Lifting tensors from orbifold quotients

TL;DR

This paper proves that symmetric 2-tensors on orbifold quotients can be lifted to invariant tensors on the original manifold, using invariant theory of reflection groups.

Contribution

It establishes a lifting property for symmetric 2-tensors from orbifold quotients to the original manifold under isometric group actions.

Findings

Any smooth orbifold symmetric 2-tensor lifts to a G-invariant tensor.

The proof uses invariant theory of finite reflection groups.

The result applies to orbifolds with polar slice representations.

Abstract

We deal with a Lie group G acting by isometries on a Riemannian manifold M, such that the quotient M/G is an orbifold, or, equivalently, all slice representations are polar. We show that any smooth orbifold symmetric 2-tensor on M/G lifts to a smooth G-invariant symmetric 2-tensor on M. The proof relies on a fact about the Invariant Theory of finite reflection groups which may be of independent interest.