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Massive bodies warp spacetime. Image courtesy NASA.
The Gromov-Hausdorff metric we explored in the previous articles lets us see compact metric spaces as themselves the points of a metric space. In this article we shall sketch two contexts where it is helpful to think of compact Riemannian manifolds as points of this space. The first is Einstein's general relativity, and the second is Perelman's proof of the Poincaré conjecture.
Newton's law of gravity says that each particle in the Universe attracts every other particle by instantaneous unmediated "action at a distance". This did not fit well with Maxwell's much later picture in which electric and magnetic attraction is mediated by the electromagnetic field. Later still the theory of special relativity made the idea of instantaneous interaction between distant particles completely untenable. Einstein had the wonderful idea that a gravitational field is a warping or curvature of space-time itself: its apparent action on particles is just the fact that they travel along geodesics in the curved manifold. His law of gravitation describes how the presence of matter produces the curvature of space-time. To put some flesh on this statement we need a digression about Riemannian manifolds and the description of curvature.
Riemannian manifolds
It is easy to imagine an
When
The tangent space
In
Among the intrinsic features of a Riemannian manifold are its geodesics. These are parametrized curves which minimize length between any two sufficiently close points on them. There is a unique geodesic emanating from any point
Curvature
The curvature of a curve in the plane is the rate at which its direction changes when one travels along the curve with unit speed. This is entirely a property of how the curve sits in the plane. But things are more subtle for a surface in
If
It turns out that part of the curvature of the surface is intrinsic. The product
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A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area. (Image by Jecowa, CC-BY-SA)
The fact that the product
To describe the curvature of a general Riemannian manifold
The sectional curvatures
A more refined measure is the Ricci curvature, which looks at the volume of the sliver of a small ball in each direction separately. It is a quadratic form
The scalar and Ricci curvatures at a point
The Ricci curvature comes into play when you are considering manifolds of three or more dimensions. For a 2-dimensional manifold the Ricci curvature gives you the same information as the scalar curvature (Ric
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Gravitational waves travelling through empty space. (Image courtesy of NASA)
Gravitation according to general relativity
We might first guess that general relativity would require the curvature of empty space to be zero. But that would be too strong: Einstein's theory allows the possibility of gravitational waves – waves of curvature – travelling through empty space just as electromagnetic radiation does. The relativistic law is that only the Ricci curvature of space vanishes in the absence of matter. Just as classical physics tells us that the motion of particles in a fixed space-time is governed by a principle of least action, the vanishing of the Ricci curvature corresponds to a principle of least action for the Riemannian metric.
The variational principle in this case is that the total scalar curvature of any compact region of space-time (i.e. the integral of the scalar curvature over the region) is minimum – or at any rate stationary – with respect to variations of the metric in the interior of the region. When we make a small change
In fact the change in the total scalar curvature caused by the change
The least action formulation gives the clue to the general law of gravitation. The distribution of matter in space-time is described by the energy-momentum tensor. This is yet another quadratic form on the tangent space at each point: its value on a unit vector
Evolving space-time
What is this telling us about how space-time evolves? For simplicity, let us suppose no matter is present. Space-time is a 4-dimensional manifold
In terms of local coordinates
Among the valid ways of looking at the situation there is one which fits in well with the study of 3-manifolds via the Ricci flow, which will be our next topic. Having fixed on the initial time-slice
The point
Perelman and the Poincaré conjecture
Our last topic is the use of the Gromov-Hausdorff metric in Perelman's proof of the Poincar\'{e} conjecture. This proof, which carried through a programme conceived by Richard Hamilton, amazed the mathematical world when it became known ten years ago. The conjecture is that any compact simply connected 3-dimensional manifold
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on"
Two-dimensional surfaces are classified by their genus, or the number of donut holes that have been removed. A sphere is a surface of genus 0 and a torus has genus of 1, and so on. (Image by Daniel Müller from The Manifold Atlas Project')"
Let us begin with the simpler case of 2-dimensional manifolds
– closed surfaces – and confine ourselves to the orientable ones. Then the topological classification is well-known: a surface is determined by its genus, or "number of handles". Furthermore, every surface can be given a metric of constant curvature, and that, in turn, implies the topological classification. If the constant curvature is positive the surface is a sphere, if it is zero the surface is a torus, and if it is negative we have a surface of genus
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A connected sum of two 2-dimensional surfaces where a small ball (circle) has been removed from each and the boundaries of these sewn together. (Image by Oleg Alexandrov – CC BY-SA 3.0)
The topological classification of closed 3-manifolds (again supposed connected and oriented) is more complicated. The first step, however, is simple: every 3-manifold has a unique decomposition into prime factors under the operation of connected sum. The connected sum of two closed manifolds is formed by removing a small open 3-ball from each, leaving each bounded by a 2-sphere, and then sewing these 2-spheres together by an orientation-reversing diffeomorphism. This operation is – up to diffeomorphism – associative and commutative, and does not depend on where one removes the 3-balls from the manifolds.
The prime factors, however, can be decomposed further. Thurston proposed that the atomic 3-manifolds are those with a geometric structure: that is, those which admit a complete Riemannian metric of finite volume for which every sufficiently small ball is identical to every other ball of the same radius. According to his geometrization conjecture, which was finally proved by Perelman's work, each prime manifold can be cut along 2-dimensional tori into such geometric pieces. Thurston showed that in three dimensions there are eight different kinds of geometric structure: besides the constant curvature metrics of positive, negative, and zero curvature, which are characterized by the fact that all small balls are isotropic (i.e., they have the group of rotations SO(3) as their isometry group), there are five kinds which are not locally isotropic. Two of these are obvious: the manifold can be locally cylindrical, like the product of a line with a surface of constant positive or negative curvature; but there are also three exotic possibilities in which the manifold looks like a 3-dimensional Lie group, such as the Heisenberg group we met in article 2.
The details of Perelman's work are very long and intricate, but the geometric picture it gives us of how Thurston's geometric structure emerges is simple and beautiful.
The picture starts with the space of isomorphism classes of compact Riemannian manifolds of a given dimension. This is a subspace of the space of all compact metric spaces, which has the Gromov-Hausdorff metric. The manifolds are not a closed subset of the space of compact metric spaces: a sphere of radius
Starting with an arbitrary Riemannian manifold, the strategy is to move it along a path in the space of manifolds so as to make its curvature more "even". It is not obvious how best to do this. Hamilton's insight was to focus on the Ricci flow: to move so that the rate of change of the inner product on each tangent space is -2 times the Ricci curvature at the point. (This defines a flow on the space of Riemannian metrics on a fixed underlying manifold, but it is equivariant with respect to diffeomorphisms of the manifold, so it induces a flow on the space of isomorphism classes.)
Why do we use the Ricci flow, rather than, say, the similar-looking gradient flow of the total scalar curvature, which is suggested by Einstein's equations? The reason is that the Ricci flow is essentially a nonlinear heat equation: its highest-order term tends to smooth out the metric by diffusion. The gradient of the Einstein action is not dispersive in this way – in fact, rather the opposite. We should not expect to be able to integrate it to define a flow on the space of Riemannian metrics.
In two dimensions the Ricci flow contracts the surface where the curvature is positive, and expands it where the curvature is negative. If we start with a dumbbell-shaped sphere the waist is expanded and the surface becomes progressively rounder. Once the sphere is positively curved everywhere the Ricci flow makes it smaller and smaller, so it is a good idea to combine the Ricci flow with an overall rescaling of the metric so that the total volume stays constant. It can be proved that any surface will then flow to one of constant curvature.
In three dimensions this does not happen. A three-dimensional manifold has waists which are 2-spheres rather than circles, and they are positively curved and contract to points in finite time, so that the manifold snaps into pieces: the first effect of the flow is to perform the decomposition of the manifold into its prime factors for the operation of connected sum. (We may, however, get a large number of 3-spheres splitting off, for these are identity elements for the connected-sum operation.) Once a component has snapped off it will continue to evolve under the Ricci flow, but we can rescale the components independently so that each has constant volume. If a component has positive Ricci curvature everywhere then Hamilton showed that it will evolve – after rescaling – to one of constant positive curvature. But, in general, large regions of negative curvature will emerge, separated by regions which will shrivel up but will not collapse completely in finite time. Each large region is bounded by a number of flat 2-dimensional tori, which become smaller and smaller under the rescaled flow. Each of the large regions tends to a manifold of constant negative curvature – a hyperbolic manifold – which has finite volume but has cusps: long narrow horns stretching out to infinity, whose cross-sections are 2-dimensional tori. (If one compactifies these by adding the points at infinity they will be singular points in the manifold, at which it looks locally like a cone whose base is a torus.) The existence of these cusps on most of the atomic 3-manifolds means that, like single oxygen atoms, they cannot occur uncombined in nature, that is, as compact 3-manifolds.
The final part of Perelman's proof is concerned with the shrivelling-up regions – bounded by tori – which remain when one removes the pieces which converge to hyperbolic manifolds. In the end, these too smooth out and acquire the more exotic geometric structures in Thurston's prophetic vision.
As a 3-manifold evolves under the Ricci flow some regions will collapse to singularities. To understand the topology of the initial manifold the crucial thing to establish is that no interesting topological feature is scrunched to a point and lost: in fact the only thing that happens topologically is that 2-spheres in the manifold contract to points in a standard way, performing the inverse of a connected-sum operation. That enables one to define the Ricci flow with surgery, which continues the Ricci flow through the singular times at which a sphere collapses. If we think of ordinary Ricci flow as analogous to flying a plane on autopilot then we need a human pilot to land the plane at each singular time, snip the singularity of the manifold away, and take off again.
The Ricci flow with surgery is a continuous path in the Gromov-Hausdorff metric which at isolated times leaves the class of smooth 3-manifolds. To construct it one needs precise control on what happens when the Ricci flow approaches a singular time. This is obtained by working in the space of 4-manifolds – "space-times" – swept out by a 3-manifold which is evolving under Ricci flow up to a time at which its scalar curvature becomes very large somewhere. It is in this space of 4-manifolds that one employs a version of the Gromov-compactness technique. More precisely, one studies a space of 4-manifolds equipped with a chosen point where the 3-dimensional curvature is large, and the idea is to consider a sequence of such space-times where the curvature at the chosen point tends to infinity as one goes along the sequence. One rescales the 3-dimensional metric of the