Holonomy groups of Lorentzian manifolds

Abstract

In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian manifolds with parallel and recurrent spinor fields, conformally flat Walker metrics and the classification of 2-symmetric Lorentzian manifolds are considered.Comment: 49 pages; this is a survey on Lorentzian holonomy groups and their application

On Lorentzian connections with parallel skew torsion

“…This implies that the holonomy algebra g of the connection ∇ is a Berger algebra with zero torsion, and consequently g is the holonomy algebra of the Levi-Civita connection on a Lorentzian manifold, see, e.g., [16]. Another conclusion is that T is parallel with respect to the Levi-Civita connection.…”

“…and the corresponding decomposition g = a ⊕ b such that a ⊂ so(1, k + 1) Rp is a weakly irreducible holonomy algebra of the Levi-Civita connection of a Lorentzian manifold admitting a parallel isotropic vector field, and b ⊂ so(n − k) is the holonomy algebra of the Levi-Civita connection of a Riemannian manifold. The subalgebra (pr so(k) a) ⊕ b ⊂ so(n) annihilates an element from ∧ 2 R n corresponding to the parallel 2-form ω on E. Using the results from [16] it is easy to see that each…”

Parallel spinors on Lorentzian Weyl spaces

“…The function F (u) does influence the curvature of the metric g [17], consequently, the coordinates may be choosen in such a way that F (u) = 0. Let us prove the inverse implication.…”

“…Let us consider a Weyl structure as in Theorem 9 and impose the Einstein-Weyl equation. The symmetric part Ric s of the connection ∇ may be written in the form Ric s = Ric + Ric, where Ric is the Ricci tensor of the metric g. The assumption on the subalgebra h ⊂ so(n) and the computations from [17] imply that…”

Holonomy Classification of Lorentz-Kähler Manifolds

“…[9,14,34]. The case of Lorentzian manifolds took the attention of geometers and theoretical physicists during the last two decades, see the reviews [3,25,26] and the references therein. In the other signatures, only partial results are known [12,13,7,23,24,28,29,35].…”