Hypersurfaces In De Sitter Space Research Articles

Locally constrained inverse curvature flow and an Alexandrov-Fenchel inequality in de Sitter space

The long-time existence and convergence are shown for the locally constrained inverse curvature flow of compact spacelike convex hypersurfaces in de Sitter space. As an application, the Alexandrov-Fenchel inequality between the quermassintegrals A0 and An−1 is established in view of their opposite monotonicity along the considered flow. With the help of the dual relationship, a geometric inequality concerning quermassintegrals of convex hypersurfaces in a hyperbolic space of even dimension follows.

Locally constrained inverse mean curvature flow in GRW spacetimes

<p style='text-indent:20px;'>We obtain the long-time existence and smooth convergence results for the locally constrained inverse mean curvature flow of compact spacelike hypersurface in certain GRW spacetimes. As an application of the convergence result, we establish a weighted Minkowski type inequality for closed spacelike mean-convex hypersurface in de Sitter space under mild rigidity assumption.</p>

Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space

Abstract Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19].

A New Characterization of Totally Umbilical Hypersurfaces in de Sitter Space

It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the kth mean curvature function Hk is a linear combination of Hk+1,…, Hn. This is a new angle to characterize round spheres.

Slant geometry of spacelike hypersurfaces in hyperbolic space and de Sitter space

We consider a one-parameter family of new extrinsic differential geometries on hypersurfaces in hyperbolic space. Recently, the second author and his collaborators have constructed a new geometry which is called horospherical geometry on hyperbolic space. There is another geometry which is the famous Gauss–Bolyai–Robechevski geometry (i.e., the hyperbolic geometry) on hyperbolic space. The slant geometry is a one-parameter family of geometries which connect these two geometries. Moreover, we construct a one-parameter family of geometries on spacelike hypersurfaces in de Sitter space.

Linear Weingarten spacelike hypersurfaces in de Sitter space

In this paper, we give a classification on spacelike linear Weingarten hypersurfaces in de Sitter space $S^{n+1}_{1}(1)$ according to the sectional curvature or the length of the second fundamental form.

SPACELIKE SUBMANIFOLDS IN DE SITTER SPACE

We investigate the differential geometry of spacelike submanifolds of codimension at least two in de Sitter space as an application of the theory of Legendrian singularities. We also discuss related geometric property of spacelike hypersurfaces in de Sitter space. Mathematics Subject Classification (2000): 53A35, 53B30, 58C25.

Spacelike submanifolds in de Sitter space

AbstractWe investigate the differential geometry of spacelike submanifolds of codimension at least two in de Sitter space as an application of the theory of Legendrian singularities. We also discuss related geometric property of spacelike hypersurfaces in de Sitter space.

A characterization of H- strictly convex hypersurfaces in de Sitter space

In this study, we introduce a H-strictly convex hypersurface in the 6-dimensional unitary de Sitter space and give a lower bound approximation for the Ricci curvature of such hypersurfaces under some appropriate conditions. Key words: Ricci curvature, H-convex submanifold, hypersurface, de Sitter space.

Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space

We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S 1 n + 1 ( c ) , n ⩾ 3 , with constant normalized scalar curvature R satisfying n − 2 n c ⩽ R ⩽ c totally umbilical?

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss-Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

Spacelike hypersurfaces in de Sitter space with constant higher‐order mean curvature

The authors apply the generalized Minkowski formula to set up a spherical theorem. It is shown that a compact connected hypersurface with positive constant higher‐order mean curvature Hr for some fixed r , 1 ≤ r ≤ n, immersed in the de Sitter space must be a sphere.

Spacelike submanifolds in the de Sitter space S p n+p (c) with constant scalar curvature

Let M n be a closed spacelike submanifold isometrically immersed in de Sitter space S p n+p (c). Denote by R, H and S the normalized scalar curvature, the mean curvature and the square of the length of the second fundamental form of M n, respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M n immersed in S p n+p (c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature” (see Manus Math, 1998, 95:499\s-505) is corrected. Moreover, the reduction of the codimension when M n is a complete submanifold in S p n+p (c) with parallel normalized mean curvature vector field is investigated.

Compact space‐like hypersurfaces in de Sitter space

We present some integral formulas for compact space‐like hypersurfaces in de Sitter space and some equivalent characterizations for totally umbilical compact space‐like hypersurfaces in de Sitter space in terms of mean curvature and higher‐order mean curvatures.

A characterization of totally umbilical hypersurfaces in de Sitter space

It is shown that a compact spacelike hypersurface contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if there exist nonnegative constants C1,C2,…,Cl−1, at least one Ci is positive, such that Hl=∑i=1l−1CiHi.This extends the previous result in [J. Geom. Phys. 39 (2001), Theorem 2].

Spacelike hypersurfaces in de Sitter space

In this paper, we give one intrinsic inequality for spacelike hypersurfaces in de Sitter space and a sufficient and necessary condition for such hypersurfaces to be totally geodesic.

Remarks on compact spacelike hypersurfaces in de Sitter space with constant higher order mean curvature

It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if one of the mean curvatures H l does not vanish and the ratio H k / H l is constant for some k, l, 1≤ l< k≤ n. This extends the previous result in [J. Geom. Phys. 31 (1999) 195, Theorem 7].

Curvature properties of compact spacelike hypersurfaces in de Sitter space

In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a lower bound for the square of its mean curvature. Our result will be a consequence of the maximum principle for the Laplacian operator. We also derive some other applications and consequences of our main result. In particular, we establish another sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for its scalar curvature, as well as in terms of the Ricci curvature and in terms of the higher order mean curvatures

On the Curvatures of Complete Spacelike Hypersurfaces in de Sitter Space

We establish some estimates for the higher-order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in de Sitter space which is contained in certain unbounded regions of the ambient space. Our results will be an application of a generalized maximum principle due to Omori.