The spectral length of a map between Riemannian manifolds

TL;DR

This paper introduces a spectral invariant derived from Dirichlet series associated with Riemannian manifolds, providing a new way to measure the 'length' of maps and characterize isometries.

Contribution

It defines a spectral length of maps between Riemannian manifolds and establishes a spectral criterion for isometries, leading to a new distance measure between manifolds.

Findings

Spectral length induces a distance between Riemannian manifolds.

Equality of spectral Dirichlet series characterizes isometries.

Spectral invariants provide a new perspective on manifold mappings.

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.