On compact Riemannian manifolds with volume-preserving symmetries (Q582599)

Riemannian manifolds with volume-preserving local geodesic symmetries (known also as D'Atri spaces) have been studied first by \textit{J. E. D'Atri} and \textit{H. K. Nickerson} [J. Differ. Geom. 3, 467-476 (1969; Zbl 0195.236)]. They have proved that every naturally reductive space is a D'Atri space [Mich. Math. J. 22, 71-76 (1975; Zbl 0317.53045)]. \textit{A. Kaplan} gave the examples of D'Atri spaces which are not naturally reductive [Bull. Lond. Math. Soc. 15, 35-42 (1984; Zbl 0521.53048)]. Other geometrical and physical properties concerning D'Atri spaces have been given by several authors, for example, \textit{O. Kowalski} and \textit{L. Vanhecke} [Proc. Am. Math. Soc. 91, 433-435 (1984; Zbl 0524.53033)], \textit{P. H. Roberts} and \textit{H. D. Ursell} [Philos. Trans. R. Soc. Lond. A 252, 317-356 (1962; Zbl 0094.319)] and so on. \textit{O. Kowalski} has proved that every 3-dimensional D'Atri space is a locally homogeneous Riemannian manifold [Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Semin. Mat., Torino, Fasc. Spec., 131-158 (1983; Zbl 0631.53033)]. The problem whether every D'Atri space is locally homogeneous remains still open for the case where the dimension of the space is greater than or equal to 4. In the present paper, the author proves some results on D'Atri spaces which support the homogeneity conjecture, at least in the compact case. For example, the author proves that the volume \(v_ t(x)\) of a geodesic ball in a compact connected analytic D'Atri space (M,g) does not depend on the point \(x\in M\). The author proves also that a compact connected analytic Riemannian manifold (M,g) is a D'Atri space if and only if the first mean-value operator on M commutes with the Riemannian Laplacian.