On real analytic orbifolds and Riemannian metrics

TL;DR

This paper proves that every real analytic orbifold admits a real analytic Riemannian metric, and extends classical results about Riemannian metrics on manifolds to the orbifold setting, including existence of complete conformal metrics.

Contribution

It establishes the existence of real analytic Riemannian metrics on orbifolds and generalizes a key manifold result to orbifolds.

Findings

Every real analytic orbifold has a real analytic Riemannian metric.

Reduced real analytic orbifolds can be represented as quotients of manifolds by compact Lie group actions.

Existence of complete Riemannian metrics conformal to a given metric on orbifolds.

Abstract

We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free action of a compact Lie group. We then extend a well-known result of Nomizu and Ozeki concerning Riemannian metrics on manifolds to the orbifold setting: Let $X$ be a smooth (real analytic) orbifold and let $\alpha$ be a smooth (real analytic) Riemannian metric on $X$. Then $X$ has a complete smooth (real analytic) Riemannian metric conformal to $\alpha$.