Extending tensors on polar manifolds

Given a polar action on a Riemannian manifold, we prove surjectivity of restriction to the section for general invariant tensors, and a sharper surjectivity result in the special case of metrics. These are related to the Chevalley Restriction Theorem and Michor’s Basic Forms Theorem. The proofs rely on results in the Invariant Theory of finite reflection groups and symmetric pairs, some of which may be of independent interest.