The Gauss map for surfaces in 4-space
This paper is concerned with the following question: Let M be a connected oriented surface and let G be the Grassmann manifold of oriented 2-planes in \({\mathbb{R}}^ 4\). Suppose we are given an immersion g: \(M\to G\); does there exist an immersion X: \(M\to {\mathbb{R}}^ 4\) with g as its Gauss map? Is even g locally a Gauss map? Also to what extent does g uniquely determine X? The results obtained are of the following sort: There is a non-existence theorem based on algebraic criteria. The paper also deals with maps g which are called hyperbolic or elliptic. In the hyperbolic case, the main theorems deal with Cauchy problems where initial data is given along curves in the surface M. In the elliptic case the existence of an immersion X is proven when M is a closed disk. Also, in this case, for arbitrary M the Gauss map g uniquely determines the immersion X when X is known on an arbitrary arc in M. The terms hyperbolic and elliptic derive from the type of the system of p.d.e.'s one must deal with. Hence a primary concern is to relate the geometry of g with the type of the system to be considered in solving the stated problems. It is shown that the type of the system depends on the sign of a first order invariant of g which is denoted by \(\delta\). It turns out that \(\delta =(K_ g-1)/4,\) where \(K_ g(m)\) is the sectional curvature in G of the tangent plane to g at m.
scientific article; zbMATH DE number 3851919