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

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Equivalence of definitions

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