On probabilistic commutative spaces
type: Item
instance of: scholarly article
title: On probabilistic commutative spaces
zbMATH Open document ID: 0671.53032
DOI: 10.1007/BF01470736
author: Friedbert Prüfer
author: Oldřich Kowalski
published in: Monatshefte für Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178412
review text: In 1960, \textit{P. H. Roberts} and \textit{H. D. Ursell} [Philos. Trans. R. Soc. Lond., A 252, 317-356 (1960; Zbl 0094.319)] worked out an interesting theory of ``commutative'' Riemannian spaces as those compact Riemannian manifolds on which ``every two random steps commute''. In particular, they have proved: 1) A compact analytic Riemannian manifold is commutative if and only if all Euclidean Laplacians on it commute. 2) Each (compact) commutative space has volume-preserving local geodesic symmetries. In the present paper, the earlier results are generalized to the non- compact case, and a completely new characterization of these spaces is given which uses an infinite set of natural two-point functions.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 4098222
zbMATH Keywords: mean-value operator
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Laplacians
zbMATH Keywords: commutative space
zbMATH Keywords: geodesic symmetries
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4098222