On compact Riemannian manifolds with volume-preserving symmetries
type: Item
instance of: scholarly article
title: On compact Riemannian manifolds with volume-preserving symmetries
zbMATH Open document ID: 0691.53033
DOI: 10.1007/BF00127863
author: Friedbert Prüfer
published in: Annals of Global Analysis and Geometry
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 4131203
zbMATH Keywords: locally homogeneous space
zbMATH Keywords: volume-preserving local geodesic symmetries
zbMATH Keywords: D'Atri spaces
zbMATH Keywords: homogeneity conjecture
zbMATH Keywords: mean-value operator
zbMATH Keywords: Laplacian
reviewed by: Kouei Sekigawa
MaRDI profile type: Publication
Wikidata QID: Q115395249
cites work: Divergence-preserving geodesic symmetries
cites work: Geodesic symmetries in spaces with special curvature tensors
cites work: Geodesic spheres and symmetries in naturally reductive spaces
cites work: Mean-value theorems for Riemannian manifolds
cites work: Sphärische Mittelwerte in kompakten harmonischen Riemannschen Mannigfaltigkeiten
cites work: On the Geometry of Groups of Heisenberg Type
cites work: Some curvature identities for commutative spaces
cites work: A Generalization of a Theorem on Naturally Reductive Homogeneous Spaces
cites work: Two-point functions on Riemannian manifolds
cites work: On probabilistic commutative spaces
cites work: Random walk on a sphere and on a Riemannian manifold
cites work: Q5345507
cites work: Volume-preserving geodesic symmetries on four-dimensional 2-Stein spaces
full work available at URL: https://doi.org/10.1007/bf00127863
OpenAlex ID: W153239368
description: scientific article; zbMATH DE number 4131203