## Gaussian Curvature

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 p 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 1993, p. 282). The Gaussian curvature can be given entirely in terms of the first Fundamental Form
 (3)

and the Discriminant
 (4)

by
 (5)

where are the Connection Coefficients. Equivalently,

 (6)

where
 (7) (8)

Writing this out,

 (9)

The Gaussian curvature is also given by

 (10)

(Gray 1993, p. 285), as well as
 (11)

where is the Levi-Civita Symbol, is the unit Normal Vector and is the unit Tangent Vector. The Gaussian curvature is also given by
 (12)

where is the Curvature Scalar, and the Principal Curvatures, and and the Principal Radii of Curvature. For a Monge Patch with ,
 (13)

The Gaussian curvature and Mean Curvature satisfy

 (14)

with equality only at Umbilic Points, since
 (15)

If p is a point on a Regular Surface and and are tangent vectors to at p, then the Gaussian curvature of at p is related to the Shape Operator by

 (16)

Let Z be a nonvanishing Vector Field on which is everywhere Perpendicular to , and let and be Vector Fields tangent to such that , then
 (17)

(Gray 1993, pp. 291-292).

For a Sphere, the Gaussian curvature is . For Euclidean Space, the Gaussian curvature is . For Gauss-Bolyai-Lobachevsky Space, the Gaussian curvature is . A Flat Surface is a Regular Surface and special class of Minimal Surface on which Gaussian curvature vanishes everywhere.

A point p on a Regular Surface is classified based on the sign of as given in the following table (Gray 1993, p. 280), 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 and Cylinder are the only Flat Surfaces of Revolution.

See also Anticlastic, Brioschi Formula, Developable Surface, Elliptic Point, Flat Surface, Hyperbolic Point, Integral Curvature, Mean Curvature, Metric Tensor, Minimal Surface, Parabolic Point, Planar Point, Synclastic, Umbilic Point

References

Geometry Center. Gaussian Curvature.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/concepts/curvatures/gauss-curv.html.

Gray, A. The Gaussian and Mean Curvatures'' and Surfaces of Constant Gaussian Curvature.'' §14.5 and Ch. 19 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 279-285 and 375-387, 1993.