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 ). 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 .
“Extrinsic” methods, for surface in
- The most concrete case is the regular surfaces in . In this case often the easiest way to do is to use the Gauss equation (often taken as the definition):
where and is the second and first fundamental form ( w.r.t local coordinates ) respectively. This form can be written in many (almost equivalent) ways:
where are othonormal tangent vectors, is the second fundamental form, and are the principal curvatures. Another way of writing it is
where is the Gauss map, by , we mean the determinant of the matrix representation w.r.t. a basis of . Basic linear algebra then shows that this is independent of our choice of basis.
The mean curvature is the trace of the shape operator , i.e.
(Unfortunately, there is no common consensus about the sign (even has been chosen already), and I often omit the factor 1/2.) In my convention, the mean curvature of the unit sphere in is 2.
- There are some cases where and are easy to compute. E.g when is a graph of function and (this can always be done locally, by implicit function theorem), then at that point and is just the determinant and the trace (i.e. laplacian) of the hessian at .
- In the special case where the surface is umbilical (at that point), i.e. , or in other words every direction is a eigenvector of , so that the second fundamental form is constant. We just have to take a normal section at that point on the surface and compute its curvature , the curvatures are just .
- (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
where for some nowhere zero function , is orthonormal and . The point is that when is suitably chosen so that is simple, then it often simplifies the computation. As an example, take , then by taking (restricted on ), then (after some slightly tricky, but not complicated computation)
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 , 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 only, so there is no need to embed it into (and an abstract surface can’t always be embedded (isometrically) in ). The
- Of course Gauss Egregium tells us that depends only on (or, for a surface, its first fundamental form=induced metric from ), so there must be a formula involving only. However, this formula is not particularly enlightening, or is it easy to remember. Anyway, there is a formula like this:
where denotes the partials of w.r.t. 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.
(which unfortunately does not always exist)(which always exists when , actually there even exists isothermal coordinates, see below), then
This vastly simplifies the formula above.
- When the metric is isothermal (=conformally flat), i.e. there is a smooth function such that
where is the Euclidean metric. Then using (1), we have the simple formula
Here This formula is useful when calculating the curvature of a Riemann surface with metric ( is the symmetric tensor product )
There is some nice property when is a surface. Actually every metric on a two dimensional surface is locally conformally flat. i.e. for some non zero . In other words is a flat metric (in some neighborhood ). To see this, we will see later that (hopefully…) the scalar curvature
where , is the laplacian in . So for a given , we solve the Poisson equation (locally) in :
On a small open set , this equation is solvable. And thus has scalar curvature , ( in 2-dimension ). If has zero Gaussian curvature locally, this is isometric to (locally) (This is a standard fact, and is true even in space form = space with constant sectional curvature).
- In (geodesic) polar coordinates, if , then using (1) again, it’s easy to see that
This formula is very useful. Actually, let , then finding a metric of constant curvature is equivalent to solving the linear partial differential equations:
Note that there are no partial derivatives w.r.t. , so if we set the reasonable initial condition
If is easy to see that
We have proved that
Proposition 1 The metric with constant Gaussian curvature in geodesic polar coordinates (in 2-dimensional) is given by
The same is true in higher dimension (will talk about it later), just replace in the above by , the standard metric of . The proof is exactly the same.
“Intrinsic methods”, higher dimensional case
- Definition of sectional curvature, given a 2 plane (2-dim subspace) in :
- General formula
- General conformal metric
- Product metric
- 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.