Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces

Abstract

Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group 𝐺 are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact, the same result holds for a compact connected homogeneous manifold 𝐺/𝘏 with 𝐺 compact connect and semi-simple.