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We show that in any harmonic space, the eigenvalue spectra of the Laplaceoperator on small geodesic spheres around a given point determine the norm$|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor inthat point. In particular, the spectra of small geodesic spheres in a harmonicspace determine whether the space is locally symmetric. For the proof we usethe first few heat invariants and consider certain coefficients in the radialpower series expansions of the curvature invariants $|R|^2$ and $|Ric|^2$ ofthe geodesic spheres. Moreover, we obtain analogous results for geodesic ballswith either Dirichlet or Neumann boundary conditions.

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