On submanifolds with zero normal torsion in Euclidean space (Q1920765)

Let \(F^n\) be an \(n\)-dimensional \((n \geq 2)\) submanifold of class \(C^3\) in a Euclidean space \(E^{n+p}\) \((p \geq 2)\). Let \(k_N(x,t)\) and \(\kappa_N(x,t)\) be the normal curvature and the normal torsion of \(F^n\) at the point \(x \in E^n\) in the direction \(t \in T_x F^n\) where \(T_x F^n\) is a tangent plane of \(F^n\) at \(x\). The author proves the following results: Let \(R\) be a set of the surfaces \(F^n\) in \(E^{n+p}\) with zero normal torsion \(\kappa_N(x,t)\equiv 0\) at every point \(x \in F^n\) and in every direction \(t\in T_xF^n\) and \(k_N(x,t) \neq 0\). Then \(F^n\) lies in \(E^{n+1}\subset E^{n+p}\) or \(F^n\) is of the form \(S^{k_1}\times \dots \times S^{k_m}\subset E^{n+m}\), \(k_1+k_2+\dots +k_m= n\), where \(S^{k_1},\dots,S^{k_m}\) are spheres in the spaces \(E^{k_1+1},\dots, E^{k_m+1}\), respectively; or \(F^n\) is of the form \(E^k\times S^{k_1} \times \dots \times S^{k_\ell} \subset E^{n+1} \subseteq E^{n+p}\), \(k_1+\dots+k_\ell= n-k\), \(1 \leq k\leq n-2\), \(2\leq \ell \leq \min\{n-k,p\}\).