The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds

Abstract

The holonomy algebra g of an indecomposable Lorentzian ( n + 2 ) -dimensional manifold M is a weakly-irreducible subalgebra of the Lorentzian algebra so 1 , n + 1 . L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible subalgebras into 4 types and associated with each such subalgebra g a subalgebra h ⊂ so n of the orthogonal Lie algebra. We give a description of the spaces R ( g ) of the curvature tensors for algebras of each type in terms of the space P ( h ) of h -valued 1-forms on R n that satisfy the Bianchi identity and reduce the classification of the holonomy algebras of Lorentzian manifolds to the classification of irreducible subalgebras h of so ( n ) with L ( P ( h ) ) = h . We prove that for n ⩽ 9 any such subalgebra h is the holonomy algebra of a Riemannian manifold. This gives a classification of the holonomy algebras for Lorentzian manifolds M of dimension ⩽11.