The Gauss map and second fundamental form of surfaces in \(\mathbb{R}^3\)
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 \(S\).
scientific article; zbMATH DE number 1505836