Surfaces in Euclidean space with planar normal connection and zero normal torsion (Q1337834)

Let \(F^m\) be a submanifold of dimension \(m\) and a class \(C^3\) in a Euclidean space \(E^n\), \(n \geq m\). Suppose that \(x \in F^m\) and \(\overline {t} \in T_x F^m\) is a tangent vector to \(F^m\). Let \(\{x, e_i, e_\alpha\}\), where \(i = 1, \dots, m\) and \(\alpha = m + 1, \dots, n\), be an orthonormal moving frame of \(F^m\) at the point \(x\), the vectors \(e_i\) and \(e_\alpha\) forming orthonormal bases of the tangent subspace \(T_x F^m\) and the normal subspace \(N_x F^m\), respectively, and \(R^\alpha_{\beta ij}\) be the curvature tensor of the normal connection \(\nabla^\perp\) in \(N_x F^m\). Denote by \(\gamma = \gamma (x; \overline {t})\) the normal cross section in the direction of \(\overline {t}\), i.e. the line of intersection of \(F^m\) and the \((n - m + 1)\)-plane which passes through the point \(x\) in the direction of the vector \(\overline {t}\) and the normal plane \(N_x F^m\) at the point \(x\). The curvature \(k_N = k_N (x; \overline {t})\) and the torsion \(\kappa_N = \kappa_N(x; \overline {t})\) of the curve \(\gamma\) in \(E^{n - m + 1}\) at \(x\) are called, respectively, the normal curvature and the normal torsion of \(F^m\) at \(x\) in the direction \(\overline {t}\). A submanifold \(F^m\) is called reducible if it is a Riemannian product of submanifolds of smaller dimensions. The authors study tangentially nondegenerate submanifolds \(F^m\) with flat normal connection \(\nabla^\perp\) (for \(R^\alpha_{\beta ij} = 0\)) and with vanishing normal torsion \(\kappa_N(x; t) = 0\). They prove that 1) if a submanifold \(F^m\) under consideration is nonreducible, then \(F^m \subset E^{m + 1} \subset E^s\), \(s \geq m + l\), \(l = 1,\dots, m\), and 2) if a submanifold \(F^m\) is reducible, then \(F^m\) can be represented as a Riemannian product of spheres. The converse is also valid: if \(F^m\) is a nonreducible tangentially nondegenerate submanifold, \(F^m \subset E^{m + 1} \subset E^s\), \(s \geq m + l\), or \(F^m\) is reducible and can be represented as a Riemannian product of spheres, then \(F^m\) has a flat normal connection \(\nabla^\perp\) and its normal torsion \(\kappa_N (x; \overline {t}) = 0\), \(\overline {t} \in T_x F^m\).