Gaussian Curvature

Gaussian curvature, sometimes also called total curvature (Kreyszig 1991, p. 131), is an intrinsic property of a space independent of the coordinate system used to describe it. The Gaussian curvature of a regular surface in at a point is formally defined as

(1)

where is the shape operator and det denotes the determinant.

If is a regular patch, then the Gaussian curvature is given by

(2)

where , , and are coefficients of the first fundamental form and , , and are coefficients of the second fundamental form (Gray 1997, p. 377). The Gaussian curvature can be given entirely in terms of the first fundamental form

(3)

and the metric discriminant

(4)

by

(5)

where are Christoffel symbols of the second kind. Equivalently,

(6)

where

			(7)
			(8)

Writing this out,

(9)

The Gaussian curvature is also given by

(10)

(Gray 1997, p. 380), as well as

(11)

where is the permutation symbol, is the unit normal vector and is the unit tangent vector. The Gaussian curvature is also given by

			(12)
			(13)
			(14)

where is the scalar curvature, and the principal curvatures, and and the principal radii of curvature. For a Monge patch with ,

(15)

For a twice continuously differentiable real-valued function , the Gaussian curvature of the surface defined implicitly by can be computed in the local Monge patch when as

(16)

(Trott 2004, pp. 1285-1286).

At any regular point, where , a formula independent of the choice of graph coordinates is

(17)

where is the gradient written as a column vector, is the Hessian of , and is its adjugate matrix (Goldman 2005, pp. 642-643). Unlike the preceding expression, this formula does not require .

The Gaussian curvature and mean curvature satisfy

(18)

with equality only at umbilic points, since

(19)

If is a point on a regular surface and and are tangent vectors to at , then the Gaussian curvature of at is related to the shape operator by

(20)

Let be a nonvanishing vector field on which is everywhere perpendicular to , and let and be vector fields tangent to such that , then

(21)

(Gray 1997, p. 410).

For a sphere, the Gaussian curvature is . For Euclidean space, the Gaussian curvature is . For Gauss-Bolyai-Lobachevsky space, the Gaussian curvature is . A developable surface has vanishing Gaussian curvature at every regular point.

A point on a regular surface is classified based on the sign of as given in the following table (Gray 1997, p. 375), where is the shape operator.

Sign	Point
	elliptic point
	hyperbolic point
but	parabolic point
and	planar point

A surface on which the Gaussian curvature is everywhere positive is called synclastic, while a surface on which is everywhere negative is called anticlastic. Surfaces with constant Gaussian curvature include the cone, cylinder, Kuen surface, plane, pseudosphere, and sphere. Of these, the cone, cylinder, and plane are developable surfaces of revolution.