Gaussian curvature

Gaussian curvature at a point on a surface is the product of its two principal curvatures. It is intrinsic: it can be determined from distances measured within the surface, without reference to how it sits in space.

Principal curvature: One of the two extreme normal curvatures of a smooth surface at a point. Their product defines Gaussian curvature.
Tangent plane: The plane that best approximates a smooth surface near a point. Principal directions and curvatures are defined relative to this plane.
Theorema Egregium: Gauss's theorem that Gaussian curvature depends only on a surface's intrinsic metric. It establishes why bending a surface without stretching preserves this curvature.
Sphere: The set of points at a fixed distance from a center in three-dimensional Euclidean space. Its two principal curvatures have the same sign, giving positive Gaussian curvature.
Shape operator: A linear map on a surface's tangent plane that encodes how its normal changes. Its eigenvalues are the principal curvatures whose product gives this quantity.
Normal curvature: The curvature of a surface curve formed by intersecting it with a plane containing the surface normal. Its extreme values in tangent directions are the principal curvatures.
Gauss–Bonnet theorem: A theorem relating integrated Gaussian curvature of a surface to its topology and boundary geometry. It turns local curvature values into a global topological constraint.
Cylinder: A surface formed by translating a plane curve along a straight direction. One principal curvature is zero, so its Gaussian curvature vanishes.
Mean curvature: The average of a surface's two principal curvatures at a point. Unlike this product, mean curvature changes when a surface bends without stretching.
First fundamental form: A quadratic form that records lengths and angles measured on a surface. Together with the second fundamental form, it yields the intrinsic curvature.

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