of 2 and n(n≥3)dimensions Einstein manifold: ρ Maybe is constant and ρ is constant, the relation between Einstein Manifold and Riemannian Manifold with constant sectional curvature: if Einstein Manifold (M, g) is connected, dimM=3, then it is Riemannian Manifold with constant sectional curvature. A few consequences of its sectional curvature. At last there is a existence theorem of nonzero parallel vector field, moreover several conclusions are given.