Contribution to the proceedings of the XIII Fall Workshop on Geometry and Physics, (Murcia, Spain, 2004); Introduction added and minor corrections made
The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using the causal character of the mean curvature vector. Trapped surfaces can thus be easily seen to be generalizations, possible in semi-Riemannian geometry, of the standard concepts of minimal surfaces, maximal hypersurfaces, geodesics, and the like. Selected applications are mentioned.
We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.
In this paper we define and study the slant and semi-slant submanifolds of a Lorentzian almost paracontact manifold. Some necessary and sufficient conditions for a submanifold of a Lorentzian almost paracontact manifold to be slant are obtained. We give some examples for the slant and semi-slant submanifolds of a Lorentzian almost paracontact manifold. We also investigate the integrability conditions for the distributions involved in the definition of a semi-slant submanifold when the ambient manifold is a Lorentzian paracosymplectic and a Lorentzian para-Sasakian manifold, respectively.
Recently Y{ü}ksel et. al. \cite{SY} shows that there doesn't exist any proper semi-slant warped product submanifolds in a Lorentzian paracosymplectic manifold. In the present article, we first define and give preparatory lemmas for a new generalize class of semi-slant submanifolds called pointwise semi-slant submanifolds in a Lorentzian paracosymplectic manifold, and then we ensure by presenting some existence results and characterization theorem that there exist a non-trivial pointwise semi-slant warped product submanifolds in a Lorentzian paracosymplectic manifold counter to warped product s…
In this paper we study slant submanifolds of Lorentzian almost contact manifolds. We have taken the submanifold as a space like and then defined the slant angle on a submanifold and thus we extended the results of A. Lotta (Slant submanifolds in contact geometry [8]) and M. A. Khan et. al. (Slant submanifolds of Lorentzian Paracontact manifolds [7]) in this new setting.
In this paper, we give the Cartan's formula for half-lightlike submanifolds of Lorentzian manifolds and use it to show that a screen homothetic half-lightlike submanifolds of a Lorentzian space form, with a conformal co-screen distribution are locally a lightlike triple product manifolds. Then we give a classification theorem for half-lightlike submanifolds of Lorentzian space form with constant screen principal curvatures. These results extend some results obtained in the case of lightlike hypersurfaces of Lorentzian manifolds ([1]).
In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\times_{f}M_{\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to $M$ is an usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifo…
In this paper I have studied about CR(Cauchy-Riemann)-submanifolds of Lorentzian Concircular Structure manifold ((LCS)n-manifold), Lorentzian Para-Sasakian(LP)-cosymplectic manifold, S-manifold and Generalized Kenmotsu (GKM) manifold. I have discussed some results regarding distribution, structure vector field, totally geodesic submanifold, leaf etc.. I have obtained results on totally umbilical contact CR-submanifold where the anti-invariant distribution has some properties. Next, I have studied some results about D-totally geodesic CR-submanifold (D is the distribution), a contact CR-submani…
Recently Hui et al. (\cite{HAP}, \cite{HAN}) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a $(LCS)_n$-manifold $\bar{M}$. In this paper we have studied the characterization for both these classes of warped product submanifolds. It is also shown that there do not exists any proper warped product bi-slant submanifold of a $(LCS)_n$-manifold. Although we constructed an example of a bi-slant submanifold of $(LCS)_n$-manifold.
We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of local solutions. We refine the description in the case of curves with null acceleration in three-dimensional spaces Robertson-Walker spaces, and in the case of codimension two submanifolds whose second fundamental form is null
For every codimension two spacelike submanifold of a Lorentz manifold and each choice of a normal lightlike vector field, we introduce a canonical way to construct a tractor conformal bundle. We characterize when the induced connection of a such submanifold defines a tractor connection and then, in this case, when this tractor conformal bundle with the induced connection is standard for the induced metric. Finally, the normality condition for this tractor conformal bundle endowed with the induced connection is characterized in terms of a strong relationship between the intrinsic and the extrin…
We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian minimal surfaces, we introduce, within the varifold class, various notions of generalized minimal timelike submanifolds of arbitrary codimension in flat Minkowski spacetime, which are global in character and admit conserved quantities, such as relativistic energy and momentum. In …
The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse's proof. This approach reduces substantially the effort required in the proofs of the Theorem given in previous articles on the subject. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Loren…
For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of \emph{total shear tensor} and \emph{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 subman…
We give explicit representation formulas for marginally trapped submanifolds of co-dimension two in pseudo-Riemannian spaces with arbitrary signature and constant sectional curvature. This paper is dedicated to the memory of Franki Dillen, 1963-2013.
In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.
A new approach to the study of spacelike submanifolds in a spherical Robertson-Walker spacetime: characterization of the stationary spacelike submanifolds as an application
In this paper, we introduce the notion of a quasi-biharmonic submanifold in a pseudo-Riemannian manifold and classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms.
Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.
Anti-de Sitter space is the Lorentzian space form with negative curvature. In this paper we consider lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension. In particular, we investigate the singularities of lightlike hypersurfaces as an application of the theory of Legendrian singularities.
The authors study the geometry of lightlike hypersurfaces on manifolds $(M, c)$ endowed with a pseudoconformal structure $c = CO (n - 1, 1)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, the authors consider the fibration of isotropic geodesics and investigate their singular points and singular submanifolds. They construct a conformally invariant normalization of a lightlike hypersurface intrinsically connected with its geometry and investigate affine connections i…
There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conformal structure of Lorentzian type. Geometry of these two types of hypersurfaces can be studied in a manner that is similar to that for hypersurfaces of a proper conformal space. Lightlike hypersurfaces are endowed with a degenerate conformal structure. This is the reason that their…
The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we investigate a special class of surfaces (i.e., marginally trapped surfaces) in Minkowski space-time from the view point of the lightlike geometry.
We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-manifold $(M,g)$, if ${\boldsymbol k}$ is a complete null vector field with geodesic flow along which $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, and if $f$ is any smooth function on $M$ with ${\boldsymbol k}(f)$ nowhere vanishing, then $dg(e^f{\boldsymbol k},\cdot)$ is a …
We study the set of trapped photons of a subcritical (a<M) Kerr spacetime as a subset of the phase space. First, we present an explicit proof that the photons of constant Boyer--Lindquist coordinate radius are the only photons in the Kerr exterior region that are trapped in the sense that they stay away both from the horizon and from spacelike infinity. We then proceed to identify the set of trapped photons as a subset of the (co-)tangent bundle of the subcritical Kerr spacetime. We give a new proof showing that this set is a smooth 5-dimensional submanifold of the (co-)tangent bundle with top…
The concept of a marginally trapped surface is important in the theory of general relativity. In the Schwarzschild black hole spacetime, its event horizon is foliated by marginally trapped surfaces. In a more general black hole spacetime, the concept of a marginally trapped surface is closely related to various sorts of horizon, for example, the apparent horizon, the trapping boundary, the isolated horizon and the dynamical horizon. In this paper, we study the set of marginally trapped surfaces in a perturbed Schwarzschild spacetime. We show that for every incoming null hypersurface which is n…
We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M,g), we consider a geodesic $γ$ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, …
