The spectral length of a map between Riemannian manifolds

Details The spectral length of a map between Riemannian manifolds The spectral length of a map between Riemannian manifolds

2010-07-06

arxiv.org/abs/1007.0907v2

Mathematics

Differential Geometry

24 pages, 3 figures, smoothness assumptions added, various small changes and clarifications

Scientific paper

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.

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