Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature $Κ$ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, $κ 1$ and $κ 2$ , at the given point:

Gaussian curvature is an intrinsic measure of curvature, depending 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.

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 or . 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.

saddle 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.

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 $κ 1$ , $κ 2$ . The Gaussian curvature is 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.

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[edit]

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 curvatures[edit]

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, $f$ , 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 the Hessian matrix of $f$ (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.

'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 called developable surfaces. Minding also raised the question of whether a closed surface with constant positive curvature is necessarily rigid.

Minding

's theorem (1900) answered Minding's question. The only regular (of class $C 2$ ) closed surfaces in $R 3$ with constant positive Gaussian curvature are spheres.^[2] 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.^[3]

Liebmann

(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. The pseudosphere has constant negative Gaussian curvature except at its boundary circle, where the gaussian curvature is not defined.

Hilbert's theorem

There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution $(\phi (v)\cos(u),\phi (v)\sin(u),\psi (v))$ where $\phi (v)=C\cos v$ , and ${\textstyle \psi (v)=\int _{0}^{v}{\sqrt {1-C^{2}\sin ^{2}v'}}\ dv'}$ (an incomplete 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 is pseudosphere.^[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]

Gaussian curvature of a surface in $R 3$ can be expressed as the ratio of the of the second and first fundamental forms $II$ and $I$ :
$K={\frac {\det(\mathrm {I\!I} )}{\det(\mathrm {I} )}}={\frac {LN-M^{2}}{EG-F^{2}}}.$

determinants

The Brioschi formula (after ) gives Gaussian curvature solely in terms of the first fundamental form:
$K={\frac {{\begin{vmatrix}-{\frac {1}{2}}E_{vv}+F_{uv}-{\frac {1}{2}}G_{uu}&{\frac {1}{2}}E_{u}&F_{u}-{\frac {1}{2}}E_{v}\\F_{v}-{\frac {1}{2}}G_{u}&E&F\\{\frac {1}{2}}G_{v}&F&G\end{vmatrix}}-{\begin{vmatrix}0&{\frac {1}{2}}E_{v}&{\frac {1}{2}}G_{u}\\{\frac {1}{2}}E_{v}&E&F\\{\frac {1}{2}}G_{u}&F&G\end{vmatrix}}}{\left(EG-F^{2}\right)^{2}}}$

Francesco Brioschi

For an parametrization ( $F = 0$ ), Gaussian curvature is:
$K=-{\frac {1}{2{\sqrt {EG}}}}\left({\frac {\partial }{\partial u}}{\frac {G_{u}}{\sqrt {EG}}}+{\frac {\partial }{\partial v}}{\frac {E_{v}}{\sqrt {EG}}}\right).$

orthogonal

For a surface described as graph of a function $z = F (x, y)$ , Gaussian curvature is:
$K={\frac {F_{xx}\cdot F_{yy}-F_{xy}^{2}}{\left(1+F_{x}^{2}+F_{y}^{2}\right)^{2}}}$

[6]

For an implicitly defined surface, $F (x, y, z) = 0$ , the Gaussian curvature can be expressed in terms of the gradient $\nabla F$ and $H (F)$ :^[7]^[8]
$K=-{\frac {\begin{vmatrix}H(F)&\nabla F^{\mathsf {T}}\\\nabla F&0\end{vmatrix}}{|\nabla F|^{4}}}=-{\frac {\begin{vmatrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\\end{vmatrix}}{|\nabla F|^{4}}}$

Hessian matrix

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 usual ):
$K=-{\frac {1}{2e^{\sigma }}}\Delta \sigma .$

Laplace operator

Gaussian curvature is the limiting difference between the of a geodesic circle and a circle in the plane:^[9]
$K=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}$

circumference

Gaussian curvature is the limiting difference between the of a geodesic disk and a disk in the plane:^[9]
$K=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}$

area

Gaussian curvature may be expressed with the :^[10]
$K=-{\frac {1}{E}}\left({\frac {\partial }{\partial u}}\Gamma _{12}^{2}-{\frac {\partial }{\partial v}}\Gamma _{11}^{2}+\Gamma _{12}^{1}\Gamma _{11}^{2}-\Gamma _{11}^{1}\Gamma _{12}^{2}+\Gamma _{12}^{2}\Gamma _{12}^{2}-\Gamma _{11}^{2}\Gamma _{22}^{2}\right)$

Christoffel symbols

Earth's Gaussian radius of curvature

Sectional curvature

Mean curvature

Gauss map

Riemann curvature tensor

Principal curvature

Grinfeld, P. (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. 978-1-4614-7866-9.

ISBN

Rovelli, Carlo (2021). General Relativity the Essentials. Cambridge University Press. 978-1-009-01369-7.

ISBN

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

Gaussian curvature

saddle point

Relation to geometries[edit]

Relation to principal curvatures[edit]

Minding

Liebmann

Hilbert's theorem

determinants

Francesco Brioschi

orthogonal

[6]

Hessian matrix

Laplace operator

circumference

area

Christoffel symbols

Earth's Gaussian radius of curvature

Sectional curvature

Mean curvature

Gauss map

Riemann curvature tensor

Principal curvature

ISBN

ISBN

"Gaussian curvature"