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.
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