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

TL;DR

This paper investigates a unique decomposition of Lie groups equipped with left invariant pseudo-Riemannian metrics into totally geodesic sub-manifolds, extending the understanding beyond classical decompositions.

Contribution

It introduces a new type of decomposition for Lie groups with pseudo-Riemannian metrics and proves its uniqueness, especially for Einstein metrics.

Findings

Decomposition into totally geodesic sub-manifolds is distinct from De Rham's.

The decomposition of Lie groups with Einstein metrics is unique up to order.

Application to Einstein metrics provides new structural insights.

Abstract

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