三维空间型中闭曲面和测地球面的比较

本文建立关于三维空间型中的可定向闭曲面的某些公式.引入积分绝对平均曲率来描述可定向闭曲面的平均弯曲程度.在此意义上,可定向闭曲面可以与作为包含该闭曲面的凸的测地球的边界测地球面进行比较.这种比较能用于说明空间型自身的某些属性.     

在不可定向的流形网格曲面上计算测地距离的一般方法

不可定向的流形曲面不仅在拓扑学中占据重要的地位,在可视化和极小曲面等问题中也有很多的应用.从拓扑学的观点来看,二流形曲面的每个局部与圆盘同胚,该性质与曲面的全局可定向性无关.但在离散化的网格表示上,可定向的二流形曲面常用半边结构来表达,而不可定向的二流形曲面大多表达成若干多边形的集合,这给以可定向网格曲面为主要研究对象的数字几何处理带来很多不便.本文提出了把不可定向的二流形网格曲面上的测地距离问题转化到可定向曲面上进行处理的一般算法框架.该框架有望在不可定向的二流形网格曲面与传统数字几何处理方法之间搭起一座桥梁.为了展示该算法框架的普适性,本文将其应用于不可定向曲面上的三个重要场合,包括测地距离的求解、离散指数映射和最远点采样.     

有限群在曲面上的自由作用

本文借助于计算机编程给出了有限群在可定向闭曲面T~(nr 1)上反向自由作用个数的上界,同时决定了反向自由作用于小亏格闭曲面T~(nr 1)上的有限群以及p-1为素数时反向自由作用于闭曲面T_p上的有限群。     

定向闭曲面上具有伪轨跟踪性质的C~r流

本文给出了定向闭曲面上具有伪轨跟踪性质的 C~r(r≥2)流的特征.     

高斯映射与3维Minkowski空间中类时曲面的表示公式

本文首先证明了3维 Minkiwski 空间中任意可定向的类时曲面的高斯映射满足一组一阶偏微分方程.其次,对于任意给定平均曲率的可定向的类时曲面,利用高斯映射给出了一种表示公式.进一步,作为上述表示公式的完全可积性条件,得到一组关于高斯映射的二阶偏微分方程.特别,当平均曲率为常数时,这条件仅意味着高斯映射应是一个调和映射.     

飞越北极

本文对“飞机从北京出发、飞越北极直达底特律的所需时间 ,可比原航线节省多少时间”的问题进行讨论 ,并将航线选择归结为寻求曲面上的最短弧 .应用“曲面上最短弧为测地线”的事实进行了讨论 .模型 (一 )假设地球是球体 ,我们可通过单位向量的点乘与夹角的关系 ,加以解决 ;对于模型 (二 )设地球是旋转椭球体 ,我们利用微分几何学中测地线方程加以解决 ,并且把球面的纬度转化为旋转椭球面纬度 .对于 4组较特殊的点 ,纬度几乎相等或相近 ,或者两者之间的经度差过大时 ,用测地线计算比较困难 ,我们用椭圆弧 (长 )代替测地线长 ,结合数学软件 Mathematica的数值积分功能 ,可求得测地线长     

类圈图的亏格分布

一个图 G 的亏格分布是指序列{g_k}, g_k表示 G 嵌入亏格为 k 的闭的可定向曲面的数目. 该文给出了标准类圈图的亏格分布的递推公式, 并得到类圈图的嵌入多项式的计算公式.     

Morse指标和闭测地线的稳定性

令ind(c)为n+1维Riemann流形M上闭测地线c的Morse指标.我们证明:对于闭测地线c,如果它是定向的且n+ind(c)是奇数,或它是非定向的且n+ind(c)为偶数,则c不稳定.Poincaré的一个著名定理说Riemann面上的极小闭测地线是不稳定的,我们的结果是该结论的一个推广.     

正定型超曲面上双曲闭特征的Maslov型指标的迭代公式

本文给出了R~(2n)中规范正定型超曲面上双曲闭特征的Maslov型指标的迭代公式.结果包含了凸超曲面和星形超曲面上已有的相应结论。     

可定向的具非负曲率完备非紧黎曼流形

本文研究了具非负曲率完备非紧黎曼流形的一些几何性质,包括闭测地线,体积等.证明了核心的余维数为奇数的可定向具非负曲率完备非紧黎曼流形在其核心的任一法测地线均为射线的条件下可等距分裂为R×N,其中N为低一维的流形.     

On the Mean Absolute Geodesic Curvature of Closed Curves in 2-Dimensional Space Forms

In this paper,we establish some formulas on closed curves in 2-dimensional space forms.Mean absolute geodesic curvature is introduced to describe the average curving of a closed curve.Inthis sense,a closed curve could be compared with a geodesic circle that is the boundary of a convex geodesic circular disk containing the closed curve.The comparison can be used to show some properties of space forms only on themselves.     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Characterizations of umbilic hypersurfaces in warped product manifolds

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.     

Decomposition of closed orientable geometric surfaces into acute geodesic triangles

We prove that for every \(g\ge 2\), a differentiable closed orientable geometric surface of genus g may be decomposed into \(16g-16\) acute geodesic triangles. We also determine the number of acute geodesic triangles needed for the sphere and the torus.     

Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.     

Pseudo-Riemannian geodesic foliations by circles

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an S ¹-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley’s Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by ${S^1 \times \mathbb{R}}$ . It follows that every Lorentzian surface contains a nonclosed geodesic.     

A representation of one-dimensional attractors ofA-diffeomorphisms by hyperbolic homeomorphisms

We give a representation for the restrictions ofA-diffeomorphisms of closed orientable surfaces of genus > 1 from a homotopy class containing a pseudo-Anosov diffeomorphism to all one-dimensional attractors that do not contain special pairs of boundary periodic points. The representation is given by the restriction of a hyperbolic homeomorphism to an invariant zero-dimensional set formed by the intersection of two transversal geodesic laminations. It is shown how this result can be generalized to the representation of the restrictions ofA-diffeomorphisms defined on a closed surface of any genus to arbitrary one-dimensional attractors. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 76–87, July, 1997. Translated by V. E. Nazaikinskii     

On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central. Received: 9 November 1999; in final form: 19 September 2000 / Published online: 25 June 2001     

Compact surfaces as configuration spaces of mechanical linkages

There exists a homeomorphism between any compact orientable closed surface and the configuration space of an appropriate mechanical linkage defined by a weighted graph embedded in the Euclidean plane.