漢米爾頓瑞曲流與可微分球定理

Abstract

在黎曼幾何中，一個令人關切的問題是如何分類有正截曲率 的流形。在 1951 年，H.E. Rauch 介紹了 curvature pinch 這個 想法，並提出如果一個簡單連通的流形他的截曲率都界在 (1, 4] 之間，那是否同胚於一個 n 維球。這問題在 1960 年，被 M.Berger 和 W. Klingenberg 利用比較技巧解決了。而之後留 下了另一個問題是這種流形是否會微分同胚於一個 n 維球， 這猜想又稱為可微分球定理。本文主要是在整理 S. Brendle 和 R. Schoen 在 2009 年利用 Hailton's Ricci flow 解決可微分球定 理的工作。

A central problem in Riemannian geometry concerns the classification of manifolds of positive sectional curvature. In 1951, H.E. Rauch introduced the notion of curvature pinching for Riemannian manifolds and posed the question of whether a simply connected manifold M^n whose sectional curvatures all lie in the interval (1, 4] is necessarily home- omorphic to the sphere S^n. This was proven by using comparison techniques due to M. Berger and W. Klingenberg around 1960. However, this theorem leaves open the ques- tion of whether M is diffeomorphic to Sn. This conjecture is known as the Differentiable Sphere Theorem. The goal of this survey is to present this work via Hamilton's Ricci flow due to S. Brendle and R. Schoen around 2009.