Holonomy groups and parallel spinors on totally reducible pseudo-Riemannian manifolds
The note deals with the following problem: What are the possible holonomy groups of simply connected pseudo-Riemannian spin manifolds \((M^{(p,q)},g)\) which admit parallel spinors? \textit{M.Y. Wang} [Ann. Global Anal. Geom. 7, No. 1, 59--68 (1989; Zbl 0688.53007)], \textit{H. Baum} and \textit{I. Kath} [Ann. Global Anal. Geom. 17, 1--17 (1999; Zbl 0928.53027)] solved the problem when \((M^{(p,q)},g)\) is irreducible and non-locally symmetric. \textit{Th. Leistner} [Holonomy and parallel spinors in Lorentzian geometry, Ph.D. Thesis, Humboldt University of Berlin, 2003] solved the problem in the Lorentzian case. The author [C. R., Math., Acad. Sci. Paris, Ser. 337, 179--184 (2003; Zbl 1052.53042)] also characterized, by their holonomy groups, the simply connected spin pseudo-Riemannian manifolds which admit parallel pure spinors. In this note, the author gives a complete answer to the above problem when \((M^{(p,q)},g)\) is totally reducible, i.e. every invariant subspace by the holonomy representation admits an invariant complementary.
scientific article; zbMATH DE number 2094047
scientific article