  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.
