For a regular surface embedded in 3-space
The Gaussian curvature of a regular surface embedded in is defined as a map:
given as follows: for , is the determinant of the shape operator at .
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.
Generalization of the abstract definition
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.