Unifying theorem for the turning number and the Gauss-Bonnet theorem

Question

The statement that the turning number of a planar curve is a multiple of $2\pi$ and the Gauss-Bonnet theorem are similar in that they both state that the integral of a curvature measure depends on the topology but not the shape of the manifold. Is there a theorem that unifies the two statements that yields the former for $n=1$ and the latter for $n=2$? What does it say for $n=3$?

Answer 1

The unifying concept is the degree of the Gauss map of a closed hypersurface. In general, if you have a smooth mapping $f\colon X\to Y$ between compact oriented (Riemannian) $n$-manifolds, the integral of its Jacobian gives the degree of the map times the volume of $Y$. More precisely, if $\omega$ is the volume form of $Y$, then $$\int_X f^*\omega = \text{deg}(f)\int_Y\omega.$$ For a detailed discussion, see $\S$9 of Chapter 4 of Guillemin & Pollack's Differential Topology.