In differential geometry, the Gaussian curvature or Gauss curvature Template:Mvar of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, Template:Math and Template:Math, at the given point: <math display="block"> K = \kappa_1 \kappa_2.</math> For example, a sphere of radius Template:Mvar has Gaussian curvature Template:Math everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Gaussian curvature is an intrinsic measure of curvature, meaning that it could in principle be measured by a 2-dimensional being living entirely within the surface, because it depends only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema Egregium.
Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema Egregium in 1827.
Informal definitionEdit
At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these Template:Math, Template:Math. The Gaussian curvature is the product of the two principal curvatures Template:Math.
The sign of the Gaussian curvature can be used to characterise the surface.
- If both principal curvatures are of the same sign: Template:Math, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
- If the principal curvatures have different signs: Template:Math, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
- If one of the principal curvatures is zero: Template:Math, the Gaussian curvature is zero and the surface is said to have a parabolic point.
Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.
Relation to geometriesEdit
When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.
When a surface has a constant positive Gaussian curvature, then the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there exist other examples as well, such as the lemon / American football.
When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.
Relation to principal curvaturesEdit
The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions from that point. We represent the surface by the implicit function theorem as the graph of a function, Template:Mvar, of two variables, in such a way that the point Template:Mvar is a critical point, that is, the gradient of Template:Mvar vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at Template:Mvar is the determinant of the Hessian matrix of Template:Mvar (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2×2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.
Alternative definitionsEdit
It is also given by <math display="block">K = \frac{\bigl\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\bigr\rangle}{\det g},</math> where Template:Math is the covariant derivative and Template:Mvar is the metric tensor.
At a point Template:Math on a regular surface in Template:Math, the Gaussian curvature is also given by <math display="block">K(\mathbf{p}) = \det S(\mathbf{p}),</math> where Template:Mvar is the shape operator.
A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.
Total curvatureEdit
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 Template:Pi. The sum of the angles of a triangle on a surface of positive curvature will exceed Template:Pi, while the sum of the angles of a triangle on a surface of negative curvature will be less than Template:Pi. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely Template:Pi radians. <math display="block">\sum_{i=1}^3 \theta_i = \pi + \iint_T K \,dA.</math> A more general result is the Gauss–Bonnet theorem.
Important theoremsEdit
Theorema egregiumEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Gauss's Theorema egregium (Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order. Equivalently, the determinant of the second fundamental form of a surface in Template:Math can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface Template:Mvar in Template:Math certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ambient space: it is an intrinsic invariant. In particular, the Gaussian curvature is invariant under isometric deformations of the surface.
In contemporary differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold. To connect this point of view with the classical theory of surfaces, such an abstract surface is embedded into Template:Math and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface Template:Mvar in Template:Math. A local isometry is a diffeomorphism Template:Math between open regions of Template:Math whose restriction to Template:Math is an isometry onto its image. Theorema egregium is then stated as follows:
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat).<ref>Template:Cite book</ref>Template:Page needed On the other hand, since a sphere of radius Template:Mvar has constant positive curvature Template:Math and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar representation of even a small part of a sphere must distort the distances. Therefore, no cartographic projection is perfect.
Gauss–Bonnet theoremEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Gauss–Bonnet theorem relates the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.
- <math>\int_M K\,dA+\int_{\partial M}k_g\,ds=2\pi\chi(M), \, </math>
Surfaces of constant curvatureEdit
- Minding's theorem (1839) states that all surfaces with the same constant curvature Template:Mvar are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. Minding also raised the question of whether a closed surface with constant positive curvature is necessarily rigid.
- Liebmann's theorem (1900) answered Minding's question. The only regular (of class Template:Math) closed surfaces in Template:Math with constant positive Gaussian curvature are spheres.<ref>Template:Cite book</ref> If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature.<ref>Template:Cite book.</ref>
- Hilbert's theorem (1901) states that there exists no complete analytic (class Template:Math) regular surface in Template:Math of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class Template:Math immersed in Template:Math, but breaks down for Template:Math-surfaces. The pseudosphere has constant negative Gaussian curvature except at its boundary circle, where the gaussian curvature is not defined.
There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution <math>(\phi(v) \cos(u), \phi(v) \sin(u), \psi(v))</math> where <math>\phi(v) = C \cos v</math>, and <math display="inline"> \psi(v) = \int_0^v \sqrt{1-C^2 \sin^2 v'}\ dv'</math> (an incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for <math>C\ne 1</math> either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is pseudosphere.<ref>Template:Cite book</ref>
There are many other possible bounded surfaces with constant Gaussian curvature. Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any such bending of a sphere with a region removed will also have constant Gaussian curvature.<ref>Template:Cite book </ref>
Alternative formulasEdit
- Gaussian curvature of a surface in Template:Math can be expressed as the ratio of the determinants of the second and first fundamental forms Template:Math and Template:Math: <math display="block">K = \frac{\det(\mathrm{I\!I})}{\det(\mathrm I)} = \frac{LN-M^2}{EG-F^2}.</math>
- The Template:Vanchor (after Francesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form: <math display="block"> 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}}{\left(EG - F^2\right)^2} </math>
- For an orthogonal parametrization (Template:Math), Gaussian curvature is: <math display="block">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).</math>
- For a surface described as graph of a function Template:Math, Gaussian curvature is:<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref> <math display="block">K = \frac{F_{xx}\cdot F_{yy}- F_{xy}^2}{\left(1+F_x^2+ F_y^2\right)^2}</math>
- For an implicitly defined surface, Template:Math, the Gaussian curvature can be expressed in terms of the gradient Template:Math and Hessian matrix Template:Math:<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> <math display="block">
K = -\frac{
\begin{vmatrix}
H(F) & \nabla F^{\mathsf T} \\ \nabla F & 0 \end{vmatrix} }{ |\nabla F|^4 } =-\frac{ \begin{vmatrix} F_{xx} & F_{xy} & F_{xz} & F_x \\ F_{xy} & F_{yy} & F_{yz} & F_y \\ F_{xz} & F_{yz} & F_{zz} & F_z \\ F_{x} & F_{y} & F_{z} & 0 \\ \end{vmatrix} }{ |\nabla F|^4 } </math>
- For a surface with metric conformal to the Euclidean one, so Template:Math and Template:Math, the Gauss curvature is given by (Template:Math being the usual Laplace operator): <math display="block"> K = -\frac{1}{2e^\sigma}\Delta \sigma.</math>
- Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:<ref name="Bertrandtheorem">Bertrand–Diquet–Puiseux theorem</ref> <math display="block"> K = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3}</math>
- Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane:<ref name="Bertrandtheorem"/> <math display="block">K = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4 } </math>
- Gaussian curvature may be expressed with the Christoffel symbols:<ref>Template:Cite book</ref> <math display="block">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)</math>
See alsoEdit
- Earth's Gaussian radius of curvature
- Sectional curvature
- Mean curvature
- Gauss map
- Riemann curvature tensor
- Principal curvature