  Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$,
with section $\Sigma\subset M$ and generalized Weyl group $W$. We show that
restriction to $\Sigma$ is a surjective map from the set of smooth
$G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on
$\Sigma$. Moreover, we show that every smooth $W$-invariant Riemannian metric
on $\Sigma$ can be extended to a smooth $G$-invariant Riemannian metric on $M$
with respect to which the $G$-action remains polar with the same section
$\Sigma$.
