局部对称流形上的数量曲率

本文讨论了无共轭点测地线上的Ｊａｃｏｂｉ声，证明了具非负数量曲率的局部对称的无共轭点流形及具非负数量曲率的具极点的局部对称的流形之数量曲率只能是零。部分解决了Ｅ．Ｈｏｐｆ猜想。     

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.     

4步次洛仑兹流形上的测地线（英文）

设(R~3,D,g)是一个次洛仑兹流形,其中D是一个4步括号生成分布,g是定义在D上的次洛仑兹度量,它是一个惯性指数为1的度量.本文首先计算了类时将来方向曲线的可达集,其次,给出了正规测地线的完全描述并计算了连接原点与任意点的正规测地线的条数.最后,证明了类时测地线不存在共轭点.     

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.     

完备Riemann流形中的测地子流形的焦点

利用沿测地线的N-Jacobi场和指标形式,得到了具非负曲率完备Riemann流形中的测地子流形为无焦点的充要条件。     

爱因斯坦空间的测地对应和黎曼空间V(K)

<正> §1.引言 设对于黎曼空间V_n,有另一黎曼空间V_n,使得V_n的测地线对应于V_n的测地线,则称V_n与V_n是相互测地对应的.大家知道,与常曲率空间测地对应的黎曼空间也是常曲率的,即常曲率空间之间能相互测地对应.但对于非常曲率的黎曼空间,则不一定存在这种对应.近年来对各种循环黎曼空间的测地对应的讨论,就说明了这个事实. 爱因斯坦空间是比常曲率空间更广泛的重要黎曼空间,这种空间之间是否存在测地对应呢?本文的第一部分就是讨论这个问题.我们给出了能相互测地对应的各种爱因斯     

一类直纹面的研究

本文给出了过任意空间Ｃｋ（ｋ≥３）类光滑曲线的直纹面是可展曲面的充要条件．同时得到了该空间曲线为相应直纹面的曲率线,测地线和渐近曲线的充要条件     

一类直纹面的研究

本给出了过任意空间C^k(k≥3)类光滑曲线的直纹面是可展曲面的充要条件．同时得到了该空间曲线为相应直纹面的曲率线,测地线和渐近曲线的充要条件。     

Hermite流形上距离函数Levi形式上界估计及其某些应用

<正> §1.引言设 M 为 m 维光滑有走向的 Riemann 流形,O,P 为 M 上两点,C:[0,ρ]→M 为连接 O,P 的极小正则测地线,C(0)=0,C(ρ)=P.假定 P 不是 C 的关于 C(0)的共轭点.则(?)ξ∈T_P,成立 Synge 公式(见陆启铿[1]或 S.Kobayashi[2]):     

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.     

Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points

In this article, we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the manifold is bounded asymptote, then the geodesic flow is entropy-expansive. Moreover, for the compact oriented surfaces without conjugate points, we prove that the geodesic flows are entropy-expansive. We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.     

Convexity of spheres in a manifold without conjugate points

For a non-compact, complete and simply connected manifoldM without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres inM is a radial function, then the geodesic spheres are convex. We also show that ifM is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma.     

Conjugate Points on a Geodesic with Random Curvature

We study conjugate points on a renewable geodesic on which the curvature is a random process. We construct the upper bound for the mean distance between neighboring conjugate points.     

Jacobi Fields along a Geodesic with Random Curvature

A notion of a renewable geodesic on which the curvature is a random process is introduced. It is shown that the modulus of the Jacobi field along such a geodesic grows exponentially. At the same time, the existence with probability 1 of infinitely many conjugate points is demonstrated.     

Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.Partially supported by NSF grant MCS77-18723(02).     

Conjugate points and closed geodesic arcs on convex surfaces

This paper discusses conjugate points on the geodesics of convex surfaces. It establishes their relationship with the cut locus. It shows the possibility of having many geodesics with conjugate points at very large distances from each other. It also shows that on many surfaces there are arbitrarily many closed geodesic arcs originating and ending at a common point. To achieve these goals, Baire category methods are employed.     

Gromov-hyperbolicity and transitivity of geodesic flows in n-dimensional Finsler manifolds

We show that the geodesic flow of a compact Finsler manifold without conjugate points is transitive provided that the universal covering satisfies the uniform Finsler visibility condition. This result is a nontrivial extension of a well known theorem due to Eberlein for Riemannian manifolds. For doing so, we introduce suitable Finsler versions of the concepts of Gromov's δ-hyperbolicity and Eberlein's visibility, and study their consequences.     

Boundaries of non-compact harmonic manifolds

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson–Sullivan measure and this measure coincides with the harmonic measure. As an application of these results we prove that the geodesic flow on a non-flat finite volume harmonic manifold without conjugate points is topologically transitive.