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A Characterization of Isoparametric Surfaces in Space Forms via Minimal Surfaces

A Characterization of Isoparametric Surfaces in Space Forms via Minimal Surfaces We give a characterization of isoparametric surfaces in three dimensional space forms with the help of minimal surfaces. Let us consider a surface in a three dimensional space form with the following property: through each point of the surface pass three curves such that the ruled orthogonal surface determined by each of them is minimal. We prove that necessarily the initial surface is isoparametric. It is also shown, that the curves are necessarily geodesics. On other hand, using the classification of isoparametric surfaces it is possible to prove that they have the above property along every geodesic. So, we have a characterization. As a preliminary result we prove that a surface with two geodesics, through every point, which are helices of the ambient, has parallel shape operator. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

A Characterization of Isoparametric Surfaces in Space Forms via Minimal Surfaces

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References (17)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-017-0061-9
Publisher site
See Article on Publisher Site

Abstract

We give a characterization of isoparametric surfaces in three dimensional space forms with the help of minimal surfaces. Let us consider a surface in a three dimensional space form with the following property: through each point of the surface pass three curves such that the ruled orthogonal surface determined by each of them is minimal. We prove that necessarily the initial surface is isoparametric. It is also shown, that the curves are necessarily geodesics. On other hand, using the classification of isoparametric surfaces it is possible to prove that they have the above property along every geodesic. So, we have a characterization. As a preliminary result we prove that a surface with two geodesics, through every point, which are helices of the ambient, has parallel shape operator.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Nov 28, 2017

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