Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres
type: Item
instance of: scholarly article
title: Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres
zbMATH Open document ID: 1246.53050
author: Teresa Arias-Marco
author: Dorothee Schueth
published in: Geometric and Functional Analysis. GAFA
publication date: 2012-05-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.1611
full work available at URL: http://hdl.handle.net/10662/2498
review text: The main result here states that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm of the covariant derivative of the Riemannian curvature tensor in that point. As a corollary the following is proved: Let \(M_1\) and \(M_2\) be harmonic spaces and assume that for any pair of points \(p_1 \in M_1\) and \(p_2 \in M_2\) there exists \(\epsilon >0\) such that for each \(r\in (0,\epsilon)\) the geodesic spheres \(S_r(p_1)\) and \(S_r(p_2)\) are isospectral, then \(M_1\) is locally symmetric iff \(M_2\) is locally symmetric. For the proof, the authors consider the power series expansion of the second heat invariant in terms of the radius of the geodesic spheres. The above results disprove certain results by \textit{Z. I. Szabo} in [Ann. Math. (2) 154, No. 2, 437--475 (2001; Zbl 1012.53034); Ann. Math. (2) 161, No. 1, 343--395 (2005; Zbl 1087.53037)]. Analogous results for geodesic balls are obtained with either Dirichlet or Neumann boundary conditions.
reviewed by: Cornelia-Livia Bejan
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J53
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 6029789
zbMATH Keywords: harmonic space
zbMATH Keywords: curvature invariants
zbMATH Keywords: second fundamental form
zbMATH Keywords: Ledger's recursion formula
zbMATH Keywords: geodesic spheres
zbMATH Keywords: geodesic balls
zbMATH Keywords: heat invariants
zbMATH Keywords: isospectral manifolds
zbMATH Keywords: Damek-Ricci spaces
MaRDI profile type: Publication
OpenAlex ID: W2962828584
arXiv ID: 1001.1611
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DOI: 10.1007/S00039-012-0146-Y
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description: scientific article; zbMATH DE number 6029789