On the notions of suborbifold and orbifold embedding

TL;DR

This paper explores the relationship between suborbifolds and orbifold embeddings, providing definitions, examples, and characterizations, and demonstrating that some suborbifolds cannot be realized as images of orbifold embeddings, with applications to geodesics.

Contribution

It introduces natural definitions for suborbifolds and orbifold embeddings, characterizes which suborbifolds can be realized as images of embeddings, and applies this to geodesic segments in Riemannian orbifolds.

Findings

Some topologically embedded suborbifolds are not images of orbifold embeddings

Characterization of suborbifolds that can arise from orbifold embeddings

Length-minimizing curves in Riemannian orbifolds can be realized as orbifold embeddings

Abstract

The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples. Surprisingly, we show that there are (topologically embedded) smooth suborbifolds which do not arise as the image of a smooth orbifold embedding. We are also able to characterize those suborbifolds which can arise as the images of orbifold embeddings. As an application, we show that a length-minimizing curve (a geodesic segment) in a Riemannian orbifold can always be realized as the image of an orbifold embedding.