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

TL;DR

This paper demonstrates that the spectra of small geodesic spheres in harmonic spaces uniquely determine the local symmetry and the covariant derivative of the curvature tensor, using heat invariants and curvature expansions.

Contribution

It establishes a spectral characterization of local symmetry in harmonic spaces based on the eigenvalues of geodesic spheres.

Findings

Spectra determine the norm of the covariant derivative of the curvature tensor.

Spectra of small geodesic spheres reveal whether the space is locally symmetric.

Results extend to geodesic balls with boundary conditions.

Abstract

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.