The G-invariant spectrum and non-orbifold singularities

TL;DR

This paper investigates the G-invariant spectrum of the Laplacian on orbit spaces, demonstrating that certain geometric features like constant curvature and non-orbifold singularities cannot be distinguished by this spectrum.

Contribution

It generalizes the Sunada-Pesce-Sutton technique to the G-invariant setting, producing isospectral non-isometric orbit spaces with different singularity types.

Findings

Constructed isospectral non-isometric orbit spaces with different singularities.

Showed that constant curvature and non-orbifold singularities are inaudible to the G-invariant spectrum.

Demonstrated the limitations of spectral geometry in detecting certain geometric features.

Abstract

We consider the $G$-invariant spectrum of the Laplacian on an orbit space $M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric to an orbifold with constant sectional curvature whereas the other admits non-orbifold singularities and therefore has unbounded sectional curvature. We therefore show that constant sectional curvature and the presence of non-orbifold singularities are inaudible to the G-invariant spectrum.