一类新的齐性空间及O(n+1)上的Einstein度量

摘要

For a given Riemannian manifold (M,g), we introduce a scaling Riemannian metric on the orthonormal frame bundle F(M) of M and then study the LeviCivita connection and curvatures of F(M) endowed with the metric. It turns out that F(M) is homogeneous Riemannian space when the underlying Riemannian manifold M is homogeneous. Furthermore the canonical vector fields are invariant under a transitive subgroup of the isometric group of F(M), which makes them more or less like the left invariant vector fields of a Lie group endowed with a left invariant metric. We use the expression of the curvatures of F(M) to explicitly express Jensen's non-standard Einstein metric on O(n + 1) (n≥3).%对给定的黎曼流形(M,g),此文在其标架丛F(M)上引入可以在纤维方向伸缩的度量,并研究其Levi-Civita联络和对应的曲率.本文证明了F(M)上的典型标架场是测地向量场.在M是齐性空间时,F(M)也是齐性空间.F(M)上曲率的一般公式还被用来显式表示O(n+1)上Jensen的非标准Einstein度量.