Surfaces in the lightlike cone

In this paper, we study surfaces in the lightlike cone. We first obtain fundamental formulas for surfaces in a lightlike cone of general dimensions and then characterize certain homogeneous surfaces in the three-dimensional lightlike cone and four-dimensional lightlike cone, respectively.     

Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.     

Almost Kenmotsu manifolds with a condition of η-parallelism

We consider almost Kenmotsu manifolds (M²ⁿ⁺¹,φ,ξ,η,g) with η-parallel tensor h^′=h○φ, 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR-integrability of the structure. Under the additional condition _ξh^′=0, the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M²ⁿ⁺¹ are introduced and studied.     

On minimal lightlike surfaces in Minkowski space–time

We distinguish two particular classes of lightlike surfaces in the Minkowski space Mⁿ, which may be viewed as natural analogues of space- and timelike minimal surfaces.     

Canonical variation of a Lorentzian metric

Given a Lorentzian manifold (M,_gL)$(M,g_L)$ and a timelike unitary vector field E  , we can construct the Riemannian metric _gR=_gL+2ω⊗ω$g_R=g_L+2\omega \otimes \omega$, ω being the metrically equivalent one form to E. We relate the curvature of both metrics, especially in the case of E   being Killing or closed, and we use the relations obtained to give some results about (M,_gL)$(M,g_L)$.     

On Convolutions and Linear Combinations of Pseudo-Isotropic Distributions

Let E be a Banach space, and let E* be its dual. For understanding the main results of this paper it is enough to consider E= ⁿ . A symmetric random vector X taking values in E is called pseudo-isotropic if all its one-dimensional projections have identical distributions up to a scale parameter, i.e., for every E* there exists a positive constant c() such that (, X) has the same distribution as c() X ₀, where X ₀ is a fixed nondegenerate symmetric random variable. The function c defines a quasi-norm on E*. Symmetric Gaussian random vectors and symmetric stable random vectors are the best known examples of pseudo-isotropic vectors. Another well known example is a family of elliptically contoured vectors which are defined as pseudo-isotropic with the quasi-norm c being a norm given by an inner product on E*. We show that if X and Y are independent, pseudo-isotropic and such that X+Y is also pseudo-isotropic, then either X and Y are both symmetric -stable, for some (0, 2], or they define the same quasi-norm c on E*. The result seems to be especially natural when restricted to elliptically contoured random vectors, namely: if X and Y are symmetric, elliptically contoured and such that X+Y is also elliptically contoured, then either X and Y are both symmetric Gaussian, or their densities have the same level curves. However, even in this simpler form, this theorem has not been proven earlier. Our proof is based upon investigation of the following functional equation:

Dynamics of Null Hypersurfaces in General Relativity

This paper shows how the structure and dynamics of a lightlike thin shell in general relativity can be obtained from a distributional approach.     

Total lightcone curvatures of spacelike submanifolds in Lorentz–Minkowski space

We introduce the totally absolute lightcone curvature for a spacelike submanifold with general codimension and investigate global properties of this curvature. One of the consequences is that the Chern–Lashof type inequality holds. Then the notion of lightlike tightness is naturally induced.