Null Hypersurfaces on Lorentzian Manifolds and Rigging Techniques

This paper is concerned with geometrical properties of null hypersurfaces in Lorentzian manifolds. Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. First, the contravariant metric cannot immediately be defined, so the connection cannot be specified uniquely in the normal way. Secondly, the normal is a null vector lying in the tangent plane, which makes it necessary to look for some other vector to rig the hypersurface, and makes it impossible to normalise the normal in the usual way. These problems are considered in this paper.

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Compact null hypersurfaces in Lorentzian manifolds

Advances in Geometry ◽

10.1515/advgeom-2021-0001 ◽

2021 ◽

Vol 21 (2) ◽

pp. 251-263

Author(s):

C. Atindogbé ◽

M. Gutiérrez ◽

R. Hounnonkpe

Keyword(s):

Global Symmetries ◽

Lorentzian Manifold ◽

Lorentzian Manifolds ◽

Geometric Properties ◽

Totally Geodesic ◽

Null Hypersurfaces ◽

Ambient Manifold ◽

The Family

Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.

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Singularities of null mean curvature flow of null hypersurfaces in Lorentzian manifolds

10.31730/osf.io/m5hc8 ◽

2019 ◽

Author(s):

Samuel Ssekajja

Keyword(s):

Mean Curvature ◽

Mean Curvature Flow ◽

Blow Up ◽

Curvature Flow ◽

Type I ◽

Type Ii ◽

Lorentzian Manifolds ◽

Null Hypersurfaces ◽

Geometric Conditions ◽

Harnack Estimates

We classify two main singularities, as type I and type II, associated with null mean curvature flow of screen conformal null hypersurfaces in Lorentzian manifolds. We prove that the flow at a type I singularity is asymptotically self-similar, whereas at a type II singularity there is a blow-up solution which is an eternal solution. For further analysis of the above two singularities, we define null translating solitons and use them to prove some Harnack estimates for null mean curvature flow under certain geometric conditions.

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Null hypersurfaces in Lorentzian manifolds II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100058412 ◽

1981 ◽

Vol 89 (3) ◽

pp. 525-532 ◽

Cited By ~ 1

Author(s):

K. Katsuno

Keyword(s):

Lorentzian Manifold ◽

Parameter Family ◽

Lorentzian Manifolds ◽

Geometrical Properties ◽

Null Hypersurfaces

This paper is a continuation of (8), and is concerned with geometrical properties of special null hypersurfaces. In particular, on a one-parameter family of null hypersurfaces in four-dimensional Lorentzian manifoldV4, we consider the relation between their normal and the Debever vectors, especially repeated ones. Throughout this paper, the same notations as those in (8) are used.

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Geometry of isoparametric null hypersurfaces of Lorentzian manifolds

10.31730/osf.io/htfj7 ◽

2019 ◽

Author(s):

Samuel Ssekajja

Keyword(s):

Ambient Space ◽

Null Hypersurface ◽

Principal Curvatures ◽

Lorentzian Manifolds ◽

Space Forms ◽

Lorentzian Space ◽

Null Hypersurfaces ◽

Shape Operators ◽

Lorentzian Space Forms

We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi-parametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

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