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| [[Image:Gaussian curvature.PNG|thumb|From left to right: a surface of negative Gaussian curvature ([[hyperboloid]]), a surface of zero Gaussian curvature ([[cylinder (geometry)|cylinder]]), and a surface of positive Gaussian curvature ([[sphere]]).]]
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| In [[differential geometry]], the '''Gaussian curvature''' or '''Gauss curvature''' of a point on a [[surface]] is the product of the [[principal curvature]]s, ''κ''<sub>1</sub> and ''κ''<sub>2</sub>, 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 isometrically [[embedding|embedded]] in space. This result is the content of [[Carl Friedrich Gauss|Gauss's]] [[Theorema egregium]].
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| Symbolically, the Gaussian [[curvature]] ''Κ'' is defined as
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| :<math> \Kappa = \kappa_1 \kappa_2,</math>
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| where κ<sub>1</sub> and κ<sub>2</sub> are the [[principal curvature]]s.
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| ==Informal definition==
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| [[File:Minimal surface curvature planes-en.svg|thumb|300px|right|[[Saddle surface]] with normal planes in directions of principal curvatures]]
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| At any point on a surface we can find a [[Normal (geometry)|normal vector]] which is at right angles to the surface. The intersection of a plane containing the normal with 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 surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the [[principal curvature]]s, call these κ<sub>1</sub>, κ<sub>2</sub>. The '''Gaussian curvature''' is the product of the two principal curvatures Κ = κ<sub>1</sub> κ<sub>2</sub>.
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| The sign of the Gaussian curvature can be used to characterise the surface.
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| *If both principal curvatures are the same sign: κ<sub>1</sub>κ<sub>2</sub> > 0, 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.
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| *If the principal curvatures have different signs: κ<sub>1</sub>κ<sub>2</sub> < 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic point. At such points the surface will be saddle shaped. For two directions the sectional curvatures will be zero giving the [[asymptotic direction]]s.
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| *If one of the principal curvature is zero: κ<sub>1</sub>κ<sub>2</sub> = 0, the Gaussian curvature is zero and the surface is said to have a parabolic point.
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| 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]].
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| ==Further informal discussion==
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| In [[differential geometry]], 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 at that point. We represent the surface by the [[implicit function theorem]] as the graph of a function, ''f'', of two variables, in such a way that 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'' (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2-by-2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between cup/cap ''versus'' saddle point.
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| ==Alternative definitions==
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| It is also given by
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| : <math>\Kappa = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g},</math>
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| where <math>\nabla_i = \nabla_{{\mathbf e}_i}</math> is the [[covariant derivative]] and ''g'' is the [[metric tensor]].
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| At a point '''p''' on a regular surface in '''R'''<sup>''3''</sup>, the Gaussian curvature is also given by
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| : <math>K(\mathbf{p}) = \det(S(\mathbf{p})),</math>
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| where ''S'' is the [[shape operator]].
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| A useful formula for the Gaussian curvature is [[Liouville equations|Liouville's equation]] in terms of the Laplacian in [[isothermal coordinates]].
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| ==Total curvature==
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| [[Image:Hyperbolic triangle.svg|thumb|The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle.]]
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| 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.
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| :<math>\sum_{i=1}^3 \theta_i = \pi + \iint_T K \,dA.</math>
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| A more general result is the [[Gauss–Bonnet theorem]].
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| ==Important theorems==
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| ===Theorema egregium===
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| {{main|Theorema Egregium}}
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| 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 derivative]]s of first and second order. Equivalently, the [[determinant]] of the [[second fundamental form]] of a surface in '''R'''<sup>3</sup> can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the ''definition'' of the Gaussian curvature of a surface ''S'' in '''R'''<sup>3</sup> 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 (mathematics)|invariant]]. In particular, the Gaussian curvature is invariant under [[isometry|isometric]] deformations of the surface.
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| In contemporary [[differential geometry]], a "surface", viewed abstractly, is a two-dimensional [[differentiable manifold]]. To connect this point of view with the [[differential geometry of surfaces|classical theory of surfaces]], such an abstract surface is [[embedding|embedded]] into '''R'''<sup>3</sup> and endowed with the [[Riemannian metric]] given by the first fundamental form. Suppose that the image of the embedding is a surface ''S'' in '''R'''<sup>3</sup>. A ''local isometry'' is a [[diffeomorphism]] ''f'': ''U'' → ''V'' between open regions of '''R'''<sup>3</sup> whose restriction to ''S'' ∩ ''U'' is an [[isometry]] onto its image. '''Theorema Egregium''' is then stated as follows:
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| : The Gaussian curvature of an embedded smooth surface in '''R'''<sup>3</sup> is invariant under the local isometries.
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| For example, the Gaussian curvature of a [[cylinder (geometry)|cylindrical]] tube is zero, the same as for the "unrolled" tube (which is flat).<ref>Porteous, I. R., ''Geometric Differentiation''. Cambridge University Press, 1994. ISBN 0-521-39063-X</ref> On the other hand, since a [[sphere]] of radius ''R'' has constant positive curvature ''R''<sup>−2</sup> and a flat plane has constant curvature 0, these two surfaces are not isometric, even locally. Thus any planar representation of even a part of a sphere must distort the distances. Therefore, no [[cartographic projection]] is perfect.
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| ===Gauss–Bonnet theorem===
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| {{main|Gauss-Bonnet theorem}}
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| 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.
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| == Surfaces of constant curvature ==
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| *'''[[Ferdinand Minding|Minding]]'s theorem''' (1839) states that all surfaces with the same constant curvature ''K'' are locally [[Isometry|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 surface]]s. Minding also raised the question whether a [[closed surface]] with constant positive curvature is necessarily rigid.
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| *'''Liebmann's theorem''' (1900) answered Minding's question. The only regular (of class ''C''<sup>2</sup>) closed surfaces in '''R'''<sup>3</sup> with constant positive Gaussian curvature are [[sphere]]s.<ref>{{cite book | last = Kühnel | first = Wolfgang | title = Differential Geometry: Curves - Surfaces - Manifolds | publisher = American Mathematical Society | year = 2006 | isbn = 0-8218-3988-8}}</ref>
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| *'''[[Hilbert's theorem (differential geometry)|Hilbert's theorem]]''' (1901) states that there exists no complete analytic (class ''C''<sup>''ω''</sup>) regular surface in '''R'''<sup>3</sup> of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class ''C''<sup>2</sup> immersed in '''R'''<sup>3</sup>, but breaks down for ''C''<sup>1</sup>-surfaces. The [[pseudosphere]] has constant negative Gaussian curvature except at its singular [[cusp (singularity)|cusp]].<ref>[http://eom.springer.de/h/h047410.htm ''Hilbert theorem''. Springer Online Reference Works.]</ref>
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| ==Alternative formulas==
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| *Gaussian curvature of a surface in '''R'''<sup>3</sup> can be expressed as the ratio of the [[determinant]]s of the [[second fundamental form|second]] and [[first fundamental form|first]] fundamental forms:
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| ::<math>K = \frac{\det II}{\det I} = \frac{LN-M^2}{EG-F^2}.</math>
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| *The '''Brioschi formula''' gives Gaussian curvature solely in terms of the first fundamental form:
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| ::<math> K =\frac{\det \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}- \det \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} </math>
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| *For an '''[[orthogonal coordinates|orthogonal]] parametrization''' (i.e., ''F'' = 0), Gaussian curvature is:
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| ::<math>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>
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| *For a surface described as graph of a function ''z'' = ''F(x, y)'', Gaussian curvature is:
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| ::<math>K = \frac{F_{xx}\cdot F_{yy}- F_{xy}^2}{(1+F_x^2+ F_y^2)^2}</math>
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| * For a surface ''F(x,y,z)'' = 0, Gaussian curvature is:<ref>[http://mathworld.wolfram.com/GaussianCurvature.html Gaussian Curvature on Wolfram MathWorld]</ref>
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| ::<math>K=\frac{[F_z(F_{xx}F_z-2F_xF_{xz})+F_x^2F_{zz}][F_z(F_{yy}F_z-2F_yF_{yz})+F_y^2F_{zz}]-[F_z(-F_xF_{yz}+F_{xy}F_z-F_{xz}F_y)+F_xF_yF_{zz}]^2}{F_z^2(F_x^2+F_y^2+F_z^2)^2}</math>
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| <!-- This one is too long, cannot figure out how to divide it nicely-->
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| * For a surface with metric conformal to the Euclidean one, so ''F'' = 0 and ''E'' = ''G'' = e<sup>σ</sup>, the Gauss curvature is given by (Δ being the usual [[Laplace operator]]):
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| ::<math> K = -\frac{1}{2e^\sigma}\Delta \sigma,</math>
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| *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>
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| ::<math> K = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3}</math>
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| *Gaussian curvature is the limiting difference between the '''[[area]] of a geodesic disk''' and a disk in the plane:<ref name="Bertrandtheorem"/>
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| ::<math>K = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4 } </math>
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| *Gaussian curvature may be expressed with the '''[[Christoffel symbols]]''':<ref>{{cite book | last = Struik | first = Dirk| title = Lectures on Classical Differential Geometry | publisher = Courier Dover
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| Publications | year = 1988 | isbn = 0-486-65609-8}}</ref>
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| ::<math>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>
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| ==See also==
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| * [[Sectional curvature]]
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| * [[Mean curvature]]
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| * [[Gauss map]]
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| == References ==
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| {{Reflist}}
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| ==External links==
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| * {{springer|title=Gaussian curvature|id=p/g043590}}
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| * [http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.3 Curvature in two spacelike dimensions]
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| {{curvature}}
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| [[Category:Curvature (mathematics)]]
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| [[Category:Differential geometry]]
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Surfaces]]
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