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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 geodesicspheres 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 factthat it contradicts some of my isospectrality examples constructed on geodesicspheres and balls of certain harmonic manifolds. The authors overlooked thatthe Lichnerowicz identity is not determined by the given spectral data, and sois the final crucial equation obtained by eliminating with the Lichnerowiczidentity. In short, the above theorem has falsely been established byspectrally undetermined identities which can not be computed (determined) bythe spectra of local geodesic spheres. More complicated spectrally undeterminedfunctions cause the problems also in case of local geodesic balls. I describealso a strong physical argument which clearly explains why the manifoldsappearing in my examples are isospectral. It must be pointed out, however, thata very remarkable new idea, namely, the asymptotic expansion of the heatinvariants $a_k(p,r)$ defined on geodesic spheres, $S_p(r)$, is introduced inthe paper. It can be used for developing both geometric uncertainty theory andglobal vs. local spectral investigations. Among my contributions to thisdeveloping field is the following statement: Average volumes, $\intvol(B_p(r))dp$ resp. $\int vol(S_p(r))dp$, of geodesic balls resp. spheres aregenerically not determined by the spectra of compact Riemann manifolds. Thistheorem interestingly contrasts the statement asserting that the volume of thewhole manifold can be determined in terms of this global spectrum. We alsohave: Music written for local drums can not be played, in general, on the wholemanifold.

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]