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Leaf space isometries of singular Riemannian foliations and their spectral properties. (English) Zbl 1473.58016

The general setting of this paper is to consider essentially leaf spaces spanned by singular Riemannian foliations (SRF) \(\mathcal{F}\) with closed leaves and show when they have the same spectra. SRF is a structure on a compact Riemannian manifold \(M\) that partitions \(M\) in connected immersed submanifolds, called leaves of \(\mathcal{F}\).
The authors initially define a smooth structure in the leaf space \(M/ \mathcal{F}\) (Definition 1.1), called smooth SRF isometry, and verify that this is very well defined. Afterwards, they ask if just the existence of this smooth structure between leaf spaces implies in isospectral(equivalent spectra) leaf spaces. It is verified, in Example 1.2, that the existence of a smooth SRF isometry on leaf spaces is not sufficient to guarantee isospectral leaf spaces. However, the main Theorem of this paper (Theorem 1.3) is dedicated to explore conditions that are verified.
Given \((M_{i}, \mathcal{F}_{i})\) a singular Riemannian foliation with mean curvature vector field \(H_{i}\), for \(i=1,2\) and \(\varphi : M_{1}/\mathcal{F}_{1} \rightarrow M_{2}/\mathcal{F}_{2}\) a smooth SRF isometry, then the mean Theorem of this paper allows to affirm that the leaf spaces are isospectral under the following conditions:
i)
\(H_{1}\) and \(H_{2}\) are basics;
ii)
\(d\varphi(H_{1\ast}) = H_{2\ast}.\)

Finally, as an application of main Theorem in Corollary 1.6, the authors show geometric conditions that guarantee isospectral of the leaf spaces.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J53 Isospectrality
22D99 Locally compact groups and their algebras
53C12 Foliations (differential geometric aspects)
32S65 Singularities of holomorphic vector fields and foliations
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References:

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