Cylinder theorems for convex hypersurfaces
Let \(F^n\) be a convex hypersurface of class \(C^k\), \(k \geq 3\), in a Euclidean space \(\mathbb{R}^{n+1}\), and let \(k_1(p), k_2(p), \dots ,k_n(p)\) be the principal curvatures of \(F^n\) at the point \(p \in F^n\), satisfying the inequalities \(k_1(p) \leq k_2 (p) \leq \dots \leq k_n(p)\). The author proves two theorems: 1) If \(\sup_{p \in F^n} k_i (p) < \inf_{p\in F^n} k_{i+1} (p)\), then \(F^n\) is a cylinder with an \(i\)-dimensional generator. 2) If there exists a two-dimensional direction at each point of \(F^n\) along which the sectional curvature of \(F^n\) vanishes, then \(F^n\) is a cylinder having a generator of at least dimension one.
scientific article; zbMATH DE number 917102