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研究生： 李雅婷
Lee, Ya-Ting
論文名稱： 三維流形上的觸結構
Contact Structures on 3-Manifolds
指導教授： 江孟蓉
Chiang, River
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 34
中文關鍵詞： 觸流形 、overtwisted 結構 、tight 結構 、characteristic foliation 、分集 、convex surface 理論 、Thurston-Bennequin 不等式
外文關鍵詞： contact manifold, overtwisted, tight contact structures, characteristic foliation, dividing set, convex surface theory, Thurston-Bennequin inequality
相關次數： 點閱：88  下載：9 
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    • 流形上的觸結構透過Yasha Eliashberg 可將其二分為overtwisted及tight. 本文主要討論此結構的存在性以及如何判別一個三維觸流形是overtwisted還是tight，其技巧提及Giroux's criterion和Thurston-Bennequin inequality。

    Contact structures on 3-dimensional manifolds are divided into two categories, overtwisted structures and tight structures, due to Eliashberg. In this thesis, we discuss their existence and the tools to distinguish them including Giroux's criterion and Thurston-Bennequin inequality.

    1 Introduction 1 2 Contact Manifolds 3 2.1 Contact manifolds......3 2.2 Reeb vector field......5 2.3 Darboux's theorem......5 2.4 Gray stability theorem......7 3 Contact Structures on 3-Manifolds 11 3.1 Existence of contact structures......11 3.2 Tight and overtwisted structures......14 3.3 Existence and classification of overtwisted structures......16 3.4 examples of tight structures......20 4 Giroux's Criterion and Thurston-Bennequin inequality 21 4.1 Characteristic foliation......21 4.2 Convex surfaces and dividing sets......25 4.3 Giroux's Criterion......30 4.4 Thurston-Bennequin inequality......31 References......33

    [1] Y. Eliashberg. Classification of overtwisted contact
    structures on 3-manifolds. Invent. Math 98,(623-637),1989.
    [2] Y. Eliashberg. Filling by holomorphic discs and its
    applications, Geometry of Low-Dimensional Manifolds, volume 2. Cambridge University Press, 1990.
    [3] J. Etnyre and K. Honda. On the nonexistence of tight
    contact structures. Annals of Math 153,(749-766),2001.
    [4] H. Geiges. An introduction to contact topology, volume 1. Cambridge University Press, 2008.
    [5] E. Giroux. Convexie en topologie de contact. Comment. Math. Helv. 66,(637-677),1991.
    [6] M. Gromov. Pseudo holomorphic curves in symplectic
    manifolds. Invent. Math 82,(307-347),1985.
    [7] Ko Honda. Note for math 599: Contact geometry. (web)http://www-bcf.usc.edu/khonda/math599/notes.pdf,(33).
    [8] Ko Honda. Note for math 599: Contact geometry.(web)http://www-bcf.usc.edu/khonda/math599/notes.pdf,(19-22).
    [9] Ko Honda. On the classification of tight contact
    structures i. Geom. topol. 4,(309-368),2000.
    [10] Steven Sivek. Note for math 273. (web)https://web.math.princeton.edu/ssivek/math273.php,2012.

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