# Gaussian curvature

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of a surface

The product of the principal curvatures (cf. Principal curvature) of a regular surface at a given point.

If

$$\textrm{ I } = ds ^ {2} = \ E du ^ {2} + 2 F du dv + G dv ^ {2}$$

is the first fundamental form of the surface and

$$\textrm{ II } = L du ^ {2} + 2 M du dv + N dv ^ {2}$$

is the second fundamental form of the surface, then the Gaussian curvature can be computed by the formula

$$K = \frac{LN - M ^ {2} }{EG - F ^ {2} } .$$

The Gaussian curvature is identical with the Jacobi determinant of the spherical map:

$$| K | _ {P _ {0} } = \ \lim\limits _ {d( s) \rightarrow 0 } \ { \frac{S}{s} } ,$$

where $P _ {0}$ is a point on the surface, $s$ is the area of a domain $U$ which contains $P _ {0}$, $S$ is the area of the spherical image of $U$, and $d$ is the diameter of the domain. The Gaussian curvature is positive at an elliptic point, negative at a hyperbolic point, and is zero at a parabolic point or a flat point. It may be expressed in terms of the coefficients of the first fundamental form and their derivatives alone (the Gauss theorem), viz.

$$K = \frac{1}{4W ^ {4} } \left | \begin{array}{ccc} E &E _ {u} &E _ {v} \\ F &F _ {u} &F _ {v} \\ G &G _ {u} &G _ {v} \\ \end{array} \ \right | + \frac{1}{2W} \left \{ \frac \partial {\partial u } \frac{F _ {v} - G _ {u} }{W } + \frac \partial {\partial v } \frac{F _ {u} - E _ {v} }{W } \right \} ,$$

where

$$W ^ {2} = EG - F ^ {2} .$$

Since the Gaussian curvature depends on the metric only, i.e. on the coefficients of the first fundamental form, the Gaussian curvature is invariant under isometric deformation (cf. Deformation, isometric). The Gaussian curvature plays a special role in the theory of surfaces, and many formulas are available for its computation, .

The concept was introduced by C.F. Gauss , and was named after him.

How to Cite This Entry:
Gaussian curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gaussian_curvature&oldid=47055
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article