Lightlike surfaces of spacelike curves in de Sitter 3-space

The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces. Work partially supported by Grant-in-Aid for formation of COE. ‘Mathematics of Nonlinear Structure via Singularities’     

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

In this paper, we introduce three kinds of tubular surfaces associated to original center curves γ lying in spacelike surfaces in Lorentz‐Minkowski 3‐space. It is demonstrated that these tubular surfaces can occur some singularities and the types of these singularities can be characterised by several invariants, respectively. Some interesting relations between the contacts of original curve γ with osculating model surfaces, the contacts of γ with slices, and the singularities of three kinds of surfaces are further revealed. Several examples are presented to explain the theoretical results.     

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.     

Space curves of constant breadth in Minkowski 3-space

In this paper, we study spacelike and timelike curves of constant breadth in Minkowski 3-space. We show that in Minkowski 3-space spacelike and timelike curves of constant breadth are normal, helices, and spherical curves in some special cases. Furthermore, we give that the total torsion of a closed spacelike curve of constant breadth is zero while the total torsion of a simple closed timelike curve is equal to ${2\pi n, (n \in Z)}$ .     

Spacelike Localization of Long-Range Fields in a Model of Asymptotic Electrodynamics

A previously proposed algebra of asymptotic fields in quantum electrodynamics is formulated as a net of algebras localized in regions which in general have unbounded spacelike extension. Electromagnetic fields may be localized in ‘symmetrical spacelike cones’, but there are strong indications this is not possible in the present model for charged fields, which have tails extending in all space directions. Nevertheless, products of appropriately ‘dressed’ fermion fields (with compensating charges) yield bi-localized observables.     

Lorentz-Minkowski空间中给定主曲率函数的旋转超曲面

给出(n,1)型orentz-Minkowski空间中给定主曲率函数的旋转类空超曲面的位置向量场,通过计算超曲面的主曲率,证明了这类超曲面的存在性.     

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).     

Position vectors of spacelike general helices in Minkowski 3-space

In this paper, the position vectors of a spacelike general helix with respect to the standard frame in Minkowski space are studied in terms of the Frenet equations. First, a vector differential equation of third order is constructed to determine the position vectors of an arbitrary spacelike general helix. In terms of solution, we determine the parametric representation of the general helices from the intrinsic equations. Moreover, we give some examples to illustrate how to find the position vectors of spacelike general helices with a spacelike and timelike principal normal vector.     

n维Anti-de Sitter空间中类空超曲面的类时Gauss像的奇点

本文利用奇点理论研究了n维Anti-deSitter空间中的类空超曲面,介绍了类空超曲面的局部微分几何,定义了类时Anti-deSitterGauss像及Anti-deSitter高度函数,并进一步的利用Anti-deSitter高度函数族和流形间的切触理论研究了类时Anti-deSitterGauss像的几何意义及类空超曲面的通有性.最后研究了类空超曲面的AdS-Monge型.     

On maximal spacelike submanifolds

In this paper, we first prove that the nonexistence of the maximal spacelike submanifolds with positive sectional curvture in the pseudo-Riemannian space form of negative sectional curvature, then give a condition under which the submanifolds are totally geodesic.I would like to thank prof. K. Ogiue for his advice and to thank also the referee for valuable suggestions.     

On bijections of Lorentz manifolds,which leave the class of spacelike paths invariant

Suppose that (M, g) and (M′, g′) are Lorentz manifolds, and that f: M → M′ is a bijection, such that f and f^-1 preserve spacelike paths (f: M → M′ has this property, if for any spacelike path γ: J → M in (M ,g), the composition fγ: J → M′ is a spacelike path in (M′, g′)). Then f is a (manifold-) homeomorphism.This statement is the ‘spacelike’ version of an analogous ‘timelike’ theorem (Hawking, King and McCarthy [6] and Göbel [2] for strongly causal, and Malament [10] for general Lorentz manifolds).With this result it is possible to prove a conjecture of Göbel [3] which states that every bijection between time-orientable n-dimensional (n ? 3) Lorentz manifolds which preserves spacelike paths is a conformal C^∞-diffeomorphism.     

On the umbilicity of complete constant mean curvature spacelike hypersurfaces

In this paper, we show that a complete spacelike hypersurface immersed with constant mean curvature either in the de Sitter space or in the anti-de Sitter space must be totally umbilical, provided that its Gauss mapping has some suitable behavior. In particular, we use an extension of Hopf’s maximum principle due to Yau in order to give new positive answers to Goddard’s conjecture.     

On closed timelike and spacelike ruled surfaces

In this study, by using the concepts and results on spherical curves in dual Lorentzian space, we give the criterions for ruled surfaces with non‐lightlike ruling to be closed (periodic). Moreover, we obtain the necessary and sufficient conditions to guarantee that a timelike or a spacelike ruled surface is closed (periodic). Copyright © 2016 John Wiley & Sons, Ltd.     

Lightlike surfaces of Darboux-like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3-space

In this paper, by introducing a new frame on spacelike curves lying in lightcone 3-space, we investigate the geometric properties of the lightlike surface of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix generated by spacelike curves in lightcone 3-space. As an extension of our previous work and an application of the singularity theory, the singularities of the lightlike surfaces of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix are classified, several new invariants of spacelike curves are discovered to be useful for characterizing these singularities, meanwhile, it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3-space and two-dimensional lightcone whose vertices are at the singularities of lightlike surfaces. One concrete example is provided to illustrate our results.     

Integral formulas for compact spacelike hypersurfaces in de sitter space and their applications to Goddard's conjecture

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS ₁ ⁿ⁺¹ (1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result. Li Haizhong is supported by NNSFC No.19701017 and Basic Science Research Foundation of Tsinghua University and Chen Weihua is supported by NNSFC No. 19571005     

The universal torsion-free image of a group

We show that the universal torsion free homomorphic image of any group given by a sufficiently ‘small’ presentation is locally indicable, and give an application to a conjecture of Levin about equations over torsion free groups. Supported by SERC Visiting Fellowship GR/F 97355.     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

On spacelike submanifolds with parallel mean curvature in an indefinite space form

In this work we use a Simon??s type inequality for a suitable tensor and apply it to obtain sharp estimates for the supremum of the scalar curvature for complete spacelike submanifold M ⁿ with parallel mean curvature vector in an indefinite space form ${N^{n+p}_p(c)}$ .     

On the rigidity of complete spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime

We study the problem of uniqueness of complete noncompact spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime, which is supposed to satisfy the so-called null convergence condition. By extending a technique of S.T. Yau and imposing a suitable restriction on the norm of the gradient of the height function of the hypersurface, we obtain rigidity theorems in such ambient spacetimes. Furthermore, we also establish nonparametric results concerning entire vertical graphs.     

Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?^″?max{H?^′,0} along M, then Mn is either maximal or a spacelike slice Mt₀={t₀}×F, for some t₀∈I.