A new cylinder theorem (Q1271221)

Affine cylinders are characterized as the only hypersurfaces of type number not greater than one and extrinsically complete. As a consequence, an affine immersion \(f:M^n\to \mathbb{R}^{n+1}\) with a torsion-free, complete connection and with type number not greater than one must be a cylinder. These results extend other cylinder type theorems [see \textit{P. Hartman} and \textit{L. Nirenberg}, Am. J. Math. 81, 901-920 (1959; Zbl 0094.16303) and \textit{K. Nomizu} and \textit{U. Pinkall} [Math. Z. 195, 165-178 (1987; Zbl 0629.53012)].