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- Publisher
- Springer Journals
- Copyright
- Copyright © 2011 by Mathematisches Seminar der Universität Hamburg and Springer
- Subject
- Mathematics; Algebra; Geometry ; Topology; Number Theory; Combinatorics; Differential Geometry
- ISSN
- 0025-5858
- eISSN
- 1865-8784
- DOI
- 10.1007/s12188-011-0052-5
- Publisher site
- See Article on Publisher Site
Abstract
An immersed surface M in N n ×ℝ is a helix if its tangent planes make constant angle with ∂ t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N n ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.
Journal
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Mar 18, 2011