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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 invariants which combines the second fundamental form of the hypersurface and the covariant derivative of the vector field. We show how these invariants can be used as obstructions to the existence of codimension one foliations with prescribed geometric properties.

Degree of the Gauss Map and Curvature Integrals for Closed Hypersurfaces