Spacelike surfaces in anti de Sitter four-space from a contact viewpoint

We define the notions of (S _t ¹ × S _s ²)-nullcone Legendrian Gauss maps and S ₊²-nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S ₊²-nullcone Lagrangian Gauss maps, we define the notion of S ₊²-nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.     

Regular space-like hypersurfaces in S<Stack><Subscript>1</Subscript><Superscript><Emphasis Type="Italic">m</Emphasis>+1</Superscript></Stack> with parallel para-Blaschke tensors

In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space S ^m+1 ₁ with parallel para-Blaschke tensors.     

Submanifolds in de sitter space-time satisfying ΔH=λH

In [3] the author initiated the study of submanifolds whose mean curvature vectorH is an eigenvector of the Laplacian Δ and proved that such submanifolds are either biharmonic or of 1-type or of null 2-type. The classification of surfaces with ΔH=λH in a Euclidean 3-space was done by the author in 1988. Moreover, in [4] the author classified such submanifolds in hyperbolic spaces. In this article we study this problem for space-like submanifolds of the Minkowski space-timeE ₁ ^m when the submanifolds lie in a de Sitter space-time. As a result, we characterize and classify such submanifolds in de Sitter space-times.     

Singularities of Anti de Sitter torus Gauss maps

We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated to a timelike surface which are called Anti de Sitter nullcone Gauss image and Anti de Sitter torus Gauss map. We also define a family of functions named Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.     

Algebraic zero mean curvature hypersurfaces in de Sitter and anti de Sitter spaces

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter N-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature hypersurfaces in the de Sitter space.     

Complete Spacelike Submanifolds in de Sitter Space with Parallel Mean Curvature Vector Satisfying H2 = 4(n − 1)/n2

Recently, Montiel [7] proved that an n-dimensional (n 3) complete spacelike hypersurface in de Sitter space S^{1 1 +1}(1) with constant mean curvature H satisfying H² = 4 (n – 1)/n² which is not connected at infinity must be, up to rigidity motion, a certain hyperbolic cylinder. In this paper, we prove that Montiel's result still holds for higher codimensional spacelike submanifolds in de Sitter space Sⁿ _p ^+p(1).     

Cohomogeneity one de Sitter space S n 1

In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.     

de Sitter空间S1m+1中具有平行Blaschke张量的正则类空超曲面

本文引入两个以de Sitter空间为模型的非齐性坐标来覆盖共形空间Q₁^m+1.利用球面S^m+1中超曲面的Möbius几何的方法,本文研究了Q₁^m+1中正则类空超曲面的共形几何.作为其结果,本文对所有具有平行Blaschke张量的正则类空超曲面进行了完全分类.     

Gluing Construction of Initial Data with Kerr–de Sitter Ends

We construct initial data sets which satisfy the vacuum constraint equations of General Relativity with positive cosmological constant. More precisely, we deform initial data with ends asymptotic to Schwarzschild–de Sitter to obtain non-trivial initial data with exactly Kerr–de Sitter ends. The method is inspired from Corvino’s gluing method. We obtain here a extension of a previous result for the time-symmetric case by Chru?ciel and Pollack (Ann H Poincaré 9(4):639–654, 2008). We also deal with the case of asymptotically Kerr–de Sitter initial data.     

On chen surfaces in a minkowski space time

A-submanifo1ds of a pseudo-Euclidean space E _s ^m+1 are considered. A characterization for them is given. A theorem onA-submanifolds contained in a de Sitter space-time S _s ⁿ or an anti-de Sitter space-time H _s–1 ⁿ is proved. A number of non-trivial examples ofA-surfaces in a Minkowski space-time E ₁ ⁴ are studied. Some classification theorems are proved forA-surfaces contained in S ₁ ³ or H³.     

On exact solutions for quantum particles with spin <Emphasis Type="Italic">S</Emphasis> = 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space–time model are examined in detail. Firstly, for scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient R_ej{R_{\epsilon j}} on the background of the de Sitter space–time model is analyzed. It is shown that the determination of R_ej{R_{\epsilon j}} requires an additional constrain on quantum numbers er/ (^h/_2p) c >> j{\epsilon \rho / \hbar c \gg j}, where ρ is a curvature radius. When taken into account of this condition, the R_ej{R_{\epsilon j}} vanishes identically. It is claimed that the calculation of the reflection coefficient R_ej{R_{\epsilon j}} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space–time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.     

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H¹(c₁)×Sn−1(c₂) or Hn−1(c₁)×S¹(c₂) in the de Sitter space .     

A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold

In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin 1/2 fermion in the de Sitter manifold, a brane with topology \({M=\mathrm {S0}(4,1)/\mathrm {S0}(3,1)}\) living in the bulk spacetime \({{\mathbb{R}^{4,1}}=(\mathring{M}=\mathbb{R}^5,\boldsymbol{\mathring{g}})}\) and equipped with a metric field \({\boldsymbol{g}:\boldsymbol{=}-\boldsymbol{i}^{\ast} \boldsymbol{\mathring{g}}}\) with \({\boldsymbol{i}:M\rightarrow \mathring{M}}\) being the inclusion map. To obtain the analog of Dirac equation in Minkowski spacetime in the structure \({\mathring{M}}\) we appropriately factorize the two Casimir invariants C ₁ and C ₂ of the Lie algebra of the de Sitter group using the constraint given in the linearization of C ₂ as input to linearize C ₁. In this way we obtain an equation that we called DHESS1, which in previous studies by other authors was simply postulated. Next we derive a wave equation (called DHESS2) for a free spin 1/2 fermion in the de Sitter manifold using a heuristic argument which is an obvious generalization of a heuristic argument (described in detail in Appendix D) permitting a derivation of the Dirac equation in Minkowski spacetime and which shows that such famous equation express nothing more than the fact that the momentum of a free particle is a constant vector field over timelike integral curves of a given velocity field. It is a remarkable fact that DHESS1 and DHESS2 coincide. One of the main ingredients in our paper is the use of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this concept and its relation with covariant Dirac spinor fields usually used by physicists.     

Particle Decays and Stability on the de Sitter Universe

We study particle decay in de Sitter space–time as given by first-order perturbation theory in a Lagrangian interacting quantum field theory. We study in detail the adiabatic limit of the perturbative amplitude and compute the “phase space” coefficient exactly in the case of two equal particles produced in the disintegration. We show that for fields with masses above a critical mass m _c there is no such thing as particle stability, so that decays forbidden in flat space–time do occur here. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with mass lower than critical have a completely different behavior: the masses of their decay products must obey quantification rules, and their lifetime is zero.     

Matter-coupled de Sitter supergravity

The de Sitter supergravity describes the interaction of supergravity with general chiral and vector multiplets and also one nilpotent chiral multiplet. The extra universal positive term in the potential, generated by the nilpotent multiplet and corresponding to the anti-D3 brane in string theory, is responsible for the de Sitter vacuum stability in these supergravity models. In the flat-space limit, these supergravity models include the Volkov–Akulov model with a nonlinearly realized supersymmetry. We generalize the rules for constructing the pure de Sitter supergravity action to the case of models containing other matter multiplets. We describe a method for deriving the closed-form general supergravity action with a given potential K, superpotential W, and vectormatrix fAB interacting with a nilpotent chiral multiplet. It has the potential V = e^K(|F²|+|DW|²-3|W|²), where F is the auxiliary field of the nilpotent multiplet and is necessarily nonzero. The de Sitter vacuums are present under the simple condition that |F²|-3|W|² > 0. We present an explicit form of the complete action in the unitary gauge.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Integral formulas for compact spaceliken-submanifolds in de Sitter spaces applications to the parallel mean curvature vector case

We introduce a new method to study compact spaceliken-submanifolds in de Sitter spacesS _q ^n+q by means of certain integral formulas which have a very clear geometric meaning. As a first application of them we obtain a Bernstein type result for complete maximal submanifolds inS _q ^n+q . As for surfaces, we also get a uniqueness result for compact spacelike surfaces inS _q ^2+q with parallel mean curvature vector field. Partially supported by a DGICYT Grant No. PB91-0705-C02-02 Partially supported by a DGICYT Grant No. PB91-0731     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(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 Sp^(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 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.     

广义de Sitter空间中的类时超曲面

利用奇点理论研究广义de Sitter空间中的类时超曲面.介绍类时超曲面的局部微分几何,定义了广义de Sitter高斯像及广义de Sitter高度函数,研究广义deSitter高度函数族的性质及广义de Sitter高斯像的几何意义,介绍了一种证明高度函数为Morse族的新方法.最后研究了类时超曲面的通有性质.