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Local Symmetry of Harmonic Spaces as Determined by The Spectra of Small Geodesic Spheres

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Abstract

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm \({{|\nabla{R}|}}\) of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants |R|2 and |Ric|2 of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Extremadura, 06006, Badajoz, Spain

    Teresa Arias-Marco

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099, Berlin, Germany

    Dorothee Schueth

Authors
  1. Teresa Arias-Marco
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  2. Dorothee Schueth
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Correspondence to Teresa Arias-Marco.

Additional information

Dorothee Schueth would like to dedicate this article to her former high-school teacher, Martin Berg. It was he who first made her see the beauty of Mathematics.

The authors were partially supported by DFG Sonderforschungsbereich 647. The first author’s work has also been supported by D.G.I. (Spain) and FEDER Project MTM2010-15444, by Junta de Extremadura and FEDER funds, and the program “Estancias de movilidad en el extranjero ‘José Castillejo’ para jóvenes doctores” of the Ministry of Education (Spain).

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Arias-Marco, T., Schueth, D. Local Symmetry of Harmonic Spaces as Determined by The Spectra of Small Geodesic Spheres. Geom. Funct. Anal. 22, 1–21 (2012). https://doi.org/10.1007/s00039-012-0146-y

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  • DOI: https://doi.org/10.1007/s00039-012-0146-y

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