Leaf Space Isometries of Singular Riemannian Foliations and Their Spectral Properties

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 $\varphi: M_1/$\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 $spe{c}_B(M_1,{\mathcal{F}}_1)$ and $spe{c}_B(M_2,{\mathcal{F}}_2)\mathrm{.}$ 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.