An Elementary Proof of Hopf's Curvatura Integra Theorem

Abstract

We provide an elementary proof of the key aspect of Hopf's curvatura integra theorem. Namely, we show that the total curvature, i.e. the surface integral of the Gauss-Kronecker curvature B, is independent of the shape of the hypersurface. We accomplish this task by using the Calculus of Moving Surfaces to show that the rate of change of the total curvature vanishes under smooth changes in shape.