黎曼几何

微分幾何中,黎曼幾何(英語:Riemannian geometry)研究具有黎曼度量的光滑流形,即流形切空間上二次形式的選擇。它特別關注于角度、弧線長度及體積。把每个微小部分加起來而得出整體的數量。

19世紀,波恩哈德·黎曼把這個概念加以推广。[1]

任意平滑流形容許黎曼度量及這個額外結構幫助解決微分拓扑問題。它成為伪黎曼流形複雜結構的入門。其中大部分都是廣義相對論的四維研究对象。

黎曼几何古典理論

伯恩哈德·黎曼

一般理論

  1. 高斯-博内定理:紧致二維黎曼流形高斯曲率的积分等於,這裡的記作M欧拉示性数
  2. 纳什嵌入定理:(两个)被稱為黎曼幾何的基礎理論。他們表明每個黎曼流形可以是嵌入歐幾里得空間Rn

理論

所有给出的定理中,都将用空间的局部行为(通常用曲率假设表述)来推出空间的整体结构的一些信息,包括流形的拓扑类型和"足够大"距离的点间的关系。

  1. 1/4-受限 球定理:若M是完备n-维黎曼流形,其截面曲率严格限制于1和4之间,则M同胚于n-球。
  1. Cheeger's有限定理:给定常数CD,只有有限个(微分同胚的流形算作一个)紧n-维黎曼流形,其截面曲率并且直径
  1. Gromov的几乎平坦流形:存在一个使得如果一个n-维黎曼流形其度量的截面曲率且直径,则其有限覆盖微分同胚于一个零流形.

正曲率

  1. 灵魂定理:若M是一个不紧的完备正曲率n-维黎曼流形,则它微分同胚于Rn.
  2. Gromov的贝蒂数定理:有一个常数C=C(n)使得若M是一个由正截面曲率的紧连通n-维黎曼流形,则它的贝蒂数之和不超过C.
  1. Myers定理:若一个紧黎曼流形有正Ricci曲率则它的基本群有限。
  1. 分裂定理:若一个完备的n-维黎曼流形有非负Ricci曲率和一条直线(在任何区间上的距离都极小的测地线)则它等度同胚于一条实直线和一个有非负Ricci曲率的完备(n-1)-维黎曼流形的直积。
  1. Bishop's不等式:半径为r的球在一个有正Ricci曲率的完备n-维黎曼流形中的体积不超过欧几里得空间中同样半径的球的体积。
  1. Gromov's紧致性定理:所有正Ricci曲率且直径不超过D的黎曼流形在Gromov-Hausdorff度量下是仿紧的。
  1. n-维环不存在有正数量曲率的度量。
  1. 若一个紧n-维黎曼流形的单射半径,则数量曲率的平均值不超过nn-1)。

負曲率

  1. 任何有非正截面曲率的单连通黎曼流形的两点有唯一的测地线连接。
  1. M是一个有负截面曲率的完备黎曼流形,则基本群的任何可交换子群同构于整数群Z
  1. 设V*是一-rank2的紧致不可约局部对称空间,设V是一截面曲率的紧致黎曼流形,若,且,则等距。
  1. 任何有负里奇曲率的紧黎曼流形有一个离散的等距同胚群
  2. 任何光滑流形可以加入有负里奇曲率的黎曼度量。

參見

參考文献

  • Marcel Berger, Riemannian Geometry During the Second Half of the Twentieth Century, (2000) University Lecture Series vol. 17, American Mathematical Society, Rhode Island, ISBN 0-8218-2052-4. (Provides a historical review and survey, including hundreds of references.)
  • Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 (Provides a formal introduction, written at the grad-student level.)
  • Peter Peterson, Riemannian Geometry, (1998) Springer-Verlag, Berlin ISBN 0-387-98212-4. (Provides an introduction, presented at an undergrad level.)


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