Gaussian curvature

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. 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

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 2 x 2 Hessian matrix of f (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is a symmetric matrix of second derivatives, which in Euclidean space is diagonalizable via the special case of the spectral theorem restricted to when H is equivalent to its conjugate transpose, i.e. H is real) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.

Alternative definitions

It is also given by $${\displaystyle K=\frac{⟨({\mathrm{\nabla }}_2{\mathrm{\nabla }}_1-{\mathrm{\nabla }}_1{\mathrm{\nabla }}_2){\mathbf{e}}_1,{\mathbf{e}}_2⟩}{detg},}$$ 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 $${\displaystyle K(\mathbf{p})=detS(\mathbf{p}),}$$ 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.

Total curvature

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. $${\displaystyle \sum _{i=1}^3{\theta }_i=\pi +{\iint }_{T}KdA.}$$ A more general result is the Gauss–Bonnet theorem.

Important theorems

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 derivatives of first and second order. Equivalently, the determinant of the second fundamental form of a surface in R³ can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R³ 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. In particular, the Gaussian curvature is invariant under 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 classical theory of surfaces, such an abstract surface is embedded into R³ and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface S in R³. A local isometry is a diffeomorphism f : U → V between open regions of R³ whose restriction to S ∩ U is an isometry onto its image. Theorema egregium is then stated as follows:

The Gaussian curvature of an embedded smooth surface in R³ is invariant under the local isometries.

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). On the other hand, since a sphere of 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, no cartographic projection can ever be perfect.

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.

${\displaystyle {\int }_{M}KdA+{\int }_{\mathrm{\partial }M}{k}_{g}ds=2\pi \chi (M),}$

Surfaces of constant curvature

• 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. 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.
• 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 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 ${\displaystyle (\varphi (v)\cos (u),\varphi (v)\sin (u),\psi (v))}$ where ${\displaystyle \varphi (v)=C\cos v}$, and $\psi (v)={\int }_0^v\sqrt{1-C^2{\sin }^2{v}^{\prime }}d{v}^{\prime }$ (an incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for ${\displaystyle C\ne 1}$ 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.

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.

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: $${\displaystyle K=\frac{det(\mathrm{I}\mathrm{I})}{det(\mathrm{I})}=\frac{LN-M^2}{EG-F^2}.}$$
• The Brioschi formula (after Francesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form: $${\displaystyle K=\frac{\left|\begin{array}{ccc}-\frac{1}{2}{E}_{vv}+F_{uv}-\frac{1}{2}{G}_{uu}& \frac{1}{2}{E}_u& F_u-\frac{1}{2}{E}_v\\ F_v-\frac{1}{2}{G}_u& E& F\\ \frac{1}{2}{G}_v& F& G\end{array}\right|-\left|\begin{array}{ccc}0& \frac{1}{2}{E}_v& \frac{1}{2}{G}_u\\ \frac{1}{2}{E}_v& E& F\\ \frac{1}{2}{G}_u& F& G\end{array}\right|}{{\left(EG-F^2\right)}^2}}$$
• For an orthogonal parametrization (F = 0), Gaussian curvature is: $${\displaystyle K=-\frac{1}{2\sqrt{EG}}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }u}\frac{G_u}{\sqrt{EG}}+\frac{\mathrm{\partial }}{\mathrm{\partial }v}\frac{E_v}{\sqrt{EG}}\right).}$$
• For a surface described as graph of a bivariate function z = F(x,y), Gaussian curvature is: $${\displaystyle K=\frac{F_{xx}\cdot F_{yy}-F_{xy}^2}{{\left(1+F_x^2+F_y^2\right)}^2}}$$
• 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): $${\displaystyle K=-\frac{\left|\begin{array}{cc}H(F)& \mathrm{\nabla }{F}^{\mathsf{T}}\\ \mathrm{\nabla }F& 0\end{array}\right|}{|\mathrm{\nabla }F{|}^4}=-\frac{\left|\begin{array}{cccc}{F}_{xx}& F_{xy}& F_{xz}& F_x\\ F_{xy}& F_{yy}& F_{yz}& F_y\\ F_{xz}& F_{yz}& F_{zz}& F_z\\ F_x& F_y& F_z& 0\end{array}\right|}{|\mathrm{\nabla }F{|}^4}}$$
• 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): $${\displaystyle K=-\frac{1}{2e^{\sigma }}\mathrm{\Delta }\sigma .}$$
• Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane: $${\displaystyle K=\underset{r\to 0^{+}}{lim}3\frac{2\pi r-C(r)}{\pi r^3}}$$
• Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane: $${\displaystyle K=\underset{r\to 0^{+}}{lim}12\frac{\pi r^2-A(r)}{\pi r^4}}$$
• Gaussian curvature may be expressed with the Christoffel symbols: $${\displaystyle K=-\frac{1}{E}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }u}{\mathrm{\Gamma }}_{12}^2-\frac{\mathrm{\partial }}{\mathrm{\partial }v}{\mathrm{\Gamma }}_{11}^2+{\mathrm{\Gamma }}_{12}^1{\mathrm{\Gamma }}_{11}^2-{\mathrm{\Gamma }}_{11}^1{\mathrm{\Gamma }}_{12}^2+{\mathrm{\Gamma }}_{12}^2{\mathrm{\Gamma }}_{12}^2-{\mathrm{\Gamma }}_{11}^2{\mathrm{\Gamma }}_{22}^2\right)}$$

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.