An affine analogue of the Hartman-Nirenberg cylinder theorem

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Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space Aⁿ (C), and r is the rank of its Gauss map γ, γ(x) = Tₓ (X). The authors prove that if 2 ≤ r ≤ n - 1, N - n ≥ 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 ⊂ Rⁿ⁺¹ and r = 1. For n ≥ 4 and r = n - 1, there exist complete connected submanifolds X ⊂ Aⁿ (C) that are not cylinders.