Studying global properties of a closed non-regular hypersurface with a bijective Gaussian mapping using the level function

The structure of closed and non-closed regular hypersurfaces with an injective Gaussian mapping is thoroughly studied. When solving some problems of differential geometry, the desired hypersurface with a bijective Gaussian mapping may prove to be non-regular. In this paper, we study global properties of non-regular closed hypersurfaces in four-dimensional Euclidean space. A singular set of these surfaces is the sum total of closed oriented two-dimensional manifolds. The paper uses the Morse theory, the properties of the polar transformation with respect to the hypersphere, the Gauss–Bonnet theorem, the methods of the classical differential geometry of hypersurfaces and surfaces, the codimension of which is greater than 1. It is proved that under certain conditions the components of the singular set of the hypersurfaces under consideration can be only tors and spheres. In this case, the convex and saddle components of the regularity are tangent relative of each other along the sphere. It is found that a closed non-regular hypersurface with “cut off” borders and with bijective Gaussian mapping consists of two locally convex components homeomorphic to a three-dimensional ball, and one saddle component homeomorphic to the topological product of a two-dimensional sphere on the closed interval. The article provides examples of closed non-convex hypersurfaces with a bijective Gaussian mapping.