Related papers: On Time-Like Class $\mathcal A$ Surfaces in a Stat…
A surface in the Lorentz-Minkowski $3$-space is generally a mixed type surface, namely, it has the lightlike locus. We study local differential geometric properties of such a locus on a mixed type surface. We define a frame field along a…
Let $M$ be a Lorentz surface and $F:M\rightarrow N$ a time-like and conformal immersion of $M$ into a 4-dimensional neutral space form $N$ with zero mean curvature vector. We see that the curvature $K$ of the induced metric on $M$ by $F$ is…
In this note, we prove the existence of one particular class of starshaped compact hypersurfaces, by deriving global curvature estimates for such hypersurfaces; this generalizes the main result in [Hypersurfaces of prescribed mixed…
We introduce a class of surfaces in euclidean space motivated by a problem posed by \'{E}lie Cartan. This class furnishes what seems to be the first examples of pairs of non-congruent surfaces in euclidean space such that, under a…
On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The…
There are examples of complete spacelike surfaces in the Lorentzian product $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K\leq -1$. In this paper, we show that there exists no complete spacelike surface in…
Using Lie symmetry methods for differential equations we have investigated the symmetries of a Lagrangian for a plane symmetric static spacetime. Perturbing this Lagrangian we explore its approximate symmetries. It has a non-trivial…
In this article and its sequel we discuss the asymptotic structure of space-times representing isolated bodies in General Relativity. Such space-times are usually required to be asymptotically flat (AF), and thus to have a prescribed type…
For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0…
In this paper, we study three types of helicoidal surfaces in a Lorentzian n--space $\mathbb{E}^n_1$. First, we find the parametrizations of spacelike loxodromes on such spacelike helicoidal surfaces in $\mathbb{E}^n_1$. Then, we make a…
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…
In this paper, we show the existence of a timelike minimal surface with an arbitrary number of weak complete ends. Then, we discuss the asymptotic behaviour of the simple ends and the topology of the singularity set of the constructed…
In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere $\s_{\varepsilon}^3$, that is the three-dimensional sphere endowed with a $1$-parameter family of Lorentzian metrics, obtained by deforming the…
The time dependent conformally-flat spherical Rindler spacetime is investigated. The geometry has an apparent horizon that coincides with the causal horizon. The scalar acceleration of a static observer is constant and equals to the…
Uniqueness and non-existence results on complete constant mean curvature spacelike hypersurfaces lying between two spacelike slice in the Einstein-de Sitter spacetime are given. They are obtained from a Liouvielle-type theorem applied to a…
What is the shape of space in a spacetime? One way of addressing this issue is to consider edgeless spacelike submanifolds of the spacetime. An alternative is to foliate the spacetime by timelike curves and consider the quotient obtained by…
In the present paper, a new type of ruled surfaces called osculating-type (OT)-ruled surface is introduced and studied. First, a new orthonormal frame is defined for OT-ruled surfaces. The Gaussian and the mean curvatures of these surfaces…
A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $\eta$. Several sufficient assumptions on such a surface with non-degenerate $\eta$-second fundamental…
We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension $3.$ We in particular obtain a representation theorem for surfaces in the $\mathbb{L}(\kappa,\tau)$ spaces. We then…
In this manuscript we investigate the intrinsically flat (space-flat) spacetimes as viable cosmological models. We show that they have a natural geometric structure which is suitable to describe inhomogeneous matter distributions forming a…