An affine analogue of the Hartman-Nirenberg cylinder theorem

Details An affine analogue of the Hartman-Nirenberg cylinder theorem An affine analogue of the Hartman-Nirenberg cylinder theorem

2000-09-24

arxiv.org/abs/math/0009211v3

Math. Ann., 322 (2002) no. 3 573-582

Mathematics

Differential Geometry

AMS-LaTeX, 9 pages; revised version (the main theorem refined, and its conditions are given in terms of the second fundamental

Scientific paper

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 \gamma, \gamma (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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