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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 149-164 (2001)
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Multi-helicoidal Euclidean Submanifolds of Constant Sectional Curvature

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Jaime Ripoll; Ruy Tojeiro

Instituto de Matematica, Universidade Federal do R. G. do Sul (UFRGS), Av. Bento Goncalves 950, 91540-000 Porto Alegre-RS, Brasil, e-mail: ripoll@mat.ufrgs.br; Departamento de Matematica, Universidade Federal de Sao Carlos (UFSCar), Rod. Washington Luiz km 235 CP 676, 13565-905 Sao Carlos-SP, Brasil, e-mail: tojeiro@dm.ufscar.br

**Abstract:** \font\msbm=msbm10 \def\R{\hbox{\msbm R}} We classify $n$-dimensional multi-helicoidal submanifolds of nonzero constant sectional curvature and cohomogeneity one in the Euclidean space $\R^{2n-1}$, that is, $n$-dimensional submanifolds of nonzero constant sectional curvature in $\R^{2n-1}$ that are invariant under the action of an $(n-1)$-parameter subgroup of isometries of $\R^{2n-1}$ with no pure translations. This is accomplished by first giving a complete description of all these subgroups and then deriving a multidimensional version of a lemma due to Bour. We also prove that such submanifolds are precisely the ones that correspond to solutions of the generalized sine-Gordon and elliptic sinh-Gordon equations that are invariant by an $(n-1)$-dimensional subgroup of translations of the symmetry group of these equations.

**Keywords:** multi-helicoidal submanifolds, constant sectional curvature, generalized sine-Gordon and elliptic sinh-Gordon equations

**Classification (MSC2000):** 53B25, 53C42, 35Q53

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