Closed orientable surfaces and fold Gauss maps

This paper describes how the elliptic and hyperbolic regions of a surface are related to stable Gauss maps on closed orientable surfaces immersed in three-dimensional space. We will show that for certain connected, closed, orientable surfaces containing a finite number of embedded circles that delineate two distinct types of regions, if all regions of one type are homeomorphic to a cylinder, then there exists an immersion \(f: M \rightarrow \mathbb {R}^3\) $f:M\to {\mathbb{R}}^3$ for which the Gauss map is a fold Gauss map.