Multi-helicoidal Euclidean submanifolds of constant sectional curvature (Q1591752)

For an isometric immersion \(f:M^n \to\mathbb{R}^{2n-1}\) one says that \(f\) is a multi-helicoidal submanifold of cohomogeneity one if it is invariant under the action of an \((n-1)\)-parameter subgroup \(F\) of \(\text{Iso} (\mathbb{R}^{2n-1})\), i.e., there exists an \((n-1)\)-parameter subgroup \(T\) of \(\text{Iso}(M^n)\) such that \(G(\varphi) \cdot f=f\cdot T(\varphi)\) for any \(\varphi\in \mathbb{R}^{n-1}\). When such an \(M\) is of constant curvature, the following theorem holds: all these submanifolds are precisely the ones that correspond to solutions of the so-called sine-Gordon and elliptic sine-Gordon equations which are invariant by an \((n-1)\)-dimensional subgroup of translations of the symmetry group of these equations.