arXiv subjects

Kordian Lärz

Publications and source records attributed to Kordian Lärz.

4 recordsLinked to original sources

arXiv2012

On the full holonomy group of special Lorentzian manifolds

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 isometries. 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, disconnected full holonomy and a parallel spinor.

math.DG↗

arXiv2010

A class of Lorentzian manifolds with indecomposable holonomy groups

We consider a class of $S^{1}$-bundles whose total space admits a nowhere vanishing recurrent lightlike vector field with respect to a Lorentzian metric. This metric can be modified such that its restricted holonomy group is indecomposable and reducible. We apply Hodge theory to construct examples with Hermitian screen holonomy. Finally, we construct complete pp-waves.

math.DG↗

arXiv2010

Riemannian Foliations and the Topology of Lorentzian Manifolds

A parallel lightlike vector field on a Lorentzian manifold $X$ naturally defines a foliation $\mathcal{F}$ of codimension one. If either all leaves of $\mathcal{F}$ are compact or $X$ itself is compact admitting a compact leaf and the (transverse) Ricci curvature is non-negative then a Bochner type argument implies that the first Betti number of $X$ is bounded by $1 \leq b_{1} \leq \dim X$ if $X$ is compact and $0 \leq b_{1} \leq \dim X -1$ otherwise. We show that these bounds are optimal and depending on the holonomy of $X$ we obtain further results. Finally, we classify the holonomy representations for those $X$ admitting a compact leaf with finite fundamental group.

math.DG↗

arXiv2008

On the normal holonomy representation of spacelike submanifolds in pseudo-Riemannian space forms

In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichnérowicz property. In particular, we derive a classification of Lorentzian normal holonomy representations.

math.DG↗