Leaf Space Isometries of Singular Riemannian Foliations and Their Spectral Properties

TL;DR

This paper investigates when leaf space isometries of singular Riemannian foliations imply spectral equivalence, showing additional geometric conditions are needed for spectral equality, with applications to orbifolds and group actions.

Contribution

The paper demonstrates that smooth isometries of leaf spaces do not necessarily imply spectral equivalence, and identifies geometric conditions that ensure spectral equality.

Findings

An example showing isometry does not guarantee spectral equality.

Additional geometric conditions ensure spectral equality.

Applications to orbifold spectral theory and group actions.

Abstract

In this paper, the authors consider leaf spaces of singular Riemannian foliations $\mathcal{F}$ on compact manifolds $M$ and the associated $\mathcal{F}$-basic spectrum on $M$, $spec_B(M, \mathcal{F}),$ counted with multiplicities. Recently, a notion of smooth isometry $\varphi: M_1/\mathcal{F}_1\rightarrow M_2/\mathcal{F}_2$ between the leaf spaces of such singular Riemannian foliations $(M_1,\mathcal{F}_1)$ and $(M_2,\mathcal{F}_2)$ has appeared in the literature. In this paper, the authors provide an example to show that the existence a smooth isometry of leaf spaces as above is not sufficient to guarantee the equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2).$ The authors then prove that if some additional conditions involving the geometry of the leaves are satisfied, then the equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2)$ is guaranteed. Consequences and applications to orbifold spectral theory, isometric group actions, and their reductions are also explored.