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研究生： 方齊佑
Fang, Chi-You
論文名稱： 二維海森堡群上擬埃爾米特子流行的基本定理
The Fundamental Theorem of Pseudohermitian Submanifolds on 2 -Dimensional Heisenberg Groups
指導教授： 邱鴻麟
Chiu, Hung-Lin
口試委員: 劉筱凡
Liu, Hsiao-Fan
陳瑞堂
Chen, Jui-Tang
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 14
中文關鍵詞： 海森堡群 、擬埃爾米特子流形 、基本定理
外文關鍵詞： Heisenber group, pseudohermitian submanifolds, Fundamental theorem
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    • 在這篇文章中主要的結果是去證明在二維海森堡群上垂直擬埃爾米特子流
    行的不變量可以由其擬埃爾米特結構確定，所以有著相同埃爾米特結構的
    垂直擬埃爾米特子流行之間最多差一個海森堡群上的鋼體運動。第二個結
    果是擬埃爾米特流行嵌入二維海森堡群的條件，當然這種嵌入是唯一的。

    The main result of this thesis is to prove that the completely invariant of the vertical pseudohermitian submanifolds in Heisenberg group H2 can be determine by
    the pseudohermitian structure. So two vertical pseudohermitian manifolds which
    have the same pseudohermitian structure at most differ by a Heisenberg rigid motion. And the second result is the condition that a pseudohermitian manifold can
    be embedded in the H2, of course that the embedding is unique in the sense that
    they are the same after a Heisenberg rigid motion.

    1 Introduction 3 2 Pseudohermitian manifold 4 3 The Heisenberg group H2 5 4 Cartan’s method 6 4.1 The group representation of PSH(n) . . . . . . . . . . . . . . . . . . 6 5 Pseudohermitian submanifolds of H2 7 5.1 The Darboux frame for pseudohermitian submanifolds. . . . . . . . . 8 5.2 The fundamental vector field ν . . . . . . . . . . . . . . . . . . . . . 8 5.3 Intergrability condition . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.4 The structure equations . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Main theorem 10

    [1] Ivey, T.A., Landsberg, J.M.: Cartan for beginners: differential geometry via moving
    frames and exterior differential systems. Graduate Studies in Mathematics, vol. 61.
    American Mathematical Society,
    Providence, RI (2003)
    [2] Lee, J.M.: The Fefferman metric and pseudohermitian invariants. Trans. Am. Math.
    Soc. 411–429 (1986)
    [3] Hung-Lin,Chiu:The Fundamental and rigidity theorems for pseudohermitian submanifolds(2018)
    [4] S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Differential
    Geom. 13 (1978), 25-41.

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