Topological invariants for closed hypersurfaces

TL;DR

This paper introduces topological invariants for closed hypersurfaces in translational manifolds, linking geometry and topology to identify obstructions to specific foliations.

Contribution

It defines new invariants based on a perturbed Gauss map that depend on the geometry of the hypersurface and ambient space.

Findings

Invariants depend on the geometry of the hypersurface and ambient space.

Obstructions to certain codimension one foliations are identified.

Provides a method to analyze hypersurface immersions using topological invariants.

Abstract

We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion that depends on the geometry of the manifold and the ambient space. We use these quantities to find obstructions to the existence of certain codimension one foliations.