Weak helix submanifolds of Euclidean spaces
type: Item
instance of: scholarly article
title: Weak helix submanifolds of Euclidean spaces
zbMATH Open document ID: 1190.53006
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2009-10-26T00:00:00Z
review text: For a given submanifold \(M \subset \mathbb R^n\) of the Euclidean space a vector \(d \in \mathbb R\) is called a `helix direction vector' of \(M\) if the angle of \(d\) to all tangent spaces \(T_P M\) is constant. The manifold is called `weak \(r\)-helix' if there exists a set \(H(M)\) of \(r\) linearly independent helix direction vectors of \(M\). If this set \(H(M)\) can be extended to an \(r\)-dimensional subspace of \(\mathbb R^n\) the manifold \(M\) is called `strong \(r\)-helix'. The author gives an example that the two properties do not have to coincide for special \(2\)-manifolds \(M\) of \(\mathbb R^4\). He can show that all strong \(2\)-helices \(M \subset \mathbb R^4\) of dimension \(2\) can be generated by extensions of helices of \(\mathbb R^3\). The author also presents an example of a weak \(2\)-helix \(M\) which does not belong to the strong \(2\)-helices of above. This is done by discussing the solutions of special PDEs.
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 5620551
zbMATH Keywords: surfaces with constant slope
zbMATH Keywords: generalized helix submanifolds
author: Antonio J. Di Scala
reviewed by: Otto Röschel
MaRDI profile type: Publication
full work available at URL: s12188-009-0017-0
OpenAlex ID: W2022541598
cites work: Q4417327
cites work: Helix submanifolds of Euclidean spaces
cites work: Q4025262
cites work: Constant angle surfaces in \({\mathbb S}^2\times {\mathbb R}\)
cites work: Shadows and convexity of surfaces
cites work: Partial differential equations. 4th ed
DOI: 10.1007/S12188-009-0017-0
description: scientific article; zbMATH DE number 5620551