Gaussian curvature: Definition and Much More from Answers.com

Wikipedia: Gaussian curvature

From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ₁ and κ₂, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way it is embedded in space. This result is the content of Gauss's Theorema egregium.

Symbolically, the Gaussian curvature Κ is defined as

$\Kappa = \kappa_1 \kappa_2 \,\!$ .

It is also given by

$\Kappa = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g},$

where $\nabla_i = \nabla_{{\mathbf e}_i}$ is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in R³, the Gaussian curvature is also given by

$K(\mathbf{p}) = \det(S(\mathbf{p})),$

where S is the shape operator.

A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.

Informal definition

We represent the surface by the implicit function theorem as the graph of a function f of 2 variables, and assume the point p is a critical point, i.e. the gradient of f vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f, i.e. the 2 by 2 matrix of second derivatives. This definition allows one immediately to grasp the distinction between cup/cap versus saddle point behavior in terms of second year calculus.

Total curvature

The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle.

The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangle equals the deviation of the sum of its angles from $π$ . The sum of the angles of a triangle on a surface of positive curvature will exceed $π$ , while the sum of the angles of a triangle on a surface of negative curvature will be less than $π$ . On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely $π$ .

$\sum_{i=1}^3 \theta_i = \pi + \iint_T K \,dA.$

Important theorems

Theorema egregium

Main article: Theorema egregium

Gauss's 1828 Theorema egregium (or remarkable theorem) states that the Gaussian curvature depends only on the first fundamental form (metric tensor) and its derivatives and not on the second fundamental form.

A corollary of this theorem is that the Gaussian curvature is invariant under isometric deformations of the surface. Hence the Gaussian curvature of a surface is an intrinsic property of the surface, and can be determined without reference to the embedding of the surface in space. For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat).^[1]

Gauss–Bonnet theorem

Main article: Gauss-Bonnet theorem

The Gauss-Bonnet theorem links the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.

Theorems on surfaces of constant curvature

Liebmann's theorem (1900) states that the sphere is the only surface (embedded in 3-space) without boundary or singularities with constant positive Gaussian curvature.^[2]

Hilbert's theorem (1901) states that there exists no complete regular surface of constant negative Gaussian curvature. The pseudosphere has constant negative Gaussian curvature except at its cusp.^[3]

Minding's theorem states that all surfaces which have the same constant curvature are isometric.

Alternative Formulas

Gaussian curvature can be expressed via the first fundamental form and second fundamental form:

$K = \frac{eg-f^2}{EG-F^2}$

The Brioschi formula gives Gaussian curvature solely in terms of the first fundamental form:

$K = \frac{\begin{vmatrix} -\frac{1}{2}E_{vv} + F_{uv} - \frac{1}{2}G_{uu} & \frac{1}{2}E_u & F_u-\frac{1}{2}E_v\\F_v-\frac{1}{2}G_u & E & F\\\frac{1}{2}G_v & F & G \end{vmatrix}-\begin{vmatrix} 0 & \frac{1}{2}E_v & \frac{1}{2}G_u\\\frac{1}{2}E_v & E & F\\\frac{1}{2}G_u & F & G \end{vmatrix}} {(EG-F^2)^2}$

For an orthogonal parametrization, Gaussian curvature is:

$K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right).$

Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:

$K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}$

Gaussian curvature is the limiting difference between the area of a geodesic circle and a circle in the plane:

$K = \lim_{r \rarr 0} (\pi r^2 - \mbox{A}(r)) \cdot \frac{12}{\pi r^4}$

Gaussian curvature may be expressed with the Christoffel symbols: ^[4]

$K = -\frac{1}{E} \left( \frac{\partial}{\partial u}\Gamma_{12}^2 - \frac{\partial}{\partial v}\Gamma_{11}^2 + \Gamma_{12}^1\Gamma_{11}^2 - \Gamma_{11}^1\Gamma_{12}^2 + \Gamma_{12}^2\Gamma_{12}^2 - \Gamma_{11}^2\Gamma_{22}^2\right)$

Paper

It is of some interest that paper, without creases, exihibits zero Gaussian curvature at every point on the surface of the page, excepting the frayed edges. Paper folding, as in Origami, only occurs along lines of zero Gaussian curvature. By extension, once the paper is creased, the paper in the crease may not exhibit zero Gaussian curvature, but the paper approaching the crease and on all sides of the crease, will. A crease can then be defined as all points on the surface of the paper, excepting the edges, that fail to preserve zero Gaussian curvature. Other substances also exihibit this property, for example, leaves of plants to a first approximation, but none so common and with such certainty as paper.

References

^ Porteous, I. R., Geometric Differentiation. Cambridge University Press, 1994. ISBN 0-521-39063-X
^ Kühnel, Wolfgang (2006). Differential Geometry: Curves - Surfaces - Manifolds. American Mathematical Society. ISBN 0821839888.
^ Hilbert theorem. Springer Online Reference Works.
^ Struik, Dirk (1988). Lectures on Classical Differential Geometry. Courier Dover Publications. ISBN 0486656098.

Gaussian curvature