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Extending tensors on polar manifolds

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