Gaussian curvature

Indifferential geometry,theGaussian curvatureorGauss curvature $Κ$ of a smoothsurfacein three-dimensional space at a point is the product of theprincipal curvatures, $κ 1$ and $κ 2$ ,at the given point: $K=\kappa _{1}\kappa _{2}.$ For example, a sphere of radius $r$ has Gaussian curvature $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/r2⁠$ 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 ahyperboloidor the inside of atorus.

Gaussian curvature is anintrinsicmeasure ofcurvature,depending only on distances that are measured “within” or along the surface, not on the way it is isometricallyembeddedin Euclidean space. This is the content of theTheorema egregium.

Gaussian curvature is named afterCarl Friedrich Gauss,who published theTheorema egregiumin 1827.

Informal definition

Saddle surfacewith normal planes in directions of principal curvatures

At any point on a surface, we can find anormal vectorthat is at right angles to the surface; planes containing the normal vector are callednormal planes.The intersection of a normal plane and the surface will form a curve called anormal sectionand the curvature of this curve is thenormal curvature.For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called theprincipal curvatures,call these $κ 1$ , $κ 2$ .TheGaussian curvatureis the product of the two principal curvatures $Κ = κ 1 κ 2$ .

The sign of the Gaussian curvature can be used to characterise the surface.

If both principal curvatures are of the same sign: $κ 1 κ 2 > 0$ ,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: $κ 1 κ 2 < 0$ ,then the Gaussian curvature is negative and the surface is said to have a hyperbolic orsaddle 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 theasymptotic curvesfor that point.
If one of the principal curvatures is zero: $κ 1 κ 2 = 0$ ,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 aparabolic line.

Relation to geometries

When a surface has a constant zero Gaussian curvature, then it is adevelopable surfaceand the geometry of the surface isEuclidean geometry.

When a surface has a constant positive Gaussian curvature, then the geometry of the surface isspherical geometry.Spheresand patches of spheres have this geometry, but there exist other examples as well, such as thelemon / American football.

When a surface has a constant negative Gaussian curvature, then it is apseudospherical surfaceand the geometry of the surface ishyperbolic geometry.

Relation to principal curvatures

The twoprincipal curvaturesat a given point of asurfaceare theeigenvaluesof theshape operatorat the point. They measure how the surface bends by different amounts in different directions from that point. We represent the surface by theimplicit function theoremas the graph of a function, $f$ ,of two variables, in such a way that the point $p$ is a critical point, that is, 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 theHessian matrixof $f$ (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 $K={\frac {{\bigl \langle }(\nabla _{2}\nabla _{1}-\nabla _{1}\nabla _{2})\mathbf {e} _{1},\mathbf {e} _{2}{\bigr \rangle }}{\det g}},$ where $\nabla i = \nabla e i$ is thecovariant derivativeand $g$ is themetric tensor.

At a point $p$ on a regular surface in $R 3$ ,the Gaussian curvature is also given by $K(\mathbf {p} )=\det S(\mathbf {p} ),$ where $S$ is theshape operator.

A useful formula for the Gaussian curvature isLiouville's equationin terms of the Laplacian inisothermal coordinates.

Total curvature

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

Thesurface integralof the Gaussian curvature over some region of a surface is called thetotal curvature.The total curvature of ageodesic triangleequals 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 theEuclidean plane,the angles will sum to precisely $π$ radians. $\sum _{i=1}^{3}\theta _{i}=\pi +\iint _{T}K\,dA.$ A more general result is theGauss–Bonnet theorem.

Important theorems

Theorema egregium

Gauss'sTheorema 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 thefirst fundamental formand expressed via the first fundamental form and itspartial derivativesof first and second order. Equivalently, thedeterminantof thesecond fundamental formof a surface in $R 3$ can be so expressed. The "remarkable", and surprising, feature of this theorem is that although thedefinitionof the Gaussian curvature of a surface $S$ in $R 3$ certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by theintrinsic metricof the surface without any further reference to the ambient space: it is anintrinsic invariant.In particular, the Gaussian curvature is invariant underisometricdeformations of the surface.

In contemporarydifferential geometry,a "surface", viewed abstractly, is a two-dimensionaldifferentiable manifold.To connect this point of view with theclassical theory of surfaces,such an abstract surface isembeddedinto $R 3$ and endowed with theRiemannian metricgiven by the first fundamental form. Suppose that the image of the embedding is a surface $S$ in $R 3$ .Alocal isometryis adiffeomorphism $f : U \to V$ between open regions of $R 3$ whose restriction to $S \cap U$ is an isometry onto its image.Theorema egregiumis then stated as follows:

The Gaussian curvature of an embedded smooth surface in $R 3$ is invariant under the local isometries.

For example, the Gaussian curvature of acylindricaltube is zero, the same as for the "unrolled" tube (which is flat).^[1]^{[page needed]}On the other hand, since asphereof radius $R$ has constant positive curvature $R -2$ 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, nocartographic projectionis perfect.

Gauss–Bonnet theorem

The Gauss–Bonnet theorem relates the total curvature of a surface to itsEuler characteristicand provides an important link between local geometric properties and global topological properties.

\int _{M}K\,dA+\int _{\partial M}k_{g}\,ds=2\pi \chi (M),\,

Surfaces of constant curvature

Two surfaces which both have constant positive Gaussian curvature but with either an open boundary or singular points.

Minding's theorem(1839) states that all surfaces with the same constant curvature $K$ 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 calleddevelopable surfaces.Minding also raised the question of whether aclosed surfacewith constant positive curvature is necessarily rigid.
Liebmann's theorem(1900) answered Minding's question. The only regular (of class $C 2$ ) closed surfaces in $R 3$ with constant positive Gaussian curvature arespheres.^[2]If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof usesHilbert's lemmathat non-umbilicalpoints of extreme principal curvature have non-positive Gaussian curvature.^[3]
Hilbert's theorem(1901) states that there exists no complete analytic (class $C ω$ ) regular surface in $R 3$ of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class $C 2$ immersed in $R 3$ ,but breaks down for $C 1$ -surfaces. Thepseudospherehas 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 Carmoconsiders surfaces of revolution $(\phi (v)\cos(u),\phi (v)\sin(u),\psi (v))$ where $\phi (v)=C\cos v$ ,and ${\textstyle \psi (v)=\int _{0}^{v}{\sqrt {1-C^{2}\sin ^{2}v'}}\ dv'}$ (anincomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for $C\neq 1$ 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 ispseudosphere.^[4]

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.^[5]

Alternative formulas

Gaussian curvature of a surface in $R 3$ can be expressed as the ratio of thedeterminantsof thesecondandfirstfundamental forms $II$ and $I$ : $K={\frac {\det(\mathrm {I\!I} )}{\det(\mathrm {I} )}}={\frac {LN-M^{2}}{EG-F^{2}}}.$
TheBrioschi formula(afterFrancesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form: $K={\frac {{\begin{vmatrix}-{\frac {1}{2}}E_{vv}+F_{uv}-{\frac {1}{2}}G_{uu}&{\frac {1}{2}}E_{u}&F_{u}-{\frac {1}{2}}E_{v}\\F_{v}-{\frac {1}{2}}G_{u}&E&F\\{\frac {1}{2}}G_{v}&F&G\end{vmatrix}}-{\begin{vmatrix}0&{\frac {1}{2}}E_{v}&{\frac {1}{2}}G_{u}\\{\frac {1}{2}}E_{v}&E&F\\{\frac {1}{2}}G_{u}&F&G\end{vmatrix}}}{\left(EG-F^{2}\right)^{2}}}$
For anorthogonalparametrization( $F = 0$ ), Gaussian curvature is: $K=-{\frac {1}{2{\sqrt {EG}}}}\left({\frac {\partial }{\partial u}}{\frac {G_{u}}{\sqrt {EG}}}+{\frac {\partial }{\partial v}}{\frac {E_{v}}{\sqrt {EG}}}\right).$
For a surface described as graph of a function $z = F (x, y)$ ,Gaussian curvature is:^[6] $K={\frac {F_{xx}\cdot F_{yy}-F_{xy}^{2}}{\left(1+F_{x}^{2}+F_{y}^{2}\right)^{2}}}$
For an implicitly defined surface, $F (x, y, z) = 0$ ,the Gaussian curvature can be expressed in terms of the gradient $\nabla F$ andHessian matrix $H (F)$ :^[7]^[8] $K=-{\frac {\begin{vmatrix}H(F)&\nabla F^{\mathsf {T}}\\\nabla F&0\end{vmatrix}}{|\nabla F|^{4}}}=-{\frac {\begin{vmatrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\\end{vmatrix}}{|\nabla F|^{4}}}$
For a surface with metric conformal to the Euclidean one, so $F = 0$ and $E = G = e σ$ ,the Gauss curvature is given by ( $Δ$ being the usualLaplace operator): $K=-{\frac {1}{2e^{\sigma }}}\Delta \sigma.$
Gaussian curvature is the limiting difference between thecircumferenceof ageodesic circleand a circle in the plane:^[9] $K=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}$
Gaussian curvature is the limiting difference between theareaof ageodesic diskand a disk in the plane:^[9] $K=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}$
Gaussian curvature may be expressed with theChristoffel symbols:^[10] $K=-{\frac {1}{E}}\left({\frac {\partial }{\partial u}}\Gamma _{12}^{2}-{\frac {\partial }{\partial v}}\Gamma _{11}^{2}+\Gamma _{12}^{1}\Gamma _{11}^{2}-\Gamma _{11}^{1}\Gamma _{12}^{2}+\Gamma _{12}^{2}\Gamma _{12}^{2}-\Gamma _{11}^{2}\Gamma _{22}^{2}\right)$

References

^Porteous, I. R. (1994).Geometric Differentiation.Cambridge University Press.ISBN 0-521-39063-X.
^Kühnel, Wolfgang (2006).Differential Geometry: Curves, Surfaces, Manifolds.American Mathematical Society.ISBN 0-8218-3988-8.
^Gray, Alfred(1997). "28.4 Hilbert's Lemma and Liebmann's Theorem".Modern Differential Geometry of Curves and Surfaces with Mathematica(2nd ed.). CRC Press. pp. 652–654.ISBN 9780849371646..
^Carmo, Manfredo Perdigão do (2016) [1976].Differential geometry of curves and surfaces(2nd ed.). Mineola, NY: Dover Publications. p. 171.ISBN 978-0-486-80699-0– via zbMATH.
^Hilbert, David;Cohn-Vossen, Stephan(1952).Geometry and the Imagination(2nd ed.). Chelsea. p. 228.ISBN 0-8284-1087-9.
^"General investigations of curved surfaces of 1827 and 1825".[Princeton] The Princeton university library. 1902.
^Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces".Computer Aided Geometric Design.22(7): 632–658.CiteSeerX10.1.1.413.3008.doi:10.1016/j.cagd.2005.06.005.
^Spivak, M. (1975).A Comprehensive Introduction to Differential Geometry.Vol. 3. Boston: Publish or Perish.
^^a ^bBertrand–Diquet–Puiseux theorem
^Struik, Dirk (1988).Lectures on Classical Differential Geometry.Courier Dover Publications.ISBN 0-486-65609-8.

Books

Grinfeld, P. (2014).Introduction to Tensor Analysis and the Calculus of Moving Surfaces.Springer.ISBN 978-1-4614-7866-9.
Rovelli, Carlo (2021).General Relativity the Essentials.Cambridge University Press.ISBN 978-1-009-01369-7.

External links

"Gaussian curvature",Encyclopedia of Mathematics,EMS Press,2001 [1994]

[1] Porteous, I. R. (1994).Geometric Differentiation.Cambridge University Press.ISBN 0-521-39063-X.

[2] Kühnel, Wolfgang (2006).Differential Geometry: Curves, Surfaces, Manifolds.American Mathematical Society.ISBN 0-8218-3988-8.

[3] Gray, Alfred(1997). "28.4 Hilbert's Lemma and Liebmann's Theorem".Modern Differential Geometry of Curves and Surfaces with Mathematica(2nd ed.). CRC Press. pp. 652–654.ISBN 9780849371646..

[4] Carmo, Manfredo Perdigão do (2016) [1976].Differential geometry of curves and surfaces(2nd ed.). Mineola, NY: Dover Publications. p. 171.ISBN 978-0-486-80699-0– via zbMATH.

[5] Hilbert, David;Cohn-Vossen, Stephan(1952).Geometry and the Imagination(2nd ed.). Chelsea. p. 228.ISBN 0-8284-1087-9.

[6] "General investigations of curved surfaces of 1827 and 1825".[Princeton] The Princeton university library. 1902.

[7] Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces".Computer Aided Geometric Design.22(7): 632–658.CiteSeerX10.1.1.413.3008.doi:10.1016/j.cagd.2005.06.005.

[8] Spivak, M. (1975).A Comprehensive Introduction to Differential Geometry.Vol. 3. Boston: Publish or Perish.

[Bertrandtheorem-9] Bertrand–Diquet–Puiseux theorem

[10] Struik, Dirk (1988).Lectures on Classical Differential Geometry.Courier Dover Publications.ISBN 0-486-65609-8.

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