Joeri Van der Veken

Marginally trapped surfaces in Lorentzian space forms with positive relative nullity

In general relativity, a spatial surface in a 4-dimensional Lorentzian manifold is called marginally trapped if its mean curvature vector is lightlike at each point. In this paper, we completely classify marginally trapped spatial surfaces with positive relative nullity in Lorentzian space forms.We consider a simple, physical approach to the problem of marginally trapped surfaces in the Nonsymmetric Gravitational Theory (NGT). We apply this approach to a particular spherically symmetric, Wyman sector gravitational field, consisting of a pulse in the antisymmetric field variable. We demonstrate that marginally trapped surfaces do exist for this choice of initial data.

Spatial and Lorentzian surfaces in Robertson-Walker space times

Let L14(f,c)=(I×fS,gfc) be a Robertson-Walker space time which does not contain any open subset of constant curvature. In this paper, we provide a general study of nondegenerate surfaces in L14(f,c). First, we prove the nonexistence of marginally trapped surfaces with positive relative nullity. Then, we classify totally geodesic submanifolds. Finally, we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector.

COMPLETE CLASSIFICATION OF PARALLEL LORENTZIAN SURFACES IN LORENTZIAN COMPLEX SPACE FORMS

A surface of a pseudo-Riemannian manifold is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. Such surfaces are fundamental since the extrinsic invariants of the surfaces do no change from point to point. In this article, we completely classify parallel Lorentzian surfaces in Lorentzian complex space forms of complex dimension two.

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.

Totally umbilical hypersurfaces of manifolds admitting a unit Killing field

We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient condition under which a Riemannian three-manifold carrying a unit Killing field admits totally geodesic surfaces and we study local and global properties of three-manifolds satisfying this condition.

On holomorphic Riemannian geometry and submanifolds of Wick-related spaces

Abstract In this article we show how holomorphic Riemannian geometry can be used to relate certain submanifolds in one pseudo-Riemannian space to submanifolds with corresponding geometric properties in other spaces. In order to do so, we shall first rephrase and extend some background theory on holomorphic Riemannian manifolds, which is essential for the later application of the presented method.

Umbilical properties of spacelike co-dimension two submanifolds

For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of total shear tensor and shear operators as the trace-free part of the corresponding second fundamental form and shape operators. The relationship between these quantities and the umbilical properties of the submanifold is shown. Several novel notions of umbilical submanifolds are then considered along with the classical concepts of totally umbilical and pseudo-umbilical submanifolds. Then we focus on the case of co-dimension 2, and we present necessary and sufficient conditions for the submanifold to be umbilical with respect to a normal direction. Moreover, we prove that the umbilical direction, if it exists, is unique —unless the submanifold is totally umbilical— and we give a formula to compute it explicitly. When the ambient manifold is Lorentzian we also provide a way of determining its causal character. We end the paper by illustrating our results on the Lorentzian geometry of the Kerr black hole.

Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field

We classify parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field. Some families of semi-parallel surfaces are also described.

ON EXTRINSICALLY SYMMETRIC HYPERSURFACES OF ℍ n ×ℝ

Totally umbilical, semi-parallel and parallel hypersurfaces of ℍ n ×ℝ are completely classified. More examples arise than in the analogous study on the ambient space 𝕊 n ×ℝ.

Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss map

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.