三维Minkowski空间R^2.1中混合型时空曲面的等温参数

对于三维Minkowski空间中的混合型时空曲面，证明了其在类空部分和类时部分上分别存在光滑的，并且连续到交界线上的等温参数．进一步，给出了混合型时空曲面存在在其类空部分和类时部分上是光滑的，而在交界线上是连续的等温参数的—个必要条件．     

R1^m+1中拟迷向类空超曲面

设f:M^m→R1^m+1是无脐点类空超曲面,则在Mm上可以定义四个基本的共形不变量:共形度量g,共形1-形式C,共形第二基本形式B,共形Blaschke张量A.如果存在光滑函数λ和常数μ,使得A+μB=Ag,则称M^m是拟迷向类空超曲面.本文不仅构造了拟迷向类空超曲面的例子,同时在相差R1^m+1的一个共形变换下,本文还完全分类了拟迷向类空超曲面.     

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.     

宇宙时空中具有给定平均曲率的类空超曲面

本文研究了宇宙时空中类空超曲面的推广平均曲率流,通过估计发展超曲面的高度函数、梯度函数和曲率函数等几何量,得到了一类极限类空超曲面,其平均曲率等于给定函数.     

三维Minkowski空间中曲面的广义Weierstrass公式

本文给出了三维Ｍｉｎｋｏｗｓｋｉ空间中一般类空曲面与类时曲面的广义Ｗｅｉｅｒ－ｓｔｒａｓｓ表示公式．     

Bcklund变换与类时极值平移曲面

首先通过选取适当的等温参数将三维Ｍｉｎｋｏｗｓｋｉ空间Ｒ２．１中的全脐点类时曲面与Ｌｉｏｕｖｉｌｅ方程相联系．其次，通过类时曲面上的类光曲线坐标将Ｒ２．１中的类时极值曲面与齐次波动方程相联系．进一步，利用Ｌｉｏｕｖｉｌｅ方程与齐次波动方程之间的Ｂａｃｋｌｕｎｄ变换，我们可以从三维Ｍｉｎｋｏｗｓｋｉ空间中一个全脐点的类时曲面得到该空间中一个类时极值平移曲面．     

第15卷B辑第4期（1994）目次和提要

第１５卷Ｂ辑第４期（１９９４）目次和提要三维Ｍｉｎｋｏｗｓｋｉ空间中混合型的完备极值曲面谷超豪本文用显示表达式构造了相当多的混合型的完备极值曲面同时，证明了具有给定数目“类时扩张”和给定数目的“喇叭口”的混合型完备极值曲面的存在性．拟线性方程组混合问...     

Minkowski空间中保高斯映射的共形形变

设Ｍｎ为Ｒｉｅｍａｎｎ流形，给定类空浸入：Ｍｎ→Ｒｎ，ｐ，如果存在另一个类空浸入：Ｍｎ→Ｒｎ，ｐ，使与在共形对应之下且对应点的地空间平行，则称类空子流形是可保高斯映射共形形变的．本文给出可保高斯映射共形形变的充要条件．对ｎ＝２，ｐ＝１的情形，如果上述形变是同向的，我们分类了曲面；如果是反向的，我们用主曲率满足的方程来描述．     

Backlund变换与类时极值平移曲面

首先通过选取适当的等温参数将三维Ｍｉｎｋｏｗｓｋｉ空间Ｒ＾２．１中的全脐点集时曲面与Ｌｉｏｕｖｉｌｌｅ方程相联系。其次，通过类时曲面上的类光曲线坐标将Ｒ＾２．１中的类时极值曲面与齐次波动方程相联系。进一步，利用Ｌｏｕｖｉｌｌｅ方程与齐次波动方程之间的Ｂａｃｋｌｕｎｄ变换，我们可以从三维Ｍｉｎｋｏｗｓｋｉ空间中一个全脐点的类时曲面得到该空间中一个类时极值平移曲面。     

共形空间中具有平行的共形第二基本形式的I型类时超曲面的分类

共形空间中具有平行的共形第二 基本形式的类空超曲面已经作了完全分类, 本文将继续类时情形的探讨并对此时的I型 类时超曲面分类.       

三维De Sitter与Anti-de Sitter空间中的等参曲面

A spacelike surface M in 3-dimensional de sitter space S13 or 3-dimensional anti-de Sitter space H13 is called isoparametric, if M has constant principal curvatures. A timelike surface is called isoparametric, if its minimal polynomial of the shape operator is constant. In this paper, we determine the spacelike isoparametric surfaces and the timelike isoparametric surfaces in S13 and H13.     

General Helices with Timelike Slope Axis in Minkowski 3-Space

In the present paper, we consider the general helices in Minkowski 3-space to have a constant timelike slope axis. As a result, we show that there exists no pseudo null helix with constant timelike slope axis or spacelike helix with timelike principal normal and constant timelike slope axis. Moreover, we obtain the parametric equations of spacelike helix with spacelike principal normal, timelike helix and null Cartan helix with timelike slope axis. We also give some related examples and their figures.     

Constant Mean Curvature Surfaces Foliated by Circles in Lorentz–Minkowski Space

We prove that a spacelike surface in L3 with nonzero constant mean curvature and foliated by pieces of circles in spacelike planes is a surface of revolution. When the planes containing the circles are timelike or null, examples of nonrotational constant mean curvature surfaces constructed by circles are presented. Finally, we prove that a nonzero constant mean curvature spacelike surface foliated by pieces of circles in parallel planes is a surface of revolution.     

Gradient-like and integrable vector fields on ℝ2

Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ² null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.Partially supported by a grant from the Weldon Springs endowment of the University of Missouri-Columbia     

On the Surface the Fermi–Walker Derivative in Minkowski 3-Space

In this paper Fermi–Walker derivative and Fermi–Walker parallelism and non-rotating frame concepts are given along the curve lying on the spacelike surface and the timelike surface in Minkowski 3-space. First, we consider a curve lying on the spacelike surface and investigate the Fermi–Walker derivative along the curve. The concepts which Fermi–Walker derivative and its theorems are analyzed along the curve lying on the spacelike surface in Minkowski 3-space. And then we consider a curve lying on the timelike surface and investigate the Fermi–Walker derivative along the curve.     

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 ${2\pi n, (n \in Z)}$ .     

Evolutes of fronts in the Minkowski plane

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.