Differential Geometry (4): Intrinsic Geometry — Theorema Egregium and Geodesics

Sun, 07 Nov 2021 09:00:00 +0000

The previous two chapters set up a clear dichotomy. Chapter 2 introduced the first fundamental form $\mathrm{I}$ — the intrinsic metric, what an ant on the surface can measure. Chapter 3 introduced the second fundamental form $\mathrm{II}$ and the shape operator — the extrinsic data, how the surface bends in $\mathbb{R}^3$ . From $\mathrm{II}$ we computed Gaussian curvature $K = \det S$ and mean curvature $H = \mathrm{tr}\,S/2$ . By all appearances, both $$K$$ and $$H$$ should depend on the embedding. Bend the surface (without stretching) and you would expect both to change.

Theorema-Egregium on Chen Kai Blog

Differential Geometry (4): Intrinsic Geometry — Theorema Egregium and Geodesics