Geometric structures over four-dimensional generalized symmetric spaces
A Walker manifold is a pseudo-Riemannian manifold \((M,g)\) with a parallel degenerate distribution \({\mathcal D}\). Walker structures naturally reflect many geometric structures of pseudo-Riemannian manifolds which have no Riemannian analogues. A natural problem is to decide whether a given pseudo-Riemannian manifold admits a Walker structure. The paper answers this question for generalized symmetric spaces of dimension 4. Among generalized symmetric spaces of dimension 4, Walker metrics are completely determined. Generalized symmetric spaces were studied in many previous works from various viewpoints. Basically, there are four types of generalized symmetric spaces, denoted by A, B, C, D. Spaces of type C have Lorentzian signature, other spaces have signatures \((0,4)\), \((2,2)\) or \((4,0)\). The main result of the paper says that generalized symmetric spaces of type A with neutral signature and spaces of type B and D admit two totally null degenerate complementary distributions. Spaces of type C admit a partially null degenerate distribution of type \((1,2)\). Hence, all four-dimensional pseudo-Riemannian generalized symmetric spaces are Walker manifolds. Further, the self-duality and anti-self-duality for types A, B, D are studied. It is shown that the only possible (anti-)self-dual case is the type B.
scientific article; zbMATH DE number 6164343
scientific article