Rigorous proofs of the Hopf Umlaufsatz seem to always entail nightmarish details about covering maps and degree theory, or careful estimates as in the discussion here. The strange thing is that Hopf's own argument in his notes "Differential Geometry in the Large" (page 42 of the Springer edition) seems confused and impossibly brief. He defines an entity he calls $V(s_1,s_2)$ and first says that it's "the argument of the vector pointing from $c(s_1)$ to $c(s_2)$". A few lines later he then says "the vector field $V(s_1,s_2)$" is "continuous" on it's domain and furthermore that "the variation of the argument of a continuous vector field around a closed path is zero", an assertion which he uses to quickly deduce the theorem with no estimates or degree theory. A few questions arise:
- What did Hopf want V to be, really? A scalar field of angles, or a vector field?
- Hopf's assertion, that a continuous vector field's argument doesn't vary around a closed loop seems false as stated; if this were true the Umlaufsatz itself would be false! He may have meant a vector field with no singularities inside the loop. But where can I find a proof of this assertion?
- Is Hopf's reasoning sound, assuming we can make sense of 1 and 2 above? Can we really just say "the angle variation is path independent" and be done in a couple of lines?
Edit: Here's an attempt along the lines of Green's theorem as suggested by Ted Shifrin:
In Do Carmo (for instance) one can find the measure of the angle variation of a unit vector field $V = (a,b)$ along a curve $c(t)$ to be $\theta = \int_c ab' - ba' dt$. Assuming $c(t)$ is parameterized by arclength, we write this integral explicitly with directional derivatives along $c$: $\theta = \int_c [(\nabla b \cdot c'(t)) a - (\nabla a \cdot c'(t)) b] dt = \int_c (a \nabla b - b \nabla a)\cdot c'(t) dt$. The last integral is a line integral around $c(t)$ against the new vector field defined by $W=(P,Q)=(ab_x-ba_x, ab_y - ba_y)$. Applying greens theorem gives $\theta = 2\iint_R (b_y a_x - a_y b_x)dydx$. But this integrand vanishes since $a = \cos \theta(x,y)$ and $b = \sin \theta(x,y)$.