一类新的齐性空间及O(n+1)上的Einstein度量(英文)

A NEW TYPE OF HOMOGENEOUS SPACES AND THE EINSTEIN METRICS ON O(n+1)

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．

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 Levi- Civita 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）.