§ Theorem Egregium / Gauss's theorem (Integrating curvature in 2D) [TODO ]

• Let $S$ be a 2 dimensional surface.
• Gauss Rodriguez map map: $N: S \to S^2$. The derivative of this map goes from $dN: T_p S \to T_p S^2$.
• Since surfaces are parametric, we can think of it as a map from $U \subset \mathbb R^n \to S \to S^2$.
• For gauss, the curvature of the surface at $p$ is $det(dN|_p)$. This tells us how small areas (on the tangent plane of $S$) is distorted (on the tangent plane of $S^2$, because it's the determinant / jacobian of the map. Thus, heuristically, it is the ratio of the area around $N(p)$ at $S^2$ to the area around $p$ at $S$

• To show that this normal curvature view really is curvature, let's compute $dN_p$ for a normal paraboloid. Wildberger says that all surfaces are like normal paraboloids upto second order.
• This fits with one of our views of curvature of a curve: one way was one over the osculating circle, the other was $k \cdot ds = d \theta$
• We had a formula like $\int k ds$ was a change in angle. Similarly, in our case, we see that if we consider $\int \int k(s) darea(s)$, we get the area of the image of $N$, because infinitesimally is the ratio of areas.
• In particular if the surface is homeomorphic to a sphere, then we get the total area of the sphere, $4 \pi$.. This is the 2D analogue of the fact that if we integrate the curvature of a closed curve, we get $2 \pi$. [area of a circle ]. This is by green's theorem.