The spectral length of a map between Riemannian manifolds
Abstract
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smooth diffeomorphism between Riemannian manifolds is an isometry, in terms of equality along pullback. We also use the invariant to define the (spectral) length of a map between Riemannian manifolds, where a map of length zero between manifolds is an isometry. We show that this length induces a distance between Riemannian manifolds up to isometry.
Keywords
Cite
@article{arxiv.1007.0907,
title = {The spectral length of a map between Riemannian manifolds},
author = {Gunther Cornelissen and Jan Willem de Jong},
journal= {arXiv preprint arXiv:1007.0907},
year = {2011}
}
Comments
24 pages, 3 figures, smoothness assumptions added, various small changes and clarifications