Differential Geometry (3): The Shape Operator — Curvature of Surfaces

Fri, 05 Nov 2021 09:00:00 +0000

The previous article gave us the intrinsic apparatus: the first fundamental form $\mathrm{I}$ , encoded as the symmetric matrix $\begin{pmatrix}E & F \\ F & G\end{pmatrix}$ . With it, an ant on the surface can measure lengths, angles, and areas without ever leaving. What an ant on a cylinder cannot do, equipped only with $\mathrm{I}$ , is detect that the cylinder is bent. The cylinder has the same first fundamental form as the plane, yet sits very differently in $\mathbb{R}^3$ .

Curvature on Chen Kai Blog

Differential Geometry (3): The Shape Operator — Curvature of Surfaces