Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres
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.
scientific article; zbMATH DE number 6029789
scientific article