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 $\mathbb{R}^3$ .

Differential Geometry (2): Surfaces and the First Fundamental Form: Intrinsic Measurements

Wed, 03 Nov 2021 09:00:00 +0000

Curves were a one-dimensional warm-up. The geometry was governed by ODEs, and a single moving frame caught everything interesting. From now on we go up a dimension and the difficulty rises in three directions at once. Tangent vectors get replaced by tangent planes. The single arc-length parameter splits into two coordinates $$(u, v)$$ , and reparametrization becomes a $2\times 2$ Jacobian matrix instead of a scalar. And — the real change — we acquire two distinct kinds of geometry: intrinsic (what an ant living on the surface can measure) and extrinsic (how the surface bends in the surrounding $\mathbb{R}^3$ ). This article is the intrinsic story. We build the first fundamental form, the $2\times 2$ matrix-valued function that lets the ant measure lengths, angles, and areas without ever leaving the surface.

Surfaces on Chen Kai Blog

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

Differential Geometry (2): Surfaces and the First Fundamental Form: Intrinsic Measurements