Morse theory to Reeb's theorem and its generalization

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Title:	Morse theory to Reeb's theorem and its generalization
Other Titles:	Reeb定理上的Morse理論及其推廣
Authors:	李容瑄;Lee, Jung-Hsuan
Contributors:	淡江大學數學學系碩士班余成義;Yu, Cherng-Yih
Keywords:	Morse引理;Morse定理;Reeb定理;Morse lemma;Morse Theorem;Reeb's Therem
Date:	2013
Issue Date:	2014-01-23 13:49:32 (UTC+8)
Abstract:	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.
Appears in Collections:	[Graduate Institute & Department of Mathematics] Thesis

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