Gaussian curvature

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.^[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 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:^[6]
  $${\displaystyle K={\frac {F_{xx}\cdot F_{yy}-F_{xy}^{2}}{\left(1+F_{x}^{2}+F_{y}^{2}\right)^{2}}}}$$

  
• 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):^[7][8]
  $${\displaystyle K=-{\frac {\begin{vmatrix}H(F)&\nabla F^{\mathsf {T}}\\\nabla F&0\end{vmatrix}}{|\nabla F|^{4}}}=-{\frac {\begin{vmatrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\\end{vmatrix}}{|\nabla F|^{4}}}}$$

  
• 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:^[9]
  $${\displaystyle K=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}}$$

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

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

  

See also

• Earth's Gaussian radius of curvature
• Sectional curvature
• Mean curvature
• Gauss map
• Riemann curvature tensor
• Principal curvature

References