Four-dimensional D'Atri-Einstein spaces are locally symmetric
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.
four-dimensional D'Atri-Einstein manifolds
scientific article; zbMATH DE number 1339072
scientific article