On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces

TL;DR

This paper demonstrates, assuming Schanuel's conjecture, that compact locally symmetric spaces of nonpositive curvature cannot be isospectral to orbifolds with singularities if they are length-commensurable, addressing a longstanding spectral geometry question.

Contribution

It establishes a conditional impossibility result linking spectral properties and geometric structures of locally symmetric spaces and orbifolds.

Findings

Shows impossibility of isospectrality under Schanuel's conjecture

Focuses on nonpositively curved locally symmetric spaces and orbifolds

Addresses the isospectral orbifold-manifold problem in a specific geometric context

Abstract

An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel's conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.