平面环面的焦点刚度
Focal Rigidity of Flat Tori
摘要：
给定一个闭黎曼流形（M，g），它的切丛TM有一个划分n_i，称为焦点分解.集合n_i与（M，g）的测地线的聚焦密切相关，即与M中连接点p和q的长度相同的测地弧正好是i的情况密切相关。本文研究了闭黎曼流形焦点分解的拓扑结构及其与流形度量结构的关系。我们的主要结果是，平坦的n-环面是焦点刚性的，在这个意义上说，如果两个平坦的环面是焦点等价的，那么环面是等距的重新缩放。
英文摘要：
Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that the flat n-tori are focally rigid, in the sense that if two flat tori are focally equivalent, then the tori are isometric up to rescaling.
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