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Curvature of Surfaces (Gauss)

"Theorema Egregium: Gaussian curvature is an intrinsic quantity – it depends only on the first fundamental form, not on how the surface is embedded." — Carl Friedrich Gauss, Disquisitiones generales circa superficies curvas, 1827.

Gaussian curvature is the historical starting point of modern differential geometry. It measures how a surface is bent at a point – and Gauss's discovery that it is intrinsic is the springboard to sectional, Ricci, and scalar curvature in higher dimensions.

1. Principal curvatures

Let \(\Sigma \subset \mathbb{R}^3\) be a smooth surface and \(p \in \Sigma\). Choose a unit normal \(\nu(p)\). Slicing \(\Sigma\) with a plane through \(p\) that contains \(\nu(p)\) yields a planar curve with a normal curvature \(\kappa\).

Rotating the slicing plane around \(\nu(p)\), \(\kappa\) varies between a minimum \(\kappa_2\) and a maximum \(\kappa_1\). These extremal values are the principal curvatures at \(p\) (Euler 1760, elaborated by Meusnier 1776).

From them one defines:

  • Gaussian curvature: \(K = \kappa_1 \kappa_2\).
  • Mean curvature: \(H = \tfrac{1}{2}(\kappa_1 + \kappa_2)\).
Surface \(K\) Sign
plane \(0\) flat
sphere of radius \(R\) \(1/R^2\) positive
cylinder \(0\) flat (one principal curvature vanishes)
saddle surface \(< 0\) hyperbolic
pseudosphere \(-1\) constant hyperbolic

2. First and second fundamental form

Locally \(\Sigma\) can be parametrised as \(\mathbf r(u, v)\). The first fundamental form describes length measurement on \(\Sigma\): $$ \mathrm{I} = E\, \mathrm{d}u^2 + 2F\, \mathrm{d}u\, \mathrm{d}v + G\, \mathrm{d}v^2, \quad E = \mathbf r_u \cdot \mathbf r_u,\ F = \mathbf r_u \cdot \mathbf r_v,\ G = \mathbf r_v \cdot \mathbf r_v. $$ It is the Riemannian metric of the surface (see Tangent Space and Tensors).

The second fundamental form measures how the normal direction changes with the point: $$ \mathrm{II} = L\, \mathrm{d}u^2 + 2M\, \mathrm{d}u\, \mathrm{d}v + N\, \mathrm{d}v^2, \quad L = \mathbf r_{uu} \cdot \nu, \dots $$ It knows the embedding of the surface.

In this language $$ K = \frac{LN - M^2}{EG - F^2}. $$

3. Theorema Egregium

Gauss's surprising discovery (1827): $$ K \text{ depends only on } E, F, G \text{ and their derivatives.} $$ That is: someone living on the surface, equipped with length and angle measurements but unaware of the surrounding \(\mathbb{R}^3\), can still compute \(K\). So \(K\) is intrinsic data of the Riemannian metric.

Consequence: a sphere cannot be unrolled into a plane without distorting distances – every world map lies.

4. Geodesic triangles and Gauss–Bonnet

On a surface with curvature \(K\) one obtains the angle-sum theorem for geodesic triangles: $$ \alpha + \beta + \gamma - \pi = \int_T K\, \mathrm{d}A. $$ On the sphere (\(K = 1/R^2 > 0\)) the angle sum exceeds \(\pi\); on a hyperbolic surface (\(K < 0\)) it is smaller.

The Gauss–Bonnet theorem generalises this to closed surfaces: $$ \int_\Sigma K\, \mathrm{d}A = 2\pi\, \chi(\Sigma), $$ where \(\chi(\Sigma)\) is the Euler characteristic. For \(\Sigma = S^2\) this gives \(\chi(S^2) = 2\), for the torus \(\chi(T^2) = 0\). This formula is the bridge from curvature (analysis) to topology (what the surface "is") – the same architectural principle that Perelman's proof of finite extinction in Act 3, Article 05 re-uses on the 2-sphere.

5. From special case to higher-dimensional curvature

In \(n\) dimensions \(K\) generalises to the sectional curvature \(\sec(P)\) of a 2-plane \(P \subset T_p M\). Averaging over all 2-planes containing a vector \(v\) yields the Ricci curvature \(\mathrm{Ric}(v, v)\). Averaging further over all directions yields the scalar curvature \(R\). Gaussian curvature is thus the two-dimensional grandfather of the entire curvature zoo (Act 2, Article 02).

Cross-references

Sources

  • do Carmo, Manfredo P. (1976). Differential Geometry of Curves and Surfaces. Prentice Hall. Ch. 3–4.
  • Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry, Vol. 2. Publish or Perish, 3rd ed.
  • Gauss, Carl Friedrich (1827). Disquisitiones generales circa superficies curvas.
  • Lee, John M. (2018). Introduction to Riemannian Manifolds. Springer GTM 176, 2nd ed. Ch. 8.