A remark on left invariant metrics on compact Lie groups

TL;DR

This paper investigates the curvature properties of left invariant metrics on compact Lie groups, revealing that stretching biinvariant metrics generally induces negative curvature unless specific algebraic conditions are met.

Contribution

It provides a criterion for when left invariant metrics obtained by stretching are free of negative sectional curvature, linking algebraic structure to geometric properties.

Findings

Stretching biinvariant metrics usually leads to negative curvature.

Negative curvature is avoided only when the semi-simple part of the subalgebra is an ideal.

The result connects algebraic substructure with geometric curvature properties.

Abstract

We show that a left invariant metric on a compact Lie group $G$ which is obtained by stretching a biinvariant metric in the direction of a subalgebra $\h$ of $\g$ always has some negative sectional curvature, unless the semi-simple part of $\h$ is an ideal of $\g$.