Theorema Egregium

Gaussian curvature is intrinsic, so isometries preserve it.

The idea

Theorem (Theorema Egregium (Gauss)).

The Gaussian curvature $K = \kappa_{1}\kappa_{2}$ of a surface depends only on its first fundamental form. Consequently every local isometry — every map between surfaces that preserves lengths and angles measured within them — carries $K$ to $K$ at corresponding points.

The principal curvatures $\kappa_{1}$ and $\kappa_{2}$ are extrinsic: they are defined by how the unit normal turns as the surface bends in the surrounding space, and neither can be recovered from measurements made inside the surface. The theorem asserts that their product can. Gauss's formula expresses $K$ in the coefficients $E, F, G$ of the first fundamental form and their derivatives alone — the ambient information cancels — so a surveyor confined to the surface, with no access to a third dimension, can determine $K$.

An isometry leaves $E, F, G$ intact, so it leaves intact everything computed from them, and $K$ is such a quantity. Curvature therefore obstructs isometry: the plane has $K = 0$ everywhere and the unit sphere has $K = 1$ everywhere, so no patch of the sphere, however small, can be laid flat without distorting distances. The cylinder and the cone have $K = 0$, which is why they unroll onto paper exactly.

Ways to work on it

• Walkthrough. What the theorem says, why a sphere cannot be flattened, and computing K from E, G.
• Practice. Use curvature to decide whether two surface patches can be locally isometric.
• Hardest. Compute the sphere's curvature intrinsically from its first fundamental form, then draw the isometry consequence.

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