Holonomy of Einstein Lorentzian manifolds. (English) Zbl 1187.83017

Galaev, Anton S.

Classical Quantum Gravity 27, No. 7, Article ID 075008, 13 p. (2010).

Summary: The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp. vacuum Einstein) Lorentzian manifold; the direct constructions are given. Also the holonomy algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The classification of the holonomy algebras of Lorentzian manifolds is reviewed and a complete description of the spaces of curvature tensors for these holonomies is given.

MSC:

83C15	Exact solutions to problems in general relativity and gravitational theory
83A05	Special relativity
58D17	Manifolds of metrics (especially Riemannian)
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C75	Space-time singularities, cosmic censorship, etc.
83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53C29	Issues of holonomy in differential geometry

Keywords:

manifolds of metrics (esp. Riemannian); Lorentz manifolds, manifolds with indefinite metrics; space-time singularities, cosmic censorship; Einstein’s equations (general structure, canonical formalism, Cauchy problems); issues of holonomy