You can hear the local orientability of an orbifold

TL;DR

This paper demonstrates that the local orientability of a Riemannian orbifold can be detected from its Laplace spectrum, showing spectral distinctions based on orientability properties.

Contribution

It introduces a method using heat invariants to determine local orientability of orbifolds from spectral data, a novel spectral characterization.

Findings

Locally orientable orbifolds are distinguishable spectrally from non-orientable ones.

Non-orientable orbifolds cannot be Laplace isospectral to orientable orbifolds.

Non-orientable orbifolds cannot be Laplace isospectral to manifolds.

Abstract

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifold locally orientable. We use heat invariants to show that a Riemannian orbifold which is locally orientable cannot be Laplace isospectral to a Riemannian orbifold which is not locally orientable. As a corollary we observe that a Riemannian orbifold that is not locally orientable cannot be Laplace isospectral to a Riemannian manifold.