  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.
