On surfaces with no conjugate points
On surfaces with no conjugate points
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在没有共轭点的曲面上
DOI：
10.4310/jdg/1214440852
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发表时间：
1987
影响因子：
2.5
通讯作者：
K. Burns
中科院分区：
文献类型：
--
作者：
W. Ballmann;M. Brin;K. Burns
如果完整的黎曼流形 M 中的任意两点由唯一的测地线连接，则该流形 M 没有共轭点。若M的截面曲率非正，则M无共轭点；即使对于紧凑的表面，反之亦然。一个自然的问题是非正截面曲率流形的性质在多大程度上对于没有共轭点的流形有效。例如，根据 Gauss-Bonnet 定理，圆环 T 上的任何非正曲率度量都是平坦的。 1943年，E. Hopf [4]证明了定理。任何没有共轭点的 T 上的度量都是平坦的。解释我们论文的目的并引入必要的符号的最好方法是给出霍普夫论点的概述。他考虑了黎卡蒂方程
A complete Riemannian manifold M has no conjugate points if any two points in its universal cover are joined by a unique geodesic. If the sectional curvature of M is nonpositive, then M has no conjugate points; the converse is not true even for compact surfaces. A natural question is to what extent properties of manifolds of nonpositive sectional curvature are valid for manifolds with no conjugate points. For example, by the Gauss-Bonnet theorem, any metric of nonpositive curvature on the torus T is flat. In 1943, E. Hopf [4] proved Theorem. Any metric onT with no conjugate points is flat. The best way to explain the purpose of our paper and to introduce the necessary notations is to give an outline of Hopf s argument. He considers the Riccati equation