VIGRE2010 REU Paper by Grant Rotskoff
Definition 3.2. A simple region $R$ of a surface $S$ is a region such that $R$ is homeomorphic to the disk.
Definition 3.3. Given a parametrization of a surface, $\mathbf{x}: U \rightarrow S$, we have the following quantities
$$
E=\left\langle x_u, x_u\right\rangle, F=\left\langle x_u, x_v\right\rangle, G=\left\langle x_v, x_v\right\rangle .
$$
These are the coefficients of the first fundamental form of a surface.
Definition 3.4. Given a parametrization of a surface,
$$
\mathrm{x}: U \to S
$$
we call that parametrization orthogonal if
$$
F=0 .
$$
Definition 3.5. The geodesic curvature $k_g$ of a curve is a measure of the amount of deviance of the curve from the shortest are between two points on a surface.
Definition 3.6. The Gaussian curvature $\kappa$ of a surface is an intrinsic measure of the curvature of a surface at a point. It is calculated by considering the maximal and minimal curvatures on the surface at a point. Formally, these values are multiplied to give $\kappa$.
Theorem 3.7 (Gauss-Bonnet, Local). Let $R \subset \mathrm{x}(U)$ be a simple region of $S$ with orthogonal parametrization, and choose $\alpha: I \rightarrow S$ such that $\alpha(I)=\partial R$. Assume that $\alpha$ is positively oriented and parametrized piecewise by arc-length $s_i$. Let $\{\theta_i\}_{i=0}^k$ be the external angles of $\alpha$ at the vertices $\{\alpha(s_i)\}_{i=0}^k$, then
$$
\sum_{i=0}^k \int_{s_i}^{s_{i+1}} k_{g(s)} d s+\iint_R \kappa d \sigma+\sum_{i=0}^k \theta_i=2 \pi .
$$
Proof. We first let $u=u(s)$ and $v=v(s)$ be the expression of $\alpha$ in the parametrization $\mathrm{x}$. We recall that
$$
k_{g(s)}=\frac{1}{2 \sqrt{E G}}\left(G_u \frac{d v}{d s}-E_v \frac{d u}{d s}\right)+\frac{d \varphi}{d s}
$$
where we denote the differentiable function that measures the positive angle from $x$ to $\alpha^{\prime}(s)$ in $\left[s_i, s_{i+1}\right]$ as $\varphi\left(s_i\right)$. We now integrate the above expression, adding up the values for each $\left[s_i, s_{i+1}\right]$ :
$$
\sum_{i=0}^k \int_{s_i}^{s_{i+1}} k_{g(s)}=\sum_{i=0}^k \int_{s_i}^{s_{i+1}} \frac{1}{2 \sqrt{E G}}\left(G_u \frac{d v}{d s}-E_v \frac{d u}{d s}\right) d s+\sum_{i=0}^k \int_{s_i}^{s_{i+1}} \frac{d \varphi_i}{d s} d s,
$$
now, using the Gauss-Green theorem in the $u v$-plane on the right hand side of the above equation, we obtain the expression:
$$
\iint_{\mathbf{x}^{-1}(R)} \frac{E_v}{2 \sqrt{E G}_v}+\frac{G_u}{2 \sqrt{E G}_u} d u d v+\sum_{i=0}^k \int_{s_i}^{s_{i+1}} \frac{d \varphi_i}{d s} d s
$$
We note that by the Gauss Formula,
$$
-\iint_{\mathbf{x}^{-1}(R)} \kappa \sqrt{E G} d u d v=-\iint_R \kappa d \sigma
$$
also, recalling the Theorem of Turning Tangents, we know that,
$$
\sum_{i=0}^k \int_{s_i}^{s_{i+1}} \frac{d \varphi_i}{d s} d s=\sum_{i=0}^k \varphi_i(s_i+1)-\varphi_i(s_i)= \pm 2 \pi-\sum_{i=0}^k \theta_i,
$$
which we get because the theorem does not account for the discontinuities along the curve at the theta values. As we have assumed a positive orientation, we have,
$$
\sum_{i=0}^k \int_{s_i}^{s_{i+1}} k_{g(s)} d s+\iint_R \kappa d \sigma+\sum_{i=0}^k \theta_i=2 \pi,
$$
we note that we can obtain the opposite sign by assuming the opposite orientation, and we thus we have proven the local case of the Gauss-Bonnet Theorem.
[…Global…]
Definition 5.7. The index $I$ of $v$ at the singular point $p$ is defined as follows. Let $\mathbf{x}$ be an orthogonal parametrization such that $\mathbf{x}(0,0)=p$ and the orientation is compatible with that of the surface $S$. Let $\alpha:[0, l]\to S$ be a closed simple regular parametrized curve such that it is the boundary of a simple region $R \subset S$, where the only singular point in $R$ is $p$. Now, we have a function $\varphi(t)$ with $t \in[0, l]$, such that it measures the angle from $\mathbf{x}_{\mathbf{u}}$ to the restriction of $v$ to $\alpha$, then,
$$
2 \pi I=\varphi(l)-\varphi(0)=\int_0^l \frac{d \varphi}{d t} d t .
$$
Proposition 5.8. The index is independent of the choice of parametrization $\mathbf{x}$.
Proposition 5.9. The index is independent of the choice of $\alpha$.
Theorem 5.10 (Poincare-Hopf Index Theorem). The sum of the indices of a differentiable vector field $v$ with isolated singular points on a compact surface $S$ is equal to the Euler Characteristic of $S$.
Proof. Let $S \subset \mathbb{R}^3$ be a compact surface and $v$ a differentiable vector field with exclusively isolated singular points. We notice that, due to compactness, these singular points must be finite in number otherwise there would exist a non-isolated singular point as a limit point for the others. We let $\{\mathbf x_a\}$ be a family of parametrizations such that each is compatible with the orientation of $S$. Now, we let $J$ be a triangulation of $S$ with the conditions that each $T$ in $J$ is contained in a coordinate neighborhood of $\{\mathbf{x_a}\}$, each triangle $T$ contains at most one singular point, and each triangle is positively oriented with no singular points on its boundary. Now, we apply the local Gauss-Bonnet theorem to each triangle and sum up the result. However, we recall that each triangle appears twice in this formulation, in opposite orientation, therefore, we have
$$
\iint_S \kappa d \sigma-2 \pi \sum_{i=1}^k I_i=0,
$$
to which we apply the most general form of Gauss-Bonnet to obtain,
$$
\sum_{i=1}^k I_i=\frac{1}{2 \pi} \iint_S \kappa d \sigma=\chi(S) .
$$
Remark 5.11. This result guarantees that a vector field on any surface homeomorphic to a sphere must have at least two isolated singular points, because the Euler characteristic of the 2-sphere is two. The solution is particularly remarkable because it shows that the sum of the indices of a vector field does not depend on $v$, but rather the topology of $S$, which is a non-intuitive idea. More tangibly, the Poincare-Hopf Index Theorem implies the previous theorem.
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