“Gaussian Curvature”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gaussian Curvature

释义

Gaussian Curvature

An 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)

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 curvaturecan be given entirely in terms of the first Fundamental Form

(3)

and the Discriminant

(4)

(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 theunit Tangent Vector. The Gaussian curvature is also given by

(12)

where

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)

with equality only at Umbilic Points, since

(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)

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 . 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.

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