We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold and a properly discontinuous group of isome- tries. Most of our examples are quotients of pp-waves with disconnected holonomy and without parallel vector field. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, dis- connected full holonomy and a parallel spinor.

On the full holonomy group of special Lorentzian manifolds

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate center. We show that the double extension process can be used to construct all Lorentzian flat Lie algebras with degenerate center generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie algebras. Finally, we give the list of Lorentzian flat Lie algebras with degenerate center up to dimension 6.

Left-invariant Lorentzian flat metrics on Lie groups

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-
 $$H$$
 -type Lie groups.

/pdf/isometric-actions-on-pseudo-riemannian-nilmanifolds-47cu4pbzgp.pdf

Isometric actions on pseudo-Riemannian nilmanifolds

For which groups G of even order 2n does a 1-factorization of the complete graph K2n exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in 4, we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor. © 2005 Wiley Periodicals, Inc. J Combin Designs

Nilpotent 1-factorizations of the complete graph†

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation; for instance the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with left-invariant pseudo-Riemannian metric we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-$H$-type groups.

In the following we show the only possible flat, connected, incomplete homogeneous spacetimes are H/A where H = {v E Rn1g(v,N) > 0}, N is a null vector, and A is a discrete subgroup of translations. In [WI] Wolf classifies all complete flat homogeneous spacetimes. He does this by observing that the universal covering of such a spacetime must be Minkowski space, reducing the problem to finding all discrete subgroups of the Poincar& group which act properly discontinuously and whose normalizer acts transitively on Minkowski space. Classification of complete compact flat spacetimes in dimension 4 is given by Fried in [F]. In [W2] Wolf remarks that there are homogeneous, flat, incomplete spacetimes, and he gives an example. In this paper we will classify all incomplete, flat, homogeneous spacetimes. The first step in such a classification is to find all the possible universal covers of these spacetimes. Let N be a null vector in n dimensional Minkowski space. We show that the subspace H of n dimensional Minkowski space defined as {v: g(v , N) > 0} is the unique (to within isometry), incomplete, homogeneous, simply connected, flat n dimensional spacetime (3.1). Next we find all discrete subgroups A of the isometry group of H which have transitive normalizers. Then every connected flat incomplete homogeneous spacetime is isometric to H/A. This result is given in Theorem 3.7. Besides finishing the classification of all homogeneous flat spaces, this result relates to some other questions. Auslander ([Au, p. 809]) conjectured that every nilpotent, simply transitive group of affine transformations must contain a one parameter subgroup of central translations. This result was shown by Scheuneman [S, p. 226]. Auslander gives an example ([Au, p. 810]) of a three dimensional solvable group that acts affinely and simply transitively on R3 and contains no translations. A corollary of our result is that every subgroup of the Poincare group having an open orbit in Minkowski space must contain a one Received by the editors July 5, 1988 and, in revised form, December 8, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C50, 53C30; Secondary 22E43. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

/pdf/homogeneous-spacetimes-of-zero-curvature-sf4cu5zkg3.pdf

Homogeneous spacetimes of zero curvature

/pdf/the-structure-of-symmetry-groups-of-almost-perfect-one-4260gr53ta.pdf

The Structure of Symmetry Groups of Almost Perfect One Factorizations

Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries are shown to be complete. Also flat pseudo-Riemannian homogeneous manifolds with non-trivial holonomy are shown to contain a complete geodesic.

Flat Pseudo-Reimannian manifolds with a nilpotent transitive group of isometries

In the following paper we classify the connected flat homogeneous spacesM of metric signature (n, 2) with translationally isotropic associated domain.U ⊃IR s n whereIR s n denotesIRn with the usual flat metric istranslationally isotropic if the set of all translations which leaveU invariant contains its perpendicular space. IfU is the image of the universal cover ofM under the development map thenU is called theassociated domain ofM.

Translationally isotropic flat homogeneous manifolds with metric signature (n, 2)

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Papers

Homogeneous spacetimes of zero curvature

The Structure of Symmetry Groups of Almost Perfect One Factorizations

Flat Pseudo-Reimannian manifolds with a nilpotent transitive group of isometries

Translationally isotropic flat homogeneous manifolds with metric signature (n, 2)