The decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness

摘要

In this paper, we discuss the decomposition of a Lie group with a leftinvariant pseudo-Riemannian metric and the uniqueness. In fact, it is adecomposition of a Lie group into totally geodesic sub-manifolds which isdifferent from the De Rham decomposition on a Lie group. As an application, wegive a decomposition of a Lie group with a left invariant pseudo-RiemannianEinstein metric, and prove that the decomposition is unique up to the order ofthe parts in the decomposition.