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

Gromov asked if the bi-invariant metric on an $n$ dimensional compact Lie group is extremal compared to any other metrics. In this note, we prove that the bi-invariant metric on an $n$ dimensional compact connected semi-simple Lie group $G$ is extremal in the sense of Gromov when compared to the left invariant metrics. In fact the same result holds for a compact connected homogeneous Riemannian manifold $G/H$ with the Lie algebra of $G$ having trivial center.