I'm trying to prove the following theorem on page 270 of Do Carmo's "Differential Geometry of Curves and Surfaces":
\begin{equation} \sum_{i=0}^k(\phi_i(t_{i+1})-\phi_i(t_i)) + \sum_{i=0}^k \theta_i = \pm 2\pi, \end{equation}
where $\alpha$ is defined in the following way:
Let Let $\alpha: [0,l] \rightarrow S$ be a continuous map from the closed interval $[0,l]$ into the regular surface $S$. $\alpha$ is called a simple, closed, piecewise regular, parametrized curve if:
- $\alpha (0) = \alpha (l)$,
- $t_1 \neq t_2, t_1,t_2 \in [0,l) \Rightarrow \alpha(t_1)\neq \alpha(t2)$
- There exists a subdivision $0=t_0<t_1<...<t_k<t_{k+1}=l$,such that $\alpha$ is differentiable and regular in each $(t_i,t_{i+1}), i =0,...,k$.
And the $\phi_i$ are the angles between $x_u$ and $\alpha'(t)$ and the $\theta_i$ are external angles on the vertices of the curve.
Another important thing to note is that we have a conformal (even isothermal) parametrization $x:U\subset R^2 \rightarrow S$ and the entirety of $\alpha$ lies in one coordinate neighbourhood. Additionally, in this case, U is homeomorphic to a disk.
I was given the hint to do this proof in two steps:
- Prove the Theorem for a curve without vertices
- Prove the general statement by approximating a curve with vertices by one without.
The first step I have already done by pulling the entire thing back onto U via $x^{-1}(\alpha)$ and using the fact that conformal maps preserve angles and then using Theorem 2 on page 402 of the same book.
How would I go about showing the second part though?
Also, for the first part I was given a hint to use Stokes' Theorem and then somehow taking the limit $l \rightarrow 0$, but I have no idea how I would go about applying it in that case, so I would also be grateful for any hints in that regard in case you have any ideas.