# Leaf space isometries of singular Riemannian foliations and their spectral properties

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## Abstract

In this note, the authors show by example that an isometry between leaf spaces of singular Riemannian foliations need not induce an equality of the basic spectra. If the leaf space isometry preserves the mean curvature vector fields, then it is proved that the basic spectra are equivalent, i.e. that the leaf spaces are isospectral. As a corollary to the main result, the authors identify geometric conditions that ensure preservation of the mean curvature vector fields, and therefore isospectrality of the leaf spaces.

## Keywords

Spectral geometry Laplace operator Orbifolds Orbit spaces Group actions## Mathematics Subject Classification

58J50 58J53 22D99 53C12## Notes

### Acknowledgements

The authors would like to thank Carolyn Gordon for many helpful conversations throughout the course of this project, as well as Emilio Lauret and Marco Radeschi for providing valuable feedback. The authors would also like to thank the reviewers for many helpful comments, including suggesting a much shorter, more elegant proof of the main corollary, and to acknowledge the support of the National Science Foundation, Grant DMS-1632786.

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