Geometric Intuition of Gaussian Curvature

Question

Curvature of a curve at a point can be understood as how rapidly the curve tries to move away from the tangent of the curve at that point. And for curved surfaces we have defined the Gaussian curvature at point $p$ to be the determinant of the Gaussian map $dN_p$. How can one intuitively describe the curvature of a surface defined by the Gaussian Map?

Answer 1

Let's first explain what is curvature of a curve in $\mathbb R^2$. Consider Gaussian map $G:\vec\gamma(s)\rightarrow\vec n(s)$ with naturally parameterized curve $\gamma(s)$, where $\vec n(s)$ is unit vector normal to the curve at the point $\gamma(s)$. We can always move $\vec n(s)$ to Gauss circle which has the center consistent with the origin. Then an infinitesimal $d\gamma(s)$ of the curve will always corresponding to another infinitesimal angle $d\theta$ on Gauss circle. Now the curvature is defined as $\kappa=\frac{d\theta}{ds}$.

Next for surface $\vec r(u,v)$ in $\mathbb R^3$, we still have Gauss sphere which is defined similarly. Then an infinitesimal area element $dA=dudv$ on the surface will correspond to another infinitesimal area element $d\sigma$ on Gauss sphere. Gaussian curvature is precisely the ratio of two areas $K=\frac{dA}{d\sigma}$.