| 释义 |
Gaussian CurvatureAn intrinsic property of a space independent of the coordinate system used to describe it. The Gaussian curvature of aRegular Surface in  at a point p is formally defined as
 | (1) |
 is the Shape Operator and det denotes the Determinant.
If  is a Regular Patch, then the Gaussian curvature is given by
 | (2) |
 ,  , and  are coefficients of the first Fundamental Form and  ,  , and  are coefficients of the second Fundamental Form (Gray 1993, p. 282). The Gaussian curvaturecan be given entirely in terms of the first Fundamental Form
 | (3) |
 | (4) |
 | (5) |
 are the Connection Coefficients. Equivalently,
 | (6) |
Writing this out,
The Gaussian curvature is also given by
 | (10) |
where  is the Levi-Civita Symbol,  is the unit Normal Vector and  is theunit Tangent Vector. The Gaussian curvature is also given by
 | (12) |
 is the Curvature Scalar,  and  the Principal Curvatures, and  and  thePrincipal Radii of Curvature. For a Monge Patch with  ,
 | (13) |
The Gaussian curvature  and Mean Curvature  satisfy
 | (14) |
 | (15) |
If p is a point on a Regular Surface  and  and  are tangentvectors to  at p, then the Gaussian curvature of at p is related to the Shape Operator by
 | (16) |
 and  be Vector Fields tangent to such that  , then
 | (17) |
For a Sphere, the Gaussian curvature is  . For Euclidean Space, the Gaussian curvature is  . ForGauss-Bolyai-Lobachevsky Space, the Gaussian curvature is  . A Flat Surface is a Regular Surfaceand 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 thefollowing 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 asurface on which is everywhere Negative is called Anticlastic. Surfaces with constant Gaussiancurvature include the Cone, Cylinder, Kuen Surface,Plane, Pseudosphere, and Sphere. Of these, the Coneand Cylinderare 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. |
