2,678 research outputs found
Towards a classification of Lorentzian holonomy groups. Part II: Semisimple, non-simple weak-Berger algebras
The holonomy group of an (n+2)-dimensional simply-connected, indecomposable
but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic
group . The main ingredient of
such a holonomy group is the SO(n)--projection and
one may ask whether it has to be a Riemannian holonomy group. In this paper we
show that this is always the case, completing our results of the first part
math.DG/0305139. We draw consequences for the existence of parallel spinors on
Lorentzian manifolds.Comment: 13 page
Screen bundles of Lorentzian manifolds and some generalisations of pp-waves
A pp-wave is a Lorentzian manifold with a parallel light-like vector field
satisfying a certain curvature condition. We introduce generalisations of
pp-waves, on one hand by allowing the vector field to be recurrent and on the
other hand by weakening the curvature condition. These generalisations are
related to the screen holonomy of the Lorentzian manifold. While pp-waves have
a trivial screen holonomy there are no restrictions on the screen holonomy of
the manifolds with the weaker curvature condition.Comment: 18 page
Lefschetz Hyperplane Theorem for Stacks
We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for
compact smooth Deligne-Mumford stacks over the site of complex manifolds. For
a hyperplane section, can be obtained from by a sequence
of deformation retracts and attachments of high-dimensional finite disc
quotients. We use this to derive more familiar statements about the relative
homotopy, homology, and cohomology groups of the pair . We also prove
some preliminary results suggesting that the Lefschetz Hyperplane Theorem holds
for Artin stacks as well. One technical innovation is to reintroduce an
inequality of {\L}ojasiewicz which allows us to prove the theorem without any
genericity or nondegeneracy hypotheses on .Comment: 16 page
Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors
We prove that the Cauchy problem for parallel null vector fields on smooth
Lorentzian manifolds is well posed. The proof is based on the derivation and
analysis of suitable hyperbolic evolution equations given in terms of the Ricci
tensor and other geometric objects. Moreover, we classify Riemannian manifolds
satisfying the constraint conditions for this Cauchy problem. It is then
possible to characterise certain holonomy reductions of globally hyperbolic
manifolds with parallel null vector in terms of flow equations for Riemannian
special holonomy metrics. For exceptional holonomy groups these flow equations
have been investigated in the literature before in other contexts. As an
application, the results provide a classification of Riemannian manifolds
admitting imaginary generalised Killing spinors. We will also give new local
normal forms for Lorentzian metrics with parallel null spinor in any dimension.Comment: 39 pages, typos corrected and minor modifications in version
Conformal pure radiation with parallel rays
We define pure radiation metrics with parallel rays to be n-dimensional
pseudo-Riemannian metrics that admit a parallel null line bundle K and whose
Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary
conditions in terms of the Weyl, Cotton and Bach tensors for a
pseudo-Riemannian metric to be conformal to a pure radiation metric with
parallel rays. Then we derive conditions in terms of tractor calculus that are
equivalent to the existence of a pure radiation metric with parallel rays in a
conformal class. We also give an analogous result for n-dimensional
pseudo-Riemannian pp-waves.Comment: 14 pages, in v2 Remark 2 about integrability conditions adde
Completeness of compact Lorentzian manifolds with Abelian holonomy
We address the problem of finding conditions under which a compact Lorentzian
manifold is geodesically complete, a property, which always holds for compact
Riemannian manifolds. It is known that a compact Lorentzian manifold is
geodesically complete if it is homogeneous, or has constant curvature, or
admits a time-like conformal vector field. We consider certain Lorentzian
manifolds with Abelian holonomy, which are locally modelled by the so called
pp-waves, and which, in general, do not satisfy any of the above conditions.
%the condition that their curvature sends vectors that are orthogonal to the
vector field to a multiple of the vector field. We show that compact pp-waves
are universally covered by a vector space, determine the metric on the
universal cover, and prove that they are geodesically complete. Using this, we
show that every Ricci-flat compact pp-wave is a plane wave.Comment: 30 pages, comments welcome; version 2 revised, references and a new
result about compact, Ricci-flat pp-waves added. Version 3 is substantially
revised with new title. We added Corollary 2 about completeness of
indecomposable, compact locally symmetric Lorentzian manifold
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