TheoremProved

Gauss's Theorema Egregium

Statement

The Gaussian curvature $K = \kappa_1 \kappa_2$ of a smooth surface $S \subset \mathbb{R}^3$ (the product of its principal curvatures $\kappa_1$ and $\kappa_2$ ) is an intrinsic invariant: it depends only on the first fundamental form $I = E\,du^2 + 2F\,du\,dv + G\,dv^2$ and its first and second partial derivatives, and is therefore preserved by every local isometry.

Why is it true?

The individual principal curvatures $\kappa_1$ and $\kappa_2$ describe how a surface bends in surrounding space $\mathbb{R}^3$ — rolling a flat sheet of paper ( $\kappa_1 = 0, \kappa_2 = 0$ ) into a cylinder of radius $r$ changes them to $\kappa_1 = 1/r$ and $\kappa_2 = 0$ without stretching or tearing the paper. Yet their product $K = \kappa_1 \kappa_2$ stays $0$ , whereas a sphere of radius $r$ has $K = 1/r^2 > 0$ and can never be flattened onto a plane without distorting distances; two-dimensional inhabitants measuring only lengths and angles inside the surface can detect $K$ without ever seeing the ambient space $\mathbb{R}^3$ .

Proof sketch

For a smooth surface $S \subset \mathbb{R}^3$ with first fundamental form coefficients $E, F, G$ and second fundamental form coefficients $e, f, g$ , the shape operator has determinant $K = \kappa_1 \kappa_2 = \dfrac{eg - f^2}{EG - F^2}$ . Expressing the third partial derivatives of the position vector $\mathbf{r}(u, v)$ via the Gauss equations in terms of the Christoffel symbols $\Gamma_{ij}^k$ (which depend only on $E, F, G$ and their first derivatives) and using the symmetry $\mathbf{r}_{uvu} = \mathbf{r}_{uuv}$ expresses $eg - f^2$ entirely in terms of $E, F, G$ and their first and second derivatives (the Brioschi formula), proving that $K$ depends only on the intrinsic metric.

Stated by

stated 1827Carl Friedrich Gauss

Proved by

proved 1827Carl Friedrich Gauss

Topics that use this theorem

Differential geometry Non-Euclidean geometry

Related theorems

Gauss–Bonnet theorem Euclid's parallel postulate

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Carl Friedrich Gauss (translated by Peter Pesic) (2005). General Investigations of Curved Surfaces
Manfredo P. do Carmo (1976). Differential Geometry of Curves and Surfaces