Theorema Egregium

Gauss’s theorem that a surface’s Gaussian curvature is determined by its metric and is preserved by local isometries, regardless of how the surface sits in space.

Gaussian curvature: The product of a surface’s principal curvatures at a point. The theorem identifies this extrinsic-looking quantity as measurable from the surface’s metric.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, geometry, statistics, and mathematical physics. He proved the theorem and introduced its enduring name.
Metric tensor: A field of inner products that defines lengths and angles on a manifold. The theorem’s intrinsic input is the surface metric tensor.
Cylinder: A surface generated by translating a line along a fixed direction around a planar curve. A cylinder is locally isometric to a plane, so both have zero Gaussian curvature.
First fundamental form: The metric on a parametrized surface, encoding lengths and angles of tangent vectors. The theorem derives Gaussian curvature from this intrinsic metric data.
Disquisitiones Generales circa Superficies Curvas: Gauss’s 1828 treatise on the differential geometry of curved surfaces. It contains the theorem’s landmark formulation and proof.
Principal curvature: An extreme normal curvature of a surface at a point, measured in a principal direction. Gaussian curvature is the product of the two principal curvatures.
Cone: A surface formed by joining every point of a planar curve to a fixed apex. Away from its apex, a cone can be flattened locally without stretching and has zero Gaussian curvature.
Christoffel symbols: Coordinate-dependent quantities that describe how coordinate basis vectors change across a manifold. They express the metric’s derivatives used to compute curvature intrinsically.
Differential geometry: The study of smooth spaces using calculus, geometric structures, and differential equations. The theorem became a foundational result in this field.

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