If the holonomy representation of an (n + 2)-dimensional simply-connected Lorentzian manifold (M, h) admits a degenerate invariant subspace its holonomy group is contained in the parabolic group (R × SO(n)) ⋉ R n . The main ingredient of such a holonomy group is the SO(n)-projection G := pr SO(n) (Hol p (M, h)) and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is the case if G ⊂ U (n/2) or if the irreducible acting components of G are simple.

Cones over pseudo-Riemannian manifolds and their holonomy By D. V. Alekseevsky1) at Edinburgh, V. Cortés2) at Hamburg, A. S. Galaev3) at Brno, and T. Leistner4) at HamburgAbstract. By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudoorthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.Alekseevsky, Corté s, Galaev and Leistner, Cones over pseudo-Riemannian manifolds Similarly, we have the following characterisation of the case when the coneM M admits (locally) a para-hyper-Kähler structure, which means that the holonomy algebraĥ h preserves two complementary isotropic subspaces T G and a skew-symmetric complex structure J such that JT þ ¼ T À . In particular, it preserves the para-hyper-complex structure ðĴ J 1 ;Ĵ J 2 ;Ĵ J 3 ¼Ĵ J 1Ĵ J 2 Þ, whereĴ J 1 j T G ¼ GId andĴ J 3 ¼ J.Theorem 8.2. Let ðM; gÞ be a pseudo-Riemannian manifold. There is a one-to-one correspondence between para-3-Sasakian structures ðM; g; T 1 ; T 2 ; T 3 Þ on ðM; gÞ and parahyper-Kähler structures ðM M;ĝ g;Ĵ J 1 ;Ĵ J 2 ;Ĵ J 3 ¼Ĵ J 1Ĵ J 2 Þ on the cone ðM M;ĝ gÞ. The correspondence is given by T a 7 !Ĵ J a :¼' 'T a .

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The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.MSC: 53C29; 53C50; 53A30

It was proved by Hitchin that any solution of his evolution equations for a half-flat SU(3)structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G 2. We give a new proof, which does not require the compactness of M . More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N , for any real form G of SL(3, C). If G is non-compact, then the holonomy group of N is a subgroup of the non-compact form G * 2 of G C 2 . Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G 2-or G * 2 -structures, as well as for the extension of cocalibrated G2-and G * 2 -structures by parallel Spin(7)-and Spin 0 (3, 4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G 2-or G * 2 -structure. For the group H3 × H3, where H 3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G 2-or G * 2 -structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G 2 and G * 2 . Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (ω, ρ) on H 3 × H3 satisfying ω(z, z) = 0, where z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special Kähler manifolds and one special para-Kähler manifold.Proposition 1.1. Let V be an n-dimensional real or complex vector space. The general linear group GL(V ) has an open orbit in Λ k V * , with 0 k n/2, if and only if k 2 or if k = 3 and n = 6, 7 or 8.Proof. The representation of GL(V ) on Λ k V * is irreducible. In the complex case the result thus follows, for instance, from the classification of irreducible complex prehomogeneous vector spaces [32]. The result in the real case follows from the complex case, since the complexification of the GL(n, R)-module Λ k R n * is an irreducible GL(n, C)-module. Remark 1.2. An open orbit is unique in the complex case, since an orbit that is open in the usual topology is also Zariski-open and Zariski-dense (see [31, Proposition 2.2]). Over the reals, the number of open orbits is finite by a well-known theorem of Whitney.Proposition 1.4. Let V be an oriented real vector space of dimension n and assume that k ∈ {2, n − 2} with n even, or k ∈ {3, n − 3} with n = 6, 7 or 8. There is a GL + (V )-equivariant mappinghomogeneous of degree n/k, which assigns a volume form to a stable k-form and which vanishes on non-stable forms. Given a stable k-form ρ, the derivative of φ in ρ defines a dual (n − k)-form ρ ∈ Λ n−k V * by the property(1.1)The dual formρ is also stable and satisfiesA stable form, it...