Gaussian curvature

Definition

For a regular surface embedded in 3-space

The Gaussian curvature of a regular surface $M$ embedded in $\mathbb {R} ^{3}$ is defined as a map:

$K:M\to \mathbb {R}$

given as follows: for $p\in M$, $K(p)$ is the determinant of the shape operator at $p$.

For an abstract 2-dimensional Riemannian manifold

Fill this in later

Equivalence of definitions

The equivalence of these definitions arises from Gauss's Theorema Egregium.

Generalizations

Generalization of the abstract definition

Further information: sectional curvature,Ricci curvature,scalar curvature

The abstract definition generalizes to the notion of sectional curvature. Sectional curvature is a two-dimensional curvature on manifolds of higher dimension: it takes as input a point on a manifold and a tangent plane at the point, it outputs the Gaussian curvature that we'd get by taking a two-dimensional submanifold with that tangent plane as tangent space.

The Ricci curvature takes a tangent line instead of a tangent plane, and gives the average sectional curvature over all tangent planes containing that tangent line. The scalar curvature averages over all tangent lines.

Generalization of the concrete definition

Further information: Gauss-Kronecker curvature

The Gauss-Kronecker curvature is valid for codimension one submanifolds (or hypersurfaces) in Euclidean space of any dimension, and uses precisely the same definition as that of the Gaussian curvature.