Remarks on AriasMarco-Sch\"uth's paper entitled: "Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres" [arXiv:1001.1611]

TL;DR

This paper critiques a previous claim that the spectra of small geodesic spheres determine certain geometric quantities on harmonic spaces, showing that some key quantities are spectrally undetermined and introducing new heat invariant expansions.

Contribution

It identifies errors in prior spectral determination results and introduces a novel asymptotic expansion of heat invariants on geodesic spheres for spectral analysis.

Findings

Quantity ||∇R||^2(p) cannot be determined by local sphere spectra

Average volumes of geodesic balls and spheres are not spectrally determined

Heat invariants expansion offers new tools for geometric spectral analysis

Abstract

The main goal in this paper is to point out that quantity $||\nabla R||^2(p)$ on a harmonic space can not be determined by the spectra of local geodesic spheres or balls, therefore the main results of [AM-S] (quoted in the title) are wrong. My strong interest in the above theorem is motivated by the fact that it contradicts some of my isospectrality examples constructed on geodesic spheres and balls of certain harmonic manifolds. The authors overlooked that the Lichnerowicz identity is not determined by the given spectral data, and so is the final crucial equation obtained by eliminating with the Lichnerowicz identity. In short, the above theorem has falsely been established by spectrally undetermined identities which can not be computed (determined) by the spectra of local geodesic spheres. More complicated spectrally undetermined functions cause the problems also in case of local geodesic balls. I describe also a strong physical argument which clearly explains why the manifolds appearing in my examples are isospectral. It must be pointed out, however, that a very remarkable new idea, namely, the asymptotic expansion of the heat invariants $a_k(p,r)$ defined on geodesic spheres, $S_p(r)$, is introduced in the paper. It can be used for developing both geometric uncertainty theory and global vs. local spectral investigations. Among my contributions to this developing field is the following statement: Average volumes, $\int vol(B_p(r))dp$ resp. $\int vol(S_p(r))dp$, of geodesic balls resp. spheres are generically not determined by the spectra of compact Riemann manifolds. This theorem interestingly contrasts the statement asserting that the volume of the whole manifold can be determined in terms of this global spectrum. We also have: Music written for local drums can not be played, in general, on the whole manifold.