International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 4, Pages 537-543
doi:10.1155/IJMMS.2005.537
A note on surfaces with prescribed oriented Euclidean Gauss map
1Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquâs de São Vicente, 225-Gávea, Rio de Janeiro, Brazil
2Centre de Mathématiques de Jussieu, Université Paris 7-Denis Diderot, 2 Place Jussieu, Paris Cedex 05 F-75251, France
Received 6 July 2004; Revised 19 November 2004
Copyright © 2005 Ricardo Sa Earp and Eric Toubiana. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We present another proof of a theorem due to Hoffman and Osserman
in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss map G and formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.