Null hypersurfaces in Lorentzian manifolds II

1981 ◽  
Vol 89 (3) ◽  
pp. 525-532 ◽  
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.

Null hypersurfaces in Lorentzian manifolds: I

1980 ◽  
Vol 88 (1) ◽  
pp. 175-182 ◽  
Author(s):  
K. Katsuno
Keyword(s):  
Tangent Plane ◽  
Null Vector ◽  
Lorentzian Manifolds ◽  
Geometrical Properties ◽  
Null Hypersurfaces

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.

Compact null hypersurfaces in Lorentzian manifolds

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.

Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior

2003 ◽  
Vol 3 (4) ◽  
Author(s):  
R. Bartolo ◽  
A.M. Candela ◽  
J.L. Flores ◽  
M. Sánchez
Keyword(s):  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Geodesic Connectedness

AbstractThe aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉

Chen-like inequalities on null hypersurfaces with closed rigging of a Lorentzian manifold

2021 ◽  
pp. 2150125
Author(s):  
Amrinder Pal Singh ◽  
Cyriaque Atindogbe ◽  
Rakesh Kumar ◽  
Varun Jain
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Lorentzian Manifold ◽  
Null Hypersurface ◽  
Lorentzian Space ◽  
Null Hypersurfaces

We study null hypersurfaces of a Lorentzian manifold with a closed rigging for the hypersurface. We derive inequalities involving Ricci tensors, scalar curvature, squared mean curvatures for a null hypersurface with a closed rigging of a Lorentzian space form and for a screen homothetic null hypersurface of a Lorentzian manifold. We also establish a generalized Chen–Ricci inequality for a screen homothetic null hypersurface of a Lorentzian manifold with a closed rigging for the hypersurface.

scholarly journals RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS

2020 ◽  
pp. 1-27
Author(s):  
Ali Feizmohammadi ◽  
Lauri Oksanen
Keyword(s):  
Wave Equations ◽  
Model Problem ◽  
Wave Interaction ◽  
Lorentzian Manifold ◽  
Gaussian Beams ◽  
Linear Wave ◽  
Lorentzian Manifolds ◽  
Solution Map ◽  
Globally Hyperbolic ◽  
Linear Wave Equations

This paper is concerned with the resolution of an inverse problem related to the recovery of a function $V$ from the source to solution map of the semi-linear equation $(\Box _{g}+V)u+u^{3}=0$ on a globally hyperbolic Lorentzian manifold $({\mathcal{M}},g)$ . We first study the simpler model problem, where $({\mathcal{M}},g)$ is the Minkowski space, and prove the unique recovery of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.

scholarly journals Null hypersurfaces in Lorentzian manifolds with the null energy condition

2020 ◽  
Vol 155 ◽  
pp. 103751
Author(s):  
Shintaro Akamine ◽  
Atsufumi Honda ◽  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Energy Condition ◽  
Null Energy Condition ◽  
Lorentzian Manifolds ◽  
Null Hypersurfaces

scholarly journals Null hypersurfaces evolved by their mean curvature in a Lorentzian manifold

2019 ◽  
Vol 157 (1) ◽  
pp. 83-106
Author(s):  
Fortuné Massamba ◽  
Samuel Ssekajja
Keyword(s):  
Mean Curvature ◽  
Lorentzian Manifold ◽  
Null Hypersurfaces

scholarly journals Curvature Estimates in Asymptotically Flat Lorentzian Manifolds

2005 ◽  
Vol 57 (4) ◽  
pp. 708-723 ◽  
Author(s):  
Felix Finster ◽  
Margarita Kraus
Keyword(s):  
Fundamental Form ◽  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Second Fundamental Form ◽  
Riemannian Curvature ◽  
Lorentzian Manifolds ◽  
Asymptotically Flat ◽  
Riemannian Curvature Tensor ◽  
Curvature Estimates

AbstractWe consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzianmanifold becomes flat in the limit when the ADM energy tends to zero.

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