Equivalence of Examples of Sacksteder and Bourgain

Abstract

Finding examples of tangentially degenerate submanifolds (submanifolds with degenerate Gauss mappings) in an Euclidean space $R^4$ that are noncylindrical and without singularities is an important problem of differential geometry. The first example of such a hypersurface was constructed by Sacksteder in 1960. In 1995 Wu published an example of a noncylindrical tangentially degenerate algebraic hypersurface in $R^4$ whose Gauss mapping is of rank 2 and which is also without singularities. This example was constructed (but not published) by Bourgain. In this paper, the authors analyze Bourgain's example, prove that, as was the case for the Sacksteder hypersurface, singular points of the Bourgain hypersurface are located in the hyperplane at infinity of the space $R^4$, and these two hypersurfaces are locally equivalent.