臺灣博碩士論文加值系統

本文先介紹可微流型之定理及各種基本性質，也初略介紹具Riemann測度之可微流型。
本文主要內容在使用Morse引理及Morse定理，重新述明Reeb定理之證明。這裡我們使用Surgery引理，處理邊界問題。

In this thesis, we want to use Morse Theorem to prove Reeb''s Theorem. Before showing the proof of these theorems, we need to review some basic properties of a differentiable manifold M with a differentiable structure. In general, if we define some functions from M (or its subspace) to real value, the difference between manifold and coordinate space should be considered. Every point we choose must send to a coordinate subspace first. So defining a coordinate system is helpful to deal with any functions on manifold M.
The main result we review is to prove Reeb''s Theorem using Morse Lemma and Morse Theorem. Here we use a surgery lemma to prove disjoint union of two spaces, matched along their common boundary.
We also show how to construct a homotopy equivalence between manifold M and a n-sphere, for all dimension n is larger or equal to 1.

Contents
0 Introduction 1
1 Di erentiable manifold 3
2 Non-degenerate Critical Point And Hessian 7
3 Smooth Vector Field on Manifold 9
4 Homotopy 13
5 Morse Lemma 15
6 Main Theorem 18
7 State of Reeb''s Theorem(construct a homeomorphism between M and S^n) 23
8 Conclusion and Remarks 26

[1] William M. Boothby. An Introduction to Differentiable Manifolds and Rieman-
nian Geometry. ACADEMIC PRESS,INC, Orlando,Florida, second edition,
1986.
[2] J.Milnor. On manifolds homeomorphic to the 7-sphere. Annals of Mathematics,
64:399{405, 1956.
[3] James R. Munkres. Elements of Algebraic Topology. rst edition.