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截面曲率.
在黎曼几何中,截面曲率是描述黎曼流形的曲率的一种方式。截面曲率
依赖于p点的切空间裡的一个二维平面
。它就定义为该截面,考慮在 p 点以平面
作为切平面的曲面
,這曲面是收集流形中某包含
的鄰域內從 p 点出發的測地線且這測地線在
點的切向量屬於截面
(換句話說就是
其中
是
里包含原點的鄰域),而截面曲率
就是曲面
在
點的高斯曲率。形式上,截面曲率是流形上的2维格拉斯曼纤维丛的光滑实值函数。
截面曲率完全决定了曲率张量,是非常有用的几何概念。
设 M 为黎曼流形,σ 为 M 上 p 点处切空间中的二维平面,u 和 v 为 σ 中两个线性无关的向量。
则关于 σ 的截面曲率定义为
![{\displaystyle K(\sigma )={\langle R(u,v)v,u\rangle \over |u|^{2}|v|^{2}-\langle u,v\rangle ^{2))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8afc3a71011a1735a0af6f12bf1409c5cf80fce4)
其中 R 是 M 的黎曼曲率张量。
常截面曲率的黎曼流形是最简单的类型。它们称为空间形式。通过缩放度量,它们有三种情况
三类几何的模型流形分别是双曲空间,欧几里得空间和单位球面。它们是对于这些给定的截面曲率唯一可能的完备单连通黎曼流形,所有其它常曲率流形是它们在某个等距映射群下的商。
- 完备黎曼空间有非负的截面曲率,当且仅当函数
对于所有点p是一个1-凹函数。
- 一个完备单连通黎曼流形有非正截面曲率,当且仅当函数
是1-凸函数。
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