Leaf Space Isometries of Singular Riemannian Foliations and Their Spectral Properties

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 $M$, $spe{c}_B(M,\mathcal{F}),$ counted with multiplicities. Recently, a notion of smooth isometry $\phi :M_1/{\mathcal{F}}_1\to M_2/{\mathcal{F}}_2$ 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 $spe{c}_B(M_1,{\mathcal{F}}_1)$ and $spe{c}_B(M_2,{\mathcal{F}}_2).$ The authors then prove that if some additional conditions involving the geometry of the leaves are satisfied, then the equality of $spe{c}_B(M_1,{\mathcal{F}}_1)$ and $spe{c}_B(M_2,{\mathcal{F}}_2)$ is guaranteed. Consequences and applications to orbifold spectral theory, isometric group actions, and their reductions are also explored.

11 pages. arXiv admin note: text overlap with arXiv:1607.05593 This version corrects a missing hypothesis and includes a cleaner presentation with a shorter proof of the main result. An appendix has also been added with the original proof of one of the main results