JournalsjncgVol. 6, No. 4pp. 721–748

The spectral length of a map between Riemannian manifolds

  • Gunther Cornelissen

    University of Utrecht, The Netherlands
  • Jan Willem de Jong

    University of Utrecht, The Netherlands
The spectral length of a map between Riemannian manifolds cover
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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.

Cite this article

Gunther Cornelissen, Jan Willem de Jong, The spectral length of a map between Riemannian manifolds. J. Noncommut. Geom. 6 (2012), no. 4, pp. 721–748

DOI 10.4171/JNCG/103