On real analytic orbifolds and Riemannian metrics

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 $\alpha$.