Four-dimensional D'Atri-Einstein spaces are locally symmetric
type: Item
instance of: scholarly article
title: Four-dimensional D'Atri-Einstein spaces are locally symmetric
zbMATH Open document ID: 0930.53028
DOI: 10.1016/S0926-2245(99)00024-8
published in: Differential Geometry and its Applications
publication date: 1999-09-22T00:00:00Z
review text: Riemannian manifolds such that all local geodesic symmetries are volume-preserving up to sign were introduced by D'Atri and Nickerson. They proved that all naturally reductive spaces have this property. Leter on, several other large classes of examples and several aspects of their geometry were discovered. These Riemannian manifolds are now called D'Atri spaces. We refer to the survey paper [\textit{O. Kowalski}, \textit{F. Prüfer} and \textit{L. Vanhecke}, Prog. Nonlinear Differ. Equ. Appl. 20, 241-284 (1996; Zbl 0862.53039)] for more information and references. The complete classification of D'Atri spaces is known only in dimensions two and three. Moreover, it turns out that all known examples are locally homogeneous and it is still an open problem whether this holds for each D'Atri space. In view of this and Jensen's classification of four-dimensional homogeneous Einstein manifolds, it is natural to expect four-dimensional D'Atri-Einstein manifolds to be locally symmetric. A positive result has been provided by K. Sekigawa and the reviewer for several special cases but the general case remained open. In this paper and by using the spinor formalism of Penrose and Rindler, the author provides a proof of the expected result for all four-dimensional D'Atri-Einstein spaces and all four-dimensional D'Atri spaces with anti-self-dual Weyl tensor.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C27
zbMATH DE Number: 1339072
zbMATH Keywords: classification of D'Atri spaces
zbMATH Keywords: four-dimensional D'Atri-Einstein manifolds
zbMATH Keywords: locally symmetric
zbMATH Keywords: spinor formalism
zbMATH Keywords: anti-selfdual Weyl tensor
Wikidata QID: Q115337415
author: K. P. Tod
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1339072