球面、柱面、锥面与非线性控制系统

我们首先从整体化的观点定义了一种建立在三维欧氏空间的二维曲面上的非线性控制系统,并给出了在曲面的局部坐标系下非线性系统状态方程的表达式,研究了非线性系统的平衡态与曲面的测地线之间的关系.刻划了球面、柱面、锥面的特殊曲面几何结构和奇异结构与非线性控制系统之间的内在联系.进一步,讨论了建立在球面、柱面和锥面上的非线性控制系统的局部和整体能控性与能观测性.     

平行映射下保测地线的曲面

利用古典微分几何方法,讨论了在平行映射下测地线不变的曲面若干局部性质;研究了该类曲面的测地坐标系,并最终得出满足该条件的曲面只有三类.     

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     

THE CLOSED CURVES ON POSITIVE-TYPE HYPERSURFACES AND THE PERIODS OF NONLINEAR OSCILLATIONS

本文在Ｒｎ＋１中定义了一类超曲面并讨论了其上闭测地线长度的下界以及Ｈａｍｉｌｔｏｎ系统的周期解的最小周期，这类超曲面比星形超曲面广泛，因而本文结论包含了文献〔１〕、〔２〕、〔３〕中的相应结论     

圆环面上测地线的稳定性

通过讨论圆环面上的测地线,研究在纤维缠绕时测地线的稳定性问题,按传统观点,测地线在曲面上是最稳定的,其意为一条弹性柔软的细线,在给定曲面上拉紧时,其形状应是测地线,且不会使曲线变形,上述提法仅在局部领域中成立,从整体角度分析,当拉紧弹性柔软细线时,将有两类不稳定测地线,第一类不稳定产生于过两点可以有多条测地线,另一类是缠绕的测地线位于曲面的凹侧,此时就会产生搭桥现象,这两类不稳定性在纤维缠绕中有着现实的意义,文章将对圆环面的测地线进行具体讨论。     

正定型超曲面上的闭曲线与非线性振荡的周期

本文在Ｒ＾ｎ＋１中定义了一类超曲面并讨论了其上闭测地线上度的下界以及Ｈａｍｉｌｔｏｎ系统的周期解的最小周期，这类超曲面比星形超曲面广泛，因而本文结论包括了文献「１」、「２」、「３」中的相应结论。     

凸曲面上的测地线的测地法线

<正> §1.引言 在正则曲面上的测地线具有这样的性质,从测地线上的每一点可引与这测地线垂直的测地线.但在任意的凸曲面上的一条测地线L可能具有这样的点,从它不能引与L垂直的测地线.(见[1]中第255—256页.)作者最近发现,L上具有这种特殊性质的点在某种意义上说是极少数.确切言之,L上不能引与L垂直的测地线的点组成一个在L上的零测度集合.本文目的就是给出这个结果的证明.     

具非负曲率完备非紧曲面的几何性质

本文证明了单连通完备非紧具非负曲率之曲面的任一测地线γ：[0,+∞)→M均趋于∞处这一几何性质,指出了一般的高维流形不具有此性质.本文还证明了单连通完备非紧具非负曲率的曲面的割迹与第一共轭轭迹是一致的;并且讨论了一般高维流形的共轭点与测地线的关系.     

仿射联络几种新的分类

在仿射联络流形中,历史上早先只将联络的曲率和测地线看作是最重要的几何对象,晚近才逐渐认识到最基本、最重要的乃是联络本身.然而,众所周知,在仿射联络流形上存在许许多多不同的联络,也就是说,每一流形上存在着许多不同前几何结构,于是,如何将不同的联络分成等价类是一现实而有趣的问题.尽管过去一些作者曾成功地利用测地线对联络进行过分类,但他们的分类方法的实质是只利用了联络的对称部分,这势必失去分类本身的一些几何意义. 在这篇简报中,首先将给出我们所证明的关于曲率的一些基本不变性定理,然后在此基础上给出联络分类的一些新方法,其特点是除具有按测地线分类之优点而外,尚使分类本身具有更加鲜明的几何意义.     

近似可展曲面的构造及应用

1引言经典微分几何中Gauss曲率为零的曲面称为可展曲面,它是一种特殊的直纹面．可展曲面有且只有三种,即锥面、柱面和切线面,它对于自由曲面造型具有重要的意义．例如,如果物体外壳是可展曲面,那么它可以没有形变地展开到平面上,从而可以用平板材料无形变地设计出来．这一性质对于造船业、航空业中的外形设计具有重要的意义．关于可展曲面的微分几何性质,可以在任何一本微分几何教材中找到,例如[3]．可展曲面可以说是微分几何中比较简单的一类曲面,但是在计算机辅助几何设计(CAGD)和计算机图形学中至今还不存在简单有效的设计方法．在[1,2,5]以及它们的参考文献中     

Geodesics for Dual Connections and Means on Symmetric Cones

Means on self-dual and homogeneous cones (, i.e., symmetric cones) are discussed from a viewpoint of differential geometry with affine connections. We first define means on symmetric cones in an axiomatic way following [8]. Next we consider dualistic differential geometry (, i.e., Riemannian metric and affine connections) [1] naturally introduced on symmetric cones. Elucidating the relation between the geodesics defined by each affine connection, and operator monotone functions that generate means, we show an important class of means are expressed by the (mid)points on geodesics. Related results on the means and submanifolds in a symmetric cone are also presented.     

Closed geodesics on piecewise smooth constant curvature surfaces of revolution

The paper develops a study of closed geodesics on piecewise smooth constant curvature surfaces of revolution initiated by I.V. Sypchenko and D. S. Timonina. The case of constant negative curvature is considered. Closed geodesics on a surface formed by a union of two Beltrami surfaces are studied. All closed geodesics without self-intersections are found and tested for stability in a certain finite-dimensional class of perturbations. Conjugate points are found partly.     

Pseudo-Riemannian geodesics and billiards

In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.     

Finsler generalization of the Tamm metric

We study manifolds of the Finsler type whose tangent (pseudo-)Riemannian spaces are invariant under the (pseudo)orthogonal group. We construct the Cartan connection and study geodesics, extremals, and also motions. We establish that if the metric tensor of the space is a homogeneous tensor of the zeroth order with respect to the coordinates of the tangent vector, then the metric of the tangent space is realized on a cone of revolution. We describe the structure of geodesics on the cone as trajectories of motion of a free particle in a central field.     

Classical motions of infinitesimal rotators on Mylar balloons

This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two-dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three-dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so-obtained results are also compared with those for the two-dimensional sphere embedded into the three-dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.     

Cones for the Moulton planes

We present a two-parameter family of affinely connected surfaces which admit the cylinder group as a collineation group of their geodesics. The Moulton Planes in the radial model of Betten, the circular cone, as well as the real affine plane, are part of this family. The Moulton Planes occur in this family in the same way as the real affine plane is contained in a family of cones with decreasing steepness.      

Lorentzian Clairaut submersions

We investigate a class of semi-Riemannian submersions satisfying a Lorentzian analogue of the classical Clairaut's relation for surfaces of revolution. We show that a Lorentzian submersion with one-dimensional fibers is Clairaut if and only if the fibers are totally umbilic with a gradient field as the normal curvature vector field. We also investigate the behavior of timelike and null geodesics in Lorentzian Clairaut submersions. In particular, every null geodesic of a Lorentzian Clairaut submersion with one-dimensional fibers projects to a pregeodesic in the base space with respect to a conformally related metric on the base space if and only if the integrability tensor of the submersion vanishes.     

动力点圆力偶作用于弹性全空间的解及其性质*

本文给出动力点圆力偶作用于弹性全空间原点的解,并讨论它的性质.将脉冲荷载沿以原点为心、α半径,在z=0平面上的圆周切向均匀分布,当a→0时经积分运算即得问题的解.当此动力点圆力偶的强度按sinωt变化时,在弹性全空间中以z轴为轴,原点为顶点的锥面在任何时刻均为零应力面.以这些锥面为边界的回转体受按sinωt而变化的扭矩作用的动力扭转问题的解可由本文的解得出.     

The type numbers of closed geodesics

This is a short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics.     

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on ?-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical ?-periodic surface with a boundary are recurrent.