On holonomy of Weyl connections in Lorentzian signature

Abstract

Connected holonomy groups of Weyl connections in Lorentzian signature are classified. IntroductionThe holonomy group of a connection is an important invariant. This motivates the classification problem for holonomy groups. There are classification results for some cases of linear connections. There is a classification of irreducible connected holonomy groups of affine torsionfree connections [25]. Important result is a classification of connected holonomy groups of Riemannian manifolds [6,7,10,22]. Lorentzian holonomy groups are classified [4,24,13,3,19, 16].There are partial results for holonomy groups of pseudo-Riemannian manifolds of other signa-

Parallel spinors on Lorentzian Weyl spaces

“…Let us summarize the recent result on the classification of the holonomy algebras of Weyl manifolds of Lorentzian signature [9]. Let (M, c, ∇) be a Weyl manifold of Lorentzian signature (1, n + 1), n 1.…”

“…Theorem 1. [9] If the holonomy algebra g ⊂ co(1, n + 1) of a non-closed Weyl connection preserves an orthogonal decomposition…”

“…Theorem 2. [9] If the holonomy algebra g ⊂ co(1, n + 1) of a non-closed Weyl connection does not preserve any non-degenerate subspace of R 1,n+1 and preserves a null line, then g is one of the following:…”

“…Theorem 3. ( [9]) Each algebraic curvature tensor R of type co(1, n + 1) Rp is uniquely determined by the following elements:…”

“…In Sections 1 and 2 we give necessary background on weighted parallel spinors and spinor modules. In Section 3 we review the recent classification of holonomy algebras of Lorentzian Weyl manifolds [9]. In Section 4 we classify the holonomy algebras of Lorentzian Weyl spaces admitting weighted parallel spinors.…”