The Gauss map and second fundamental form of surfaces in \(\mathbb{R}^3\)
type: Item
instance of: scholarly article
title: The Gauss map and second fundamental form of surfaces in \(\mathbb{R}^3\)
zbMATH Open document ID: 0965.53007
DOI: 10.1023/A:1005240426098
author: N. Delaunay
published in: Geometriae Dedicata
publication date: 2001-07-15T00:00:00Z
review text: The properties of the Gauss map on a submanifold in \(\mathbb{R}^n\) and the extent to which the Gauss map determines the immersion of the submanifold have been of great interest. The existence and uniqueness of an immersion of a surface or a hypersurface into \(\mathbb{R}^n\) with a given metric (or conformal structure) and a given Gauss map has been studied by several authors. In this paper, the authors' object is to study the properties of the Gauss map of a surface immersed in \(\mathbb{R}^3\), particularly those related to the geometry of the immersion and the conformal structure determined by its second fundamental form. More precisely, the main problem considered by the authors is the existence and uniqueness of an immersion \( x\colon S\to \mathbb{R}^3\) of a surface \(S\) with prescribed conformal structure that yields a given Gauss map and for which the second fundamental form is a conformal metric on \(S\).
reviewed by: Christos Baikoussis
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 1505836
zbMATH Keywords: Gauss map
zbMATH Keywords: second fundamental form
zbMATH Keywords: Weierstrass representation
zbMATH Keywords: harmonic map
zbMATH Keywords: Christoffel's problem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1505836