Extending tensors on polar manifolds

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