Multi-helicoidal Euclidean submanifolds of constant sectional curvature
type: Item
instance of: scholarly article
title: Multi-helicoidal Euclidean submanifolds of constant sectional curvature
zbMATH Open document ID: 0980.53074
author: Ruy Tojeiro
author: Jaime Ripoll
published in: Beiträge zur Algebra und Geometrie
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/224912
full work available at URL: 9.html
review text: 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.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 1549907
zbMATH Keywords: multi-helicoidal submanifolds
zbMATH Keywords: constant sectional curvature
zbMATH Keywords: generalized sine-Gordon and elliptic sinh-Gordon equations
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1549907