comparison theorems

以曲率來刻劃完備黎曼流形的拓撲性質是近代微分幾何發展的路徑之一。

首先是用曲率做一些幾何性質的比較，即對兩黎曼流形的曲率關係比較相應的微分幾何與拓撲性質，然後研究標準空間(例如常曲率流形)的性質，這方面的結果稱為比較定理。  

In Riemannian geometry，the comparison results in terms of sectional curvature of Rauch，Toponogov，Morse-Schoenberg and others are basic tools used to prove resucherlts such as sphere theorem，the Bonnet-Myers theorem，and the maximal diameter theorem of Toponogov。

1. Harry E. Rauch 1925-1979 A Contribution to Diffreential Geometry in the Large
   Lectures on the Ricci flow  Peter Topping [Using Rauch comparison theorem to get an estimation of two metric]
2. Victor Andreevich Toponogov 1903-2004 [Toponogov's theorem and Applications]
   [International conference on Geometry in the Large] [A relative Toponogov comparison theorem]

More recently，comparison theorems in terms of the Ricci curvature such as Bishop-Gromov volume comparison theorem have played an important role leading to such results as the Chern maximal diameter theorem。

Hermann Karcher :Riemannian comparison constructions

1. Scalar curvature
2. Ricci curvature          Bishop-Gromov relative volume comparison theorem
3. Sectional curvature   Rauch  Toponogov   Morse-Schoenberg theorem

1. Hadamard-Cartan theorem
2. Bonnet-Meyers theorem
3. Morse-Schonberg comparison theorem
4. Rauch comparison theorem
5. Bishop-Gromov relative volume comparison theorem