Degree of the Gauss map and curvature integrals for closed hypersurfaces

TL;DR

This paper introduces a new approach to relate the Gauss map degree and curvature integrals on closed hypersurfaces using a unit vector field, revealing topological invariants that obstruct certain foliations.

Contribution

It defines a map from hypersurfaces to spheres based on a vector field, linking curvature and topology in a novel way.

Findings

Identifies topological invariants combining second fundamental form and vector fields.

Shows these invariants serve as obstructions to codimension one foliations.

Provides a new perspective on curvature integrals and topology of hypersurfaces.

Abstract

Given a unit vector field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second fundamental form of the hypersurface and the vector field itself. We show how these invariants can be used as obstructions to certain codimension one foliations.