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Weak helix submanifolds of euclidean spaces

Weak helix submanifolds of euclidean spaces Let M⊂ℝ n be a submanifold of a euclidean space. A vector d∈ℝ n is called a helix direction of M if the angle between d and any tangent space T p M is constant. Let ℋ(M) be the set of helix directions of M. If the set ℋ(M) contains r linearly independent vectors we say that M is a weak r-helix. We say that M is a strong r-helix if ℋ(M) is a r-dimensional linear subspace of ℝ n . For curves and hypersurfaces both definitions agree. The object of this article is to show that these definitions are not equivalent. Namely, we construct (non strong) weak 2-helix surfaces of ℝ4. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (13)

Publisher
Springer Journals
Copyright
Copyright © 2009 by Mathematisches Seminar der Universität Hamburg and Springer
Subject
Mathematics; Geometry ; Topology; Number Theory; Combinatorics; Differential Geometry; Algebra
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-009-0017-0
Publisher site
See Article on Publisher Site

Abstract

Let M⊂ℝ n be a submanifold of a euclidean space. A vector d∈ℝ n is called a helix direction of M if the angle between d and any tangent space T p M is constant. Let ℋ(M) be the set of helix directions of M. If the set ℋ(M) contains r linearly independent vectors we say that M is a weak r-helix. We say that M is a strong r-helix if ℋ(M) is a r-dimensional linear subspace of ℝ n . For curves and hypersurfaces both definitions agree. The object of this article is to show that these definitions are not equivalent. Namely, we construct (non strong) weak 2-helix surfaces of ℝ4.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Feb 13, 2009

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