在這篇論文中,我們考慮具有Ricci曲率下界,直徑上界,以及近乎最大
體積的所有封閉黎曼流形.利用Ricci曲率的比較定理與扭積空間(warped
product spaces)逼近的性質,我們可以證明這類流形的微分同胚型態完全
被它們的基本群之同構型態所決定.這樣的結果推廣了著名的 Mostow 剛
性定理以及 Cheeger 有限性定理. 另外,我們也考慮有限群有效作用(
effective action)於緊緻負曲率黎曼流形的問題.一個有趣的結果是,至
少有一個軌道(orbit)具有僅與維度有關的直徑.這個結果同時也證明了一
個關於基本群之有限擴張的剛性定理.

In this dissertation, we consider a class of closed
Riemannian manifolds with lower Ricci curvature bound, upper
diameter bound and almost maximal volume and show that the
isomorphism types of fundamental groups characterize the
diffeomorphism types of manifolds in such a class. The main
tools are the Ricci curvature comparison arguments and the
warped product spaces approximations. In particular, our result
can be viewed as generalizationsof the well-known Mostow''s
rigidity theorem and Cheeger''s finitenesstheorem. We also
consider finite groups acting effectively on a compact
negativelycurved manifold and show that there is at least one
orbit with big diameter.It can also be regarded as a rigidity
result for the finite exetensions offundamental groups of
compact negatively curved manifolds.

Cover
Abstrat
Contents
Chapter 1
Chapter 2
2.1 Closed hyperbolic manifolds
2.2 Ricci curvature comparison theorems
2.3 Grimiv-Hausdorff topology
2.4 Alexandrov spaces of curvature bounded below
Chapter 3 Rigity and Finitenes of Fundamental Groups
3.1 Fintie group actions on compact negatively curved manifolds
3.2 Finiteness of fundamentl groups
Chapter 4
4.1 Definition and distance functions
4.2 Convergence of warped product spaces
Chapter 5
5.1 Integral estimates on warping functions
5.2 Annuli with almost maximal volumes
5.3 Approximation under integral estimates
Chapter 6
Bibliography