Gaussian curvature

Saddle surface with normal planes in directions of principal curvatures

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 κ₁, κ₂. The Gaussian curvature is the product of the two principal curvatures Κ = κ₁κ₂.

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

• If both principal curvatures are of the same sign: κ₁κ₂ > 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: κ₁κ₂ < 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 curves for that point.
• If one of the principal curvatures is zero: κ₁κ₂ = 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.

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 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.

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, 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.

It is also given by

$${\displaystyle K={\frac {{\bigl \langle }(\nabla _{2}\nabla _{1}-\nabla _{1}\nabla _{2})\mathbf {e} _{1},\mathbf {e} _{2}{\bigr \rangle }}{\det g}},}$$

where ∇_i = ∇_ei is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in R³, the Gaussian curvature is also given by

$${\displaystyle K(\mathbf {p} )=\det S(\mathbf {p} ),}$$

where S is the shape operator.

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

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 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 radians.

$${\displaystyle \sum _{i=1}^{3}\theta _{i}=\pi +\iint _{T}K\,dA.}$$

A more general result is the Gauss–Bonnet theorem.

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 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 C²) closed surfaces in R³ 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]
• Hilbert's theorem (1901) states that there exists no complete analytic (class C^ω) regular surface in R³ of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class C² immersed in R³, but breaks down for C¹-surfaces. The pseudosphere has constant negative Gaussian curvature except at its singular cusp.[4]

There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution ${\displaystyle (\phi (v)\cos(u),\phi (v)\sin(u),\psi (v))}$ where ${\displaystyle \phi (v)=C\cos v}$, 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 ${\displaystyle C\neq 1}$ 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.[5]

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.[6]

• Gaussian curvature of a surface in R³ can be expressed as the ratio of the determinants of the second and first fundamental forms II and I:
  $${\displaystyle K={\frac {\det(\mathrm {I\!I} )}{\det(\mathrm {I} )}}={\frac {LN-M^{2}}{EG-F^{2}}}.}$$

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

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

  
• For a surface described as graph of a function z = F(x,y), Gaussian curvature is:[7]
  $${\displaystyle K={\frac {F_{xx}\cdot F_{yy}-F_{xy}^{2}}{\left(1+F_{x}^{2}+F_{y}^{2}\right)^{2}}}}$$

  
• For an implicitly defined surface, F(x,y,z) = 0, the Gaussian curvature can be expressed in terms of the gradient ∇F and Hessian matrix H(F):[8][9]
  $${\displaystyle K=-{\frac {\begin{vmatrix}H(F)&\nabla F^{\mathsf {T}}\\\nabla F&0\end{vmatrix}}{|\nabla F|^{4}}}=-{\frac {\begin{vmatrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\\end{vmatrix}}{|\nabla F|^{4}}}}$$

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

  
• Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:[10]
  $${\displaystyle K=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}}$$

  
• Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane:[10]
  $${\displaystyle K=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}}$$

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

  

• 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. ISBN 978-1-4614-7866-9.
• Rovelli, Carlo (2021). General Relativity the Essentials. Cambridge University Press. ISBN 978-1-009-01369-7.

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