An affine analogue of the Hartman-Nirenberg cylinder theorem

Authors

Maks A. Akivis, Jerusalem College of Technology
Vladislav V. Goldberg, New Jersey Institute of Technology

Document Type

Article

Publication Date

12-1-2002

Abstract

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) = 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 ⊂ A^N (ℂ) that are not cylinders.

Identifier

0036013585 (Scopus)

Publication Title

Mathematische Annalen

External Full Text Location

https://doi.org/10.1007/s002080200006

ISSN

00255831

First Page

573

Last Page

582

Issue

3

Volume

322

Recommended Citation

Akivis, Maks A. and Goldberg, Vladislav V., "An affine analogue of the Hartman-Nirenberg cylinder theorem" (2002). Faculty Publications. 14538.
https://digitalcommons.njit.edu/fac_pubs/14538