Hartman-Nirenberg柱面定理的仿射类比
An affine analogue of the Hartman-Nirenberg cylinder theorem
摘要：
设X是复仿射空间A^N（C）中n < N维光滑完备连通子流形，r是其高斯映射γ的秩，γ（x）= T_x（X）.证明了：若2 \leq r \leq n - 1，N-n \geq 2，且在X的第二基本型束中，有两种形式定义一个正则束，且它们的特征值不同，则子流形X是一个在秩为r，维数为r的光滑完备连通子流形Y上具有（n-r）维平面生成元的圆柱.这个结果是哈特曼-尼伦伯格柱面定理的仿射类比，证明了X \subset R^{n+1}和r = 1。对n个\geq 4和r = n - 1，存在完备连通子流形X \subset A^N（C）不是柱面.
英文摘要：
Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map γ, γ(x) = T_x (X). The authors prove that if 2 \leq r \leq n - 1, N - n \geq 2, and in the pencil of the second fundamental forms of X, there are two forms defining a regular pencil all eigenvalues of which are distinct, then the submanifold X is a cylinder with (n-r)-dimensional plane generators erected over a smooth, complete, connected submanifold Y of rank r and dimension r. This result is an affine analogue of the Hartman-Nirenberg cylinder theorem proved for X \subset R^{n+1} and r = 1. For n \geq 4 and r = n - 1, there exist complete connected submanifolds X \subset A^N (C) that are not cylinders.
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