## Computing curvatures

This note is a summary of what I know about the methods of computing curvatures. By curvatures I usually mean Gaussian curvature (of an abstract surface) or sectional curvatures (of a plane in higher dimension), and sometimes perhaps, the mean curvature of some special submanifold (e.g. surface in ${\mathbb{R}^3}$). Of course this is not exhaustive (there are so many methods that I don’t know or not familiar), and I will not talk about how curvatures affect the manifold itself. We will often denote a manifold (or a surface) by ${M}$.

“Extrinsic” methods, for surface in ${\mathbb{R}^3}$

• The most concrete case is the regular surfaces in ${\mathbb{R}^3}$. In this case often the easiest way to do is to use the Gauss equation (often taken as the definition):

$\displaystyle K=\frac{ln-m^2}{EG-F^2}$

where ${\begin{pmatrix} l &m\\ m &n \end{pmatrix}}$ and ${\begin{pmatrix} E&F\\ F &G \end{pmatrix}}$ is the second and first fundamental form ( w.r.t local coordinates ${\{u, v\}}$) respectively. This form can be written in many (almost equivalent) ways:

$\displaystyle K=h(X, X)h(Y, Y)-h(X, Y)^2=k_1 k_2$

where ${X, Y}$ are othonormal tangent vectors, ${h}$ is the second fundamental form, and ${k_i}$ are the principal curvatures. Another way of writing it is

$\displaystyle K=\det (dN)$

where ${N: M\rightarrow \mathbb{S}^2}$ is the Gauss map, by ${\det dN}$, we mean the determinant of the matrix representation ${[dN]_\beta}$ w.r.t. a basis ${\beta}$ of ${T_pM}$. Basic linear algebra then shows that this is independent of our choice of basis.

The mean curvature is the trace of the shape operator ${dN}$, i.e.

$\displaystyle H=tr(dN)$

(Unfortunately, there is no common consensus about the sign (even ${N}$ has been chosen already), and I often omit the factor 1/2.) In my convention, the mean curvature of the unit sphere in ${\mathbb{R}^3}$ is 2.

• There are some cases where ${K}$ and ${H}$ are easy to compute. E.g when ${M}$ is a graph of function ${f:\mathbb{R}^2\rightarrow R}$ and ${\nabla f(0, 0)=0}$ (this can always be done locally, by implicit function theorem), then at that point ${K}$ and ${H}$ is just the determinant and the trace (i.e. laplacian) of the hessian ${\frac{\partial^2 f}{\partial x\partial y}}$ at ${0}$.
• In the special case where the surface is umbilical (at that point), i.e. ${k_1=k_2=k}$, or in other words every direction is a eigenvector of ${dN}$, so that the second fundamental form is constant. We just have to take a normal section ${\gamma}$ at that point on the surface and compute its curvature ${k}$, the curvatures are just ${K=k^2, H=2k}$.
• (Other methods) There are some other very special methods which I can’t always remember. E.g. it can be shown (I learned it from doCarmo) that

$\displaystyle K=\frac{\langle d\tilde N(u)\times d\tilde N(v), \tilde N \rangle}{f^3}$

where ${\tilde N=fN}$ for some nowhere zero function ${f}$, ${u, v\in T_pM}$ is orthonormal and ${u\times v=N}$. The point is that when ${f}$ is suitably chosen so that ${\tilde N}$ is simple, then it often simplifies the computation. As an example, take ${M=\{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}}$, then by taking ${f=\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}}$ (restricted on ${M}$), then (after some slightly tricky, but not complicated computation)

$\displaystyle K=\frac{1}{a^2b^2c^2 f^4}.$

This can be regarded as a variation of the Gauss equation.

“Intrinsic” methods, for abstract surface.
In the following I will describe some other methods in computing ${K}$, or more generally the sectional curvatures. Often I will concentrate on the surface case, ie. dimension =2. However, they are intrinsic is nature, i.e. depends on the metric ${g}$ only, so there is no need to embed it into ${\mathbb{R}^3}$ (and an abstract surface can’t always be embedded (isometrically) in ${\mathbb{R}^3}$). The

• Of course Gauss Egregium tells us that ${K}$ depends only on ${g}$ (or, for a surface, its first fundamental form=induced metric from ${\mathbb{R}^3}$), so there must be a formula involving ${E, F, G}$ only. However, this formula is not particularly enlightening, or is it easy to remember. Anyway, there is a formula like this:

$\displaystyle -EK=(\Gamma_{12}^2)_{u}- (\Gamma_{11}^2)_{v} + \Gamma_{12}^1 \Gamma_{11}^2 +\Gamma_{12}^2 \Gamma_{12}^2 -\Gamma_{11}^2 \Gamma_{22}^2 -\Gamma_{11}^1 \Gamma_{12}^2$

where ${f_{u}}$ denotes the partials of ${f}$ w.r.t. ${u}$ etc. Well, I think I can safely say that not many people will border to remember this ugly looking formula anyway. (And the indices don’t seem to have a very good pattern, actually there is a more systematic way to write it (but still slightly complicated), but let’s wait until higher dimensional case ).

• Here comes a nicer formula, which sometimes is quite useful. If there is an orthogonal parametrization, i.e. ${F=0}$ (which unfortunately does not always exist) (which always exists when $dim M=2$, actually there even exists isothermal coordinates, see below), then

$\displaystyle K=-\frac{1}{\sqrt{EG}}\left( \left (\frac{E_v}{\sqrt{EG}}\right)_v - \left(\frac{G_u}{\sqrt{EG}}\right)_u \right) \ \ \ \ \ (1)$

This vastly simplifies the formula above.

• When the metric is isothermal (=conformally flat), i.e. there is a smooth function ${\lambda}$ such that

$\displaystyle g=\lambda^2 \delta$

where ${\delta=dx^2+dy^2}$ is the Euclidean metric. Then using (1), we have the simple formula

$\displaystyle K=-\frac{1}{\lambda}\Delta (\log \lambda).$

Here ${\Delta f=\frac{\partial ^2 f}{\partial u^2}+\frac{\partial ^2 f}{\partial v^2}}$This formula is useful when calculating the curvature of a Riemann surface with metric ${\lambda(z)^2 dzd\bar z}$ (${dzd\bar z}$ is the symmetric tensor product ${dz\otimes_s d\bar z}$)

• There is some nice property when ${M}$ is a surface. Actually every metric ${g}$ on a two dimensional surface is locally conformally flat. i.e. ${g=v \delta}$ for some non zero ${v}$. In other words ${v^{-1}g}$ is a flat metric (in some neighborhood ${U}$). To see this, we will see later that (hopefully…) the scalar curvature

$\displaystyle R_{\tilde g}=e^{-u}(R_g -\Delta_g u)$

where ${\tilde g=e^u g}$, ${\Delta_g}$ is the laplacian in ${g}$. So for a given ${g}$, we solve the Poisson equation (locally) in ${u}$:

$\displaystyle \Delta_g u=R_g.$

On a small open set ${U}$, this equation is solvable. And thus ${e^{u}g}$ has scalar curvature ${R({e^u g})=2K({e^u g})=0}$, (${R=2K}$ in 2-dimension ). If ${M^2}$ has zero Gaussian curvature locally, this is isometric to ${\mathbb{R}^2}$ (locally) (This is a standard fact, and is true even in space form = space with constant sectional curvature).

• In (geodesic) polar coordinates, if ${g=dr^2 +G(r, \theta)d\theta^2}$, then using (1) again, it’s easy to see that

$\displaystyle K=-\frac{(\sqrt{G})_{rr}}{\sqrt{G}}.$

This formula is very useful. Actually, let ${f(r, \theta)=\sqrt{G}}$, then finding a metric ${g}$ of constant curvature ${K}$ is equivalent to solving the linear partial differential equations:

$\displaystyle f_{rr}+K f=0$

Note that there are no partial derivatives w.r.t. ${\theta}$, so if we set the reasonable initial condition

$\displaystyle f(0, \theta)=0, f_r(0, \theta)=1$

If is easy to see that

$\displaystyle f=\left\{\begin{matrix}\sin r&\text{ if K=1,}\\ r&\text{ if K=0,}\\\sinh r&\text{ if K=-1.}\end{matrix}\right.$

We have proved that

Proposition 1 The metric ${g}$ with constant Gaussian curvature ${K}$ in geodesic polar coordinates (in 2-dimensional) is given by

$\displaystyle g=\left\{\begin{matrix}dr^2+\sin^2 r d\theta^2 &\text{ if K=1,}\\dr^2+r^2d\theta^2 &\text{ if K=0,}\\dr^2+\sinh^2 rd\theta^2&\text{ if K=-1.}\end{matrix}\right.$

The same is true in higher dimension (will talk about it later), just replace ${d\theta^2}$ in the above by ${g_{\mathbb{S}^{n-1}}}$, the standard metric of ${\mathbb{S}^{n-1}}$. The proof is exactly the same.

“Intrinsic methods”, higher dimensional case

• Definition of sectional curvature, given a 2 plane (2-dim subspace) in ${T_pM}$:

$\displaystyle K(v, w)=...$

• General formula

$\displaystyle R_{ijk}^l=\partial\Gamma-\partial\Gamma+\Gamma\Gamma-\Gamma\Gamma$

• General conformal metric ${\tilde g=e^{2u}g}$
• Product metric ${M\times N}$
• More general polar coordinates
• Moving plane methods. This is the method I am most unfamiliar with!!! @.@
How’s curvature 2-form related to the original definition of curvatures?

• Space form (3 cases)/ rotationally symmetric metric. Sectional curvatures. Mean curvature of coordinate sphere in this case.
• Gauss equation. Moving frame adapted to a hypersurface. Mean curvature.

• General coordinates expansion repressed in term of curvatures. This is hard… Volume of small balls and area of small sphere.