View source for Gaussian curvature

{{Short description|Product of the principal curvatures of a surface}}
[[Image:Gaussian curvature.svg|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]]).]]
[[File:Torus Positive and negative curvature.png|thumb|Some points on the torus have positive, some have negative, and some have zero Gaussian curvature.]]

In [[differential geometry]], the '''Gaussian curvature''' or '''Gauss curvature''' {{mvar|Κ}} of a smooth [[Surface (topology)|surface]] in three-dimensional space at a point is the product of the [[principal curvature]]s, {{math|''κ''<sub>1</sub>}} and {{math|''κ''<sub>2</sub>}}, at the given point:
<math display="block"> K = \kappa_1 \kappa_2.</math>
For example, a sphere of radius {{mvar|r}} has Gaussian curvature {{math|{{sfrac|1|''r''<sup>2</sup>}}}} 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 [[embedding|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 definition==
[[File:Minimal surface curvature planes-en.svg|thumb|300px|right|[[Saddle surface]] with normal planes in directions of principal curvatures]]

At any point on a surface, we can find a [[Normal (geometry)|normal vector]] that is at right angles to the surface; planes containing the normal vector are called ''[[normal plane (geometry)|normal plane]]s''. 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 curvature]]s, call these {{math|''κ''<sub>1</sub>}}, {{math|''κ''<sub>2</sub>}}. The '''Gaussian curvature''' is the product of the two principal curvatures {{math|''Κ'' {{=}} ''κ''<sub>1</sub>''κ''<sub>2</sub>}}.

The sign of the Gaussian curvature can be used to characterise the surface.
*If both principal curvatures are of the same sign: {{math|''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. 
*If the principal curvatures have different signs: {{math|''κ''<sub>1</sub>''κ''<sub>2</sub> < 0}}, 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 curve]]s for that point.
*If one of the principal curvatures is zero: {{math|''κ''<sub>1</sub>''κ''<sub>2</sub> {{=}} 0}}, 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 geometries==

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]]. [[Sphere]]s and patches of spheres have this geometry, but there exist other examples as well, such as the [[lemon (geometry)|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 curvatures==
The two '''principal curvatures''' at a given point of a [[Surface (topology)|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, {{mvar|f}}, of two variables, in such a way that the point {{mvar|p}} is a critical point, that is, the gradient of {{mvar|f}} vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at {{mvar|p}} is the determinant of the [[Hessian matrix]] of {{mvar|f}} (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 definitions==
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 {{math|∇<sub>''i''</sub> {{=}} ∇<sub>'''e'''<sub>''i''</sub></sub>}} is the [[covariant derivative]] and {{mvar|g}} is the [[metric tensor]].

At a point {{math|'''p'''}} on a regular surface in {{math|'''R'''<sup>3</sup>}}, the Gaussian curvature is also given by
<math display="block">K(\mathbf{p}) = \det S(\mathbf{p}),</math>
where {{mvar|S}} is the [[shape operator]].

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

==Total curvature==
[[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.]]

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 {{pi}}. The sum of the angles of a triangle on a surface of positive curvature will exceed {{pi}}, while the sum of the angles of a triangle on a surface of negative curvature will be less than {{pi}}. On a surface of zero curvature, such as the [[Euclidean plane]], the angles will sum to precisely {{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 theorems==

===''Theorema egregium''===
{{main|Theorema egregium}}
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 {{math|'''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 {{mvar|S}} in {{math|'''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 (Riemannian geometry)|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 [[differential geometry of surfaces|classical theory of surfaces]], such an abstract surface is [[embedding|embedded]] into {{math|'''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 {{mvar|S}} in {{math|'''R'''<sup>3</sup>}}. A ''local isometry'' is a [[diffeomorphism]] {{math|''f'' : ''U'' → ''V''}} between open regions of {{math|'''R'''<sup>3</sup>}} whose restriction to {{math|''S'' ∩ ''U''}} is an isometry onto its image. ''Theorema egregium'' is then stated as follows:

{{Equation box 1|indent=:|equation=The Gaussian curvature of an embedded smooth surface in {{math|'''R'''<sup>3</sup>}} is invariant under the local isometries.}}

For example, the Gaussian curvature of a [[cylinder (geometry)|cylindrical]] tube is zero, the same as for the "unrolled" tube (which is flat).<ref>{{cite book|last=Porteous|first=I. R.|title=Geometric Differentiation|url=https://archive.org/details/geometricdiffere0000port|url-access=registration|publisher=Cambridge University Press|date=1994|isbn=0-521-39063-X}}</ref>{{page needed|date=August 2019}} On the other hand, since a [[sphere]] of radius {{mvar|R}} has constant positive curvature {{math|''R''<sup>−2</sup>}} 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 theorem===
{{main|Gauss–Bonnet theorem}}
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 curvature ==
[[File:Surfaces of constant positive Gaussian curvature.png|thumb|Two surfaces which both have constant positive Gaussian 
curvature but with either an open boundary or singular points.]]
*'''[[Ferdinand Minding|Minding]]'s theorem''' (1839) states that all surfaces with the same constant curvature {{mvar|K}} 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 surface]]s. Minding also raised the question of whether a [[closed surface]] with constant positive curvature is necessarily rigid.
*'''[[Heinrich Liebmann|Liebmann]]'s theorem''' (1900) answered Minding's question. The only regular (of class {{math|''C''<sup>2</sup>}}) closed surfaces in {{math|'''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> 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 point|umbilical]] points of extreme principal curvature have non-positive Gaussian curvature.<ref>{{cite book|title=Modern Differential Geometry of Curves and Surfaces with Mathematica|first=Alfred|last=Gray|author-link=Alfred Gray (mathematician)|edition=2nd|publisher=CRC Press|year=1997|isbn=9780849371646|contribution=28.4 Hilbert's Lemma and Liebmann's Theorem|pages=652–654|url=https://books.google.com/books?id=-LRumtTimYgC&pg=PA652}}.</ref>
*'''[[Hilbert's theorem (differential geometry)|Hilbert's theorem]]''' (1901) states that there exists no complete analytic (class {{math|''C''<sup>''ω''</sup>}}) regular surface in {{math|'''R'''<sup>3</sup>}} of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class {{math|''C''<sup>2</sup>}} immersed in {{math|'''R'''<sup>3</sup>}}, but breaks down for {{math|''C''<sup>1</sup>}}-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 [[Elliptic integral#Incomplete elliptic integral of the second kind|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>{{cite book|first1=Manfredo Perdigão do|last1=Carmo |title=Differential geometry of curves and surfaces|url=https://zbmath.org/?q=an%3A1352.53002|publisher=Dover Publications |location=Mineola, NY|isbn=978-0-486-80699-0|via=zbMATH|page=171|orig-date=1976|date=2016|edition=2nd}}</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>{{cite book
  |author1=Hilbert, David
  |author-link1=David Hilbert
  |author2=Cohn-Vossen, Stephan
  |author-link2=Stefan Cohn-Vossen
  | title = Geometry and the Imagination
  | url = https://archive.org/details/geometryimaginat00davi_0 | url-access = registration| edition = 2nd
  | year = 1952
  | publisher = Chelsea
  | isbn = 0-8284-1087-9
  | page = 228}}
</ref>

==Alternative formulas==
*Gaussian curvature of a surface in {{math|'''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 {{math|II}} and {{math|I}}: <math display="block">K = \frac{\det(\mathrm{I\!I})}{\det(\mathrm I)} = \frac{LN-M^2}{EG-F^2}.</math>
*The '''{{vanchor|Brioschi formula}}''' (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 coordinates|orthogonal]] parametrization'' ({{math|''F'' {{=}} 0}}), 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 {{math|''z'' {{=}} ''F''(''x'',''y'')}}, Gaussian curvature is:<ref>{{Cite web|url=https://archive.org/details/cu31924001557226/page/n25/mode/2up|title = General investigations of curved surfaces of 1827 and 1825|year = 1902|publisher = [Princeton] The Princeton university library}}</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, {{math|''F''(''x'',''y'',''z'') {{=}} 0}}, the Gaussian curvature can be expressed in terms of the gradient {{math|∇''F''}} and [[Hessian matrix]] {{math|''H''(''F'')}}:<ref>{{Cite journal | doi = 10.1016/j.cagd.2005.06.005 | title = Curvature formulas for implicit curves and surfaces| journal = Computer Aided Geometric Design| volume = 22| issue = 7| pages = 632–658| year = 2005| last1 = Goldman | first1 = R. |citeseerx=10.1.1.413.3008}}</ref><ref>{{cite book|last=Spivak|first=M.|year=1975|title=A Comprehensive Introduction to Differential Geometry|volume=3|publisher=Publish or Perish|location=Boston}}</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 {{math|''F'' {{=}} 0}} and {{math|''E'' {{=}} ''G'' {{=}} ''e<sup>σ</sup>''}}, the Gauss curvature is given by ({{math|Δ}} 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>{{cite book | last = Struik | first = Dirk| title = Lectures on Classical Differential Geometry | publisher = Courier Dover Publications | year = 1988 | isbn = 0-486-65609-8}}</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 also==
* [[Earth radius#Gaussian radius of curvature|Earth's Gaussian radius of curvature]]
* [[Sectional curvature]]
* [[Mean curvature]]
* [[Gauss map]]
* [[Riemann curvature tensor]]
* [[Principal curvature]]

== References ==
{{Reflist}}

== Books ==
*{{cite book| first=P.|last=Grinfeld| title=Introduction to Tensor Analysis and the Calculus of Moving Surfaces | publisher=Springer| year=2014 | isbn=978-1-4614-7866-9}}
*{{cite book| first=Carlo|last=Rovelli| title=General Relativity the Essentials | publisher=Cambridge University Press | year=2021 | isbn=978-1-009-01369-7}}

==External links==
* {{springer|title=Gaussian curvature|id=p/g043590}}

{{curvature}}
{{Carl Friedrich Gauss}}

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