We consider a Riemannian homogeneous space $(R\times S^n, g)$. Suppose that the Lie algebra of the Killing fields has a natural splitting (compatible with the product) as $\operatorname{isom}(R)\oplus \operatorname{isom}(S^n)$. Can we claim that the homogeneous Riemannian metric has a natural splitting $g = dt^2 \oplus h$, where $h$ is a homogeneous metric on $S^n$?
$\begingroup$
$\endgroup$
Add a comment
|