Related papers: Totally umbilic null hypersurfaces in generalized …
In this paper, we give some new characterizations of umbilic hypersurfaces in general warped product manifolds, which can be viewed as generalizations of the work in \cite{KLP18} and \cite{WX14}. Firstly, we prove the rigidity for…
We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped…
We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We…
The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic…
Following ideas of Choe and Fernandez-do Carmo, we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for…
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…
In this paper, we classify Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form and three distinct principal curvatures. This completes the classification of such hypersurfaces initiated by Hu, Xie and…
By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with…
We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary.…
In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what…
In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is…
In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…
We define and study totally geodesic null hypersurfaces in Finsler spacetimes. We prove that the null convergence condition and a certain mild gravitational equation $\chi_\alpha=0$, imply the vanishing of the restriction of the Ricci…
The Robinson-Trautman solution in the Einstein-Maxwell-$\Lambda$ system admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. Restricting to the case where the Maxwell field is aligned, i.e., the…
For many purposes, a three-dimensional foliation of spacetime is more advantageous to understanding its light cone structure. We derive the equations describing such foliations for the Kerr geometry with non-zero cosmological constant, and…
In this article, we study space-like and time-like surfaces in a Robertson-Walker space-time,, denoted by $L^4_1(f,c)$, having positive relative nullity. First, we give the necessary and sufficient conditions for such space-like and…
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a…
The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic…
We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already…
A local description of the non-flat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos \cite{DaVl}. From their classification, it follows that there is an abundance of infinitesimally bendable…