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 $\Bbb{R}^3$ at a point p is formally defined as

$$K({\bf p})={\textstyle{1\over 2}}\mathop{\rm det}(S({\bf p})),$$ (1)

where $S$ is the Shape Operator and det denotes the Determinant.

If ${\bf x}:U\to\Bbb{R}^3$ is a Regular Patch, then the Gaussian curvature is given by

$$K={eg-f^2\over EG-F^2},$$ (2)

where $E$, $F$, and $G$ are coefficients of the first Fundamental Form and $e$, $f$, and $g$ 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

$$ds^2=E\,du^2+2F\,du\,dv+G\,dv^2$$ (3)

and the Discriminant

$$g\equiv EG-F^2$$ (4)

by

$$K ={1\over\sqrt{g}}\left[{{\partial\over\partial v}\left({{\... ...rtial u}\left({{\sqrt{g}\over E}\Gamma^2_{12}}\right)}\right],$$ (5)

where $\Gamma^k_{ij}$ are the Connection Coefficients. Equivalently,

$$K={1\over g^2}\left\vert\matrix{E & F & {\partial F\over\par... ...ver\partial v} & 0\cr}\right\vert,\hrule width 0pt height 6.pt$$ (6)

where

$$k_{23} \equiv {\partial F\over\partial u}-{1\over 2}{\partial E\over \partial v}$$ (7)

$$k_{33} \equiv -{1\over 2}{\partial^2 E\over \partial v^2}+{\partial^2F\over \partial u\partial v}-{1\over 2}{\partial^2 G\over \partial u^2}.$$ (8)

Writing this out,

$$K = {1\over 2g}\left[{2{\partial^2F\over\partial u\partial v}-{\parti... ...al G\over\partial u}}\right)-\left({\partial E\over\partial v}\right)^2}\right]$$

$$\phantom{=} + {F\over 4g_2}\left[{{\partial E\over\partial u}{\partial G\over... ...\left({2{\partial F\over\partial v}-{\partial G\over\partial u}}\right)}\right]$$

$$\phantom{=} -{E\over 4g^2}\left[{{\partial G\over\partial v}\left({2{\partial... ...l E\over\partial v}}\right)-\left({\partial G\over\partial u}\right)^2}\right].$$ (9)

The Gaussian curvature is also given by

$$K={\mathop{\rm det}({\bf x}_{uu} {\bf x}_u {\bf x}_v)\mathop... ...u\vert^2 \vert{\bf x}_v\vert^2-({\bf x}_u\cdot{\bf x}_v)^2]^2}$$ (10)

(Gray 1993, p. 285), as well as

$$K = {[\hat{\bf N}\,\hat{\bf N}_1\,\hat{\bf N}_2]\over\sqrt{g}} = {\epsilon^{ij} [\hat{\bf N}\,\hat{\bf T}\,\hat{\bf T}_i]_j\over \sqrt{g}},$$ (11)

where $\epsilon^{ij}$ is the Levi-Civita Symbol, $\hat{\bf N}$ is the unit Normal Vector and $\hat{\bf T}$ is the unit Tangent Vector. The Gaussian curvature is also given by

$$K=-{R\over 2} = \kappa_1 \kappa_2 = {1\over R_1R_2},$$ (12)

where $R$ is the Curvature Scalar, $\kappa_1$ and $\kappa_2$ the Principal Curvatures, and $R_1$ and $R_2$ the Principal Radii of Curvature. For a Monge Patch with $z=h(u,v)$,

$$K={h_{uu}h_{vv}-{h_{uv}}^2\over(1+{h_u}^2+{h_v}^2)^2}.$$ (13)

The Gaussian curvature $K$ and Mean Curvature $H$ satisfy

$$H^2\geq K,$$ (14)

with equality only at Umbilic Points, since

$$H^2-K^2={\textstyle{1\over 4}}(\kappa_1-\kappa_2)^2.$$ (15)

If p is a point on a Regular Surface $M\subset\Bbb{R}^3$ and ${\bf v}_{\bf p}$ and ${\bf w}_{\bf p}$ are tangent vectors to $M$ at p, then the Gaussian curvature of $M$ at p is related to the Shape Operator $S$ by

$$S({\bf v}_{\bf p})\times S({\bf w}_{\bf p})=K({\bf p}) {\bf v}_{\bf p}\times {\bf w}_{\bf p}.$$ (16)

Let Z be a nonvanishing Vector Field on $M$ which is everywhere Perpendicular to $M$, and let $V$ and $W$ be Vector Fields tangent to $M$ such that $V\times W={\bf Z}$, then

$$K={{\bf Z}\cdot(D_V{\bf Z}\times D_W{\bf Z})\over 2\vert{\bf Z}\vert^4}$$ (17)

(Gray 1993, pp. 291-292).

For a Sphere, the Gaussian curvature is $K=1/a^2$. For Euclidean Space, the Gaussian curvature is $K=0$. For Gauss-Bolyai-Lobachevsky Space, the Gaussian curvature is $K=-1/a^2$. 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 $M\in\Bbb{R}^3$ is classified based on the sign of $K({\bf p})$ as given in the following table (Gray 1993, p. 280), where $S$ is the Shape Operator.

Sign	Point
$K({\bf p})>0$	Elliptic Point
$K({\bf p})<0$	Hyperbolic Point
$K({\bf p})=0$ but $S({\bf p})\not=0$	Parabolic Point
$K({\bf p})=0$ and $S({\bf p})=0$	Planar Point

A surface on which the Gaussian curvature $K$ is everywhere Positive is called Synclastic, while a surface on which $K$ 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.