在一般的歐氏空間裡頭，我們對於兩個同維度的子流形可以藉由計算第一基本型，第二基本型，法連絡來判斷兩個子流形是否只相差一個剛體運動，稱為歐氏空間中子流形的基本定理。我們先使用Élie Cartan的活動標架法、李群理論證明了歐氏空間中子流形的基本定理。賴馨華博士\cite{ref3}，用此方法得到海森堡群中的Legendrian子流形的基本定理。這篇論文將此定理推廣到更一般的情況。

In the Euclidean space, we can calculate the first fundamental form, the second fundamental form and normal connection of submanifold. With these three forms, we can know when two submanifold are the same with respect to rigid motion.It is called the fundamental theorem of submanifold in the Euclidean space. We first prove these theorem by the Élie Cartan's method of moving frames and the theory of Lie groups. Sin-Hua Lai \cite{ref3} also obtains the fundamental theorem of Legendrian submanifold in the Hesenberg groups by the same way. In this paper, we generalize the theorem into higher codimension.

Acknowledgements
摘要----i
Abstract----iii
1 Introduction----1
2 Calculus on Lie Group----5
3 Local Geometry of Submanifold in R^n-----11
4 Introduction of Heisenberg group and its rigid motion----17
5 the fundamental theorem of the Legendrain sunmanifold----21

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Hung-Lin Chiu,bSin-Hua Lai: The fundamental theorem for hypersurfaces in Heisenberg groups.Calc. Var.54(2015)

Sin-Hua Lai: Perelman's Entropy Formula on Pseudohermitian Manifolds and Fundamental Theorem on Heisenberg Groups.