Orbifolds and manifold quotients with upper curvature bounds

TL;DR

This paper characterizes Riemannian orbifolds with an upper curvature bound as reflectofolds, linking local reflection-generated groups to Alexandrov curvature bounds, and characterizes quotients of manifolds by isometry groups as reflectofolds.

Contribution

It provides a new characterization of orbifolds with upper curvature bounds as reflectofolds, connecting local group structure to curvature properties.

Findings

Riemannian orbifolds with upper curvature bounds are reflectofolds.

Quotients of Riemannian manifolds by isometry groups are reflectofolds if and only if they have bounded curvature.

The result combines with Lytchak--Thorbergsson to characterize such quotients.

Abstract

We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak--Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.