Extending tensors on polar manifolds

TL;DR

This paper demonstrates that $G$-invariant tensors on a polar manifold $M$ can be fully characterized by their restrictions to a section $\Sigma$, and that any $W$-invariant metric on $\Sigma$ can be extended to a $G$-invariant metric on $M$ preserving the polar action.

Contribution

It establishes a surjective correspondence between $G$-invariant tensors on $M$ and $W$-invariant tensors on $\Sigma$, and shows the extendability of $W$-invariant metrics to $G$-invariant metrics on $M$.

Findings

Restriction to $\Sigma$ is surjective from $G$-invariant tensors on $M$ to $W$-invariant tensors on $\Sigma$.

Every smooth $W$-invariant Riemannian metric on $\Sigma$ can be extended to a $G$-invariant metric on $M$.

The $G$-action remains polar with the same section after extension.

Abstract

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 $\Sigma$.