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.Comment: 18 pages, LaTeX. Added a few lines in the introduction, corrected a
  few typos. Final version. Accepted for publication in GAF

Arias-Marco, Teresa

Schueth, Dorothee

English

arXiv

Publicada en:  	Geom. Funct. Anal. 22 (2012), no. 1, 1 - 21   doi:10.1007/s00039-012-0146-ySe demuestra que en cualquier espacio armónico, los espectros de valor propio del operador de Laplace en pequeñas esferas geodésicas alrededor de un punto dado determinan la norma | ∇R | de la derivada la covariante del tensor de curvatura de Riemann en ese punto. En particular, los espectros de pequeñas esferas geodésicas en un espacio armónico determinan si el espacio es localmente simétrico. Para la prueba se utilizan las primeras invariantes de calor y consideramos ciertos coeficientes en los desarrollos en serie de potencia radiales del invariantes de curvatura | R | 2 | y Ric | 2 de las esferas geodésicas. Por otra parte, se obtienen resultados análogos para bolas geodésicas, ya sea con Dirichlet o con las condiciones de frontera de Neumann. También abordamos la relevancia de estos resultados para construcciones de Z. Szabo.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 |∇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. Szabo.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 ‘Jos é Castillejo’ para jóvenes doctores” of the Ministry of Education (Spain)

Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres

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