I would like to ask if any of you have a couple of good illustrative examples of curves on surfaces to be used as example of the Gauss-Bonnes theorem for simple closed curves?
To specify I do not need the calculations, just the idea of where to start.
Sorry for my bad english. If there need to be explained anything then just ask.
Is there an other name for Gauss-Bonned for simple closed curves?, when i google then there is a lot there is a bit off subject.
Kind regards Jakob
The Gauss-Bonnet (with a t at the end) theorem is one of the most important theorem in the differential geometry of surfaces. The Gauss-Bonnet theorem comes in local and global version. The global version say that given a regular oriented surface S of class $C^3$ , and let R be a compact region of S with boundary $\partial R$, assuming that $\partial R$ is a simple, closed, piecewise regular, positively oriented curve there exist a relationship linking the Gaussian curvature K, the geodesic curvature $k_g$, and the $\theta_i$ the external angles of the vertices of $\partial R$ to the Euler characteristic $\chi(R)$. \begin{equation} \sum_{i=1}^{n} \int_{C_i} k_g ds+ \iint_{R} K dS+ \sum_{i=1}^{n} \theta_i=2 \pi \chi(R) \end{equation} where the boundary $\partial R$ is composed by n regular arcs $C_i$. Consider now the following examples:
A simple closed curve $\Gamma$ separate the surface of the sphere in two simply connected region I and II. By applying the Gauss-Bonnet theorem to each region you have: \begin{equation} \int_{\Gamma} k_g ds+ \iint_{I} K dS=2 \pi \end{equation} and \begin{equation} -\int_{\Gamma} k_g ds+ \iint_{II} K dS=2 \pi \end{equation} since in the second case the curve is described in the reverse direction. Adding this two equation you get \begin{equation} \iint_{S} K dS=4 \pi \end{equation} In this example we considered a simple closed curve, so there are no external angles and the Euler characteristic of R (equal to #(vertices) − #(edges) + #(faces)) is 2.
As another example, please consider the Gauss-Bonnet Theorem applied to plane, spherical, and hyperbolic geometries. In the first case, a plane is a surface with zero Gaussian curvature and curves in the plane belong to the surface. Consider now a region R in the plane such that the boundary $\partial R$ is a piecewise regular, simple, closed curve. It is easy to calculate that the Euler characteristic of a region that is homeomorphic to a disk is $\chi(R) = 1$ (you can have a look here https://en.wikipedia.org/wiki/Euler_characteristic). As a first case, let $\partial R$ be a polygon. Since the regular arcs are straight lines, their geodesic curvature is zero. Gauss-Bonnet’s formula then reduces: \begin{equation} \sum_{i=0}^{n} \theta_i=2 \pi \end{equation} Since the exterior angle $\theta_i$ at any corner is $\pi - \alpha_i$, where $\alpha_i$ is the interior angle, we can write \begin{equation} \sum_{i=1}^{n} \alpha_i=(n-2) \pi \end{equation} Thus for a triangle $n=3$, and the sum of the interior angles of a triangle is $\pi$ radians. Consider now a triangle T on a sphere of radius R. By definition, $\partial T$ has three vertices, and its regular arcs are geodesic curves. The Gaussian curvature of a sphere of radius R is $\frac{1}{R^2}$ . Then the Gauss-Bonnet theorem reduces to: \begin{equation} \sum_{i=1}^{3} \alpha_i=\pi+\frac{A}{R^2} \end{equation} where as in the previous case $\alpha_i$ are the interior angles. This shows that in a spherical triangle, the sum of the interior angles is larger then $\pi$. In contrast to spherical geometry, hyperbolic geometry is the geometry of a surface with constant negative Gaussian curvature. Consider for instance a pseudo-sphere with $K=-1$. In this particular case, the Gauss-Bonnet Theorem applied to a triangle shows that: \begin{equation} \sum_{i=1}^{3} \alpha_i=\pi-A \end{equation} i.e. the sum of the interior angles is lower then $\pi$. The Gauss-Bonnet theorem has really alot of applications. I hope this simple example are helpful.