PreprintCC BY-SA 4.0arXiv
Sub-Riemannian spectral distance
Abstract
We study eigenvalues and eigenfunctions of the ``div-grad type" sub-Laplacian with respect to Popp's volume on a compact equiregular sub-Riemannian manifold M. Since Popp's volume is canonically determined by the sub-Riemannian structure of M, the spetra of the sub-Laplacian carry geometric meanings. In this paper, we first embed M into the Hilbert space of square-summable sequences using eigenfunctions and then define a spectral distance between two compact equiregular sub-Riemannian manifolds. Our result is a sub-Riemannian analogue of Berard-Besson-Gallot's classical work in the Riemannian case.
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BibTeX
@misc{inahama2026subriemannian,
title = {{Sub-Riemannian spectral distance}},
author = {Yuzuru Inahama},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.12804},
eprint = {2606.12804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12804},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Sub-Riemannian spectral distance
AU - Yuzuru Inahama
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.12804
UR - https://arxiv.org/abs/2606.12804
AB - We study eigenvalues and eigenfunctions of the ``div-grad type" sub-Laplacian with respect to Popp's volume on a compact equiregular sub-Riemannian manifold M. Since Popp's volume is canonically determined by the sub-Riemannian structure of M, the spetra of the sub-Laplacian carry geometric meanings. In this paper, we first embed M into the Hilbert space of square-summable sequences using eigenfunctions and then define a spectral distance between two compact equiregular sub-Riemannian manifolds. Our result is a sub-Riemannian analogue of Berard-Besson-Gallot's classical work in the Riemannian case.
ER -