Hopf theorem

In the following we will assume that the term “smooth” implies just $C^1$ (once continuously differentiable). By smooth homotopy we will that the homotopy mapping is itself continuously differentiable

Theorem.

Suppose that $M$ is a connected, oriented (http://planetmath.org/Orientation2) smooth manifold without boundary of dimension $m$ and suppose $f\mathrm{,}g\mathrm{:}M\mathrm{\to }{S}^m$ are smooth mappings to the $m$-sphere. Then $f$ and $g$ are smoothly homotopic if and only if $f$ and $g$ have the same Brouwer degree.

When $M$ is not orientable, then we can always “flip” the orientation by following a closed loop on the manifold and one can then prove the followingresult.

Theorem.

Suppose that $M$ is not orientable, connected smooth manifold without boundaryof dimension $m$, and suppose $f\mathrm{,}g\mathrm{:}M\mathrm{\to }{S}^m$ are smooth mappings tothe $m$-sphere.Then $f$ and $g$ are smoothly homotopic if and only if $f$ and $g$ have thesame degree mod 2.