# Gracious Living

Differential Geometry and the Sphere Theorem
December 16, 2010, 20:38
Filed under: Topology, Uncategorized | Tags: , , , ,

I just wrote a paper about the Sphere Theorem for my differential geometry class.  Since I can’t really get it out of my head, I thought it’d be fun to use to give a brief overview of differential/Riemannian geometry, in more or less layman’s terms.  This is somewhat of a departure from my normal posting style: though you’ll get more out of this if you’ve understood the topology posts so far, I’ll try to write in broad enough strokes that calculus is the only prerequisite.  Really, the most complicated idea behind this is the idea of a map from $\mathbb{R}^m$ to $\mathbb{R}^n$ being differentiable.  I also refer to vectors, but I think that that’s a pretty intuitive concept, in general.

#### Smooth Manifolds

I’ve mentioned manifolds off-hand a couple of times, but they’re really basic and interesting structures.  In the terms of topology, a topological $n$-manifold is a topological space such that every point has a neighborhood that’s homeomorphic to $\mathbb{R}^n$, or equivalently, to an open ball in $\mathbb{R}^n$.  If that doesn’t make any sense, you can just think of an $n$-manifold as an object where you can “flatten out” small areas of it to look like $\mathbb{R}^n$.  For example, a circle is a 1-manifold: it’s not a line, but if you zoom in on it, it looks like a line in any small area, and for geometrical purposes, you can pretty much pretend that it is a line when you’re talking about small pieces of it.  On the other hand, a figure eight isn’t a manifold: at the point where it intersects itself, no matter how far you zoom in, it won’t look like a line, and no amount of “flattening” can change that.

A smooth manifold is one that locally looks Euclidean “smoothly.”  Smooth here means “infinitely differentiable,” and you have to do a bit of trickery to come up with a way to talk about such a thing on your manifold.  But we don’t really need to worry about the definition, only about what it gives us.  You can think of a smooth manifold as one that doesn’t have any weird cusps or points — a circle, say, but not a triangle.  By saying that bits of it smoothly look like Euclidean space, we can now do calculus in those bits — we just pretend that they are Euclidean space, and then it’s easy to find derivatives or whatever.  One effect of this is that we can talk about a map between smooth manifolds being differentiable/smooth itself, instead of just continuous.  A smooth one-to-one and onto map whose inverse is also smooth is called, somewhat goofily, a diffeomorphism.

### Tangent Vectors

Another of the pieces of additional structure we get in a smooth manifold is what’re called tangent spaces.  These are vector spaces $TM_p$ attached to every point $p$ of the manifold $M$ that sort of tell you the different paths you can take away from that point. It’s pretty easy to see how this works in $\mathbb{R}^n$ because the tangent spaces are the same everywhere.  The tangent space at a point is just the standard $n$-dimensional vector space where all the vectors start at that point.  But a smooth manifold looks like $\mathbb{R}^n$, so we can just carry the tangent spaces from $\mathbb{R}^n$ forward to its points.

A good example is the sphere because we can treat its points as vectors — the vectors from the center of the sphere to those points. Then the tangent space to a point is just the plane of vectors perpendicular to the vector corresponding to that point.

In a way, all of these tangent spaces are trivial.  They’re all $n$-dimensional real vector spaces, which we know are isomorphic to $\mathbb{R}^n$.  But when we consider all of them at the same time, we get more interesting structure given by the global strangeness of the manifold.  The union of all the tangent spaces, each attached to a single point, is called the tangent bundle of the manifold, or $TM$.  This is a very interesting object and important for doing any sort of manifold geometry.  For example, if we locally pretend we’re on $\mathbb{R}^n$, then all the tangent spaces become locally the same, and so we can talk about tangent vectors changing smoothly as they move from point to point.  A function smoothly assigning a tangent vector to every point in the manifold is called a vector field.  As it turns out, $TS^2$, the tangent bundle of the sphere, is highly nontrivial — there’s a theorem called the “hairy ball theorem” that says that it’s impossible to give it a vector field which is never zero.  (“You can’t comb a hairy ball flat.”)

#### Metrics

Sometimes we want extra structure on our tangent bundles.  One important example is the ability to measure vectors.  The problem as it stands is that on a smooth manifold, though we know each of the tangent spaces are standard $n$-dimensional vector spaces and though we can pretend that the manifold is $\mathbb{R}^n$ in small areas, there’s no good way to assign axes (or bases) to nearby tangent spaces.  So we can’t really say how long a vector is, what angle two vectors tangent to the same point make, et cetera.  The solution to this is a Riemannian metric — yes, the same guy as the Riemann hypothesis — and no, not the sort of metrics that measure distance.  A Riemannian metric is a smooth choice of dot product at every tangent space.  The dot product, you might remember, lets us measure length and angles easily, but since the tangent spaces move around as you move around the manifold, you can’t just assign “the same” dot product to every space — it has to sort of move with the point on the manifold.  The study of manifolds with metrics is called Riemannian geometry.

(How do Riemannian metrics measure distance?  By doing even more than that — you can measure the length of a path by integrating the length of its tangent vector.  Then you measure the distance between two points as the length of the shortest path between them, called a geodesic.)

#### Parallel Transport and Curvature

One of the other things metrics do for you is tell you when vectors in nearby vector spaces can be considered “the same vector.”  That is, if you have a path and a vector at one end of it, you can move it to the other end while always keeping it “parallel.”  This is called parallel transport, and the basic structure that lines up tangent spaces like this is called a connection.  You don’t need metrics for connections, but each metric gives you a standard connection, which is generally easier to work with.

Now, the interesting thing about manifolds is that parallel transport doesn’t always work out quite right.  The example at right is on a sphere: the tangent vector moves from $C$ to $B$ to $A$ and back to $C$ and ends up at right angles from where it started.  This happens because the sphere is curved and one of the awesome ideas the early differential geometers had is that you can use the failure of parallel transport to find the curvature.  Specifically, you can parallel transport around smaller and smaller triangles (or other shapes) and the end vector will get closer to the beginning vector; then you can take the derivative of the difference between the two, and use that to describe the curvature.

Curvature is an intrinsic property of a manifold: it depends on the metric and/or connection, but not on how that manifold is embedded in space.  We usually think of 2-spheres as sitting in 3-space, but it’s better to think about studying them from the point of view of living on them and not being able to see the bigger picture.  (As a matter of fact, Gauss actually measured the curvature of the Earth this way: he got two other guys to stand on top of three mountains in Germany, and measured the angles of the triangle they formed.  By seeing how much more than 180 degrees this was and calculating the triangle’s area, he could find out how curved the Earth was at that triangle.  It’s also important to remember that until forty years ago, nobody knew what the Earth looked like from space, so its sphericity, while factual, was something difficult to imagine.)

#### Sectional Curvature

The problem is that the above sort of depends on the two-dimensionality of the sphere.  We moved a vector around a triangle, but on a higher-dimensional manifold, there will be many triangles to choose from.  The answer to this is sectional curvature — you pick out a two-dimensional subspace of your tangent space and calculate the curvature as you move only in those directions.  This is a number, and you get a range of numbers as you rotate and change your plane.  Ultimately, though, from the sectional curvatures of different planes at a point, you can recover the general curvature, which is a more complicated object.

#### The Sphere Theorem

The Sphere Theorem is simply this: if the sectional curvatures of every point in a connected, simply-connected, complete $n$-manifold are all in the interval $(1,4]$, then the manifold is homeomorphic to the $n$-sphere $S^n$.

Some explanation: “connected” means what you think it means.  “Simply-connected” means that every loop can be contracted to a point; on a torus, for example, which is the surface of a donut, you can draw a circle around the hole in the center and not be able to shrink it down.  “Complete” means that there’s a shortest path between any two points; the plane minus a point, for example, is a manifold but two points diametrically opposite that missing point have no shortest path between them.  And “homeomorphic” means that there’s a continuous invertible way to go from one to the other — in essence, you can bend one to get to the other.

Now, this probably looks a little weird, and you might have some questions.  Why do we care?  Because sectional curvature is something you can measure, while “being a sphere” is something you can’t.  Mathematicians like having fixed, numerical properties of things that lead directly to more important properties; they’re often called invariants.  What does $(1,4]$ have to do with it?  The short answer is that the sphere of radius $r$ has constant sectional curvature $1/r^2$, so that range of curvatures includes the spheres with radii in $(1,1/2]$, and thus circumferences in $(2\pi,\pi]$.  Surprisingly, the interval can’t be closed off — there’s a manifold called the “complex projective plane”, $\mathbb{CP}^2$, that has sectional curvatures of the form $1+3\cos^2\phi$, which are thus in the range $[1,4]$.  But this isn’t a sphere at all!

#### How To Prove It

One of the key theorems of differential geometry is called the Rauch comparison theorem.  It’s difficult to state, but it looks at pairs of geodesics (which are locally “shortest paths,” that manifold’s straight lines, remember) on different manifolds.  Basically, on more positively curved manifolds, we expect geodesics coming out from the same point to get closer together than on more negatively curved manifolds.  Example: on the 2-sphere, the geodesics are “great circles,” like our longitude lines.  After they pass the equator, they start converging again, but on a plane with zero curvature, they would never converge.

Though geodesics are shortest paths locally, they aren’t necessarily so globally.  On a sphere, we could go around a great circle as many times as we want and always be travelling in a “straight line”, but at a certain point, we’d be coming closer to our starting point again.  The farthest you can go on a geodesic without coming closer to yourself is called the injectivity radius of the manifold.  Basically, we use Rauch comparison to get bounds on the injectivity radius by comparing our curvatures to those of spheres.

Once that’s done, it’s a simple matter to prove the manifold and a sphere have the same “geodesic structure”: we can take two points of maximal distance apart, look at the set of points around each one which are closer to that one along geodesics than the other one, and show that these two sets are discs that intersect in a circle.  Because manifolds are nice, we then get diffeomorphisms from each set to one hemisphere of a sphere, and we can glue them along the boundary to get a homeomorphism from the whole manifold to a sphere.

#### Can We Do Better?

If you were reading that last paragraph closely, you might have noticed something was up.  Each of the halves are diffeomorphic to half a sphere… but the whole manifold was only homeomorphic.  This is because, though continuity is easy to guarantee just by connecting in the right spots, differentiability can fail at the boundary — things might be measured differently on the other side.  Shortly after the Sphere Theorem was proved, Stephen Smale of sphere eversion fame showed that homeomorphisms couldn’t always be made into diffeomorphisms — indeed, there’s a way of making the 7-sphere into a smooth manifold that’s not diffeomorphic to the real 7-sphere!  So clearly we have a problem: how do we know we have the right sphere?

The answer to this question was open until literally last year, when Simon Brendle and Richard Schoen proved not only that the Sphere Theorem holds in the diffeomorphism case, but also that it holds if the sectional curvatures are within $(K,4K]$ at each point.  We can easily change the curvature by a constant multiple on the entire manifold by rescaling the metric, but in this case, we could have a different $K$ at each point and so the curvature over the whole manifold could be in any interval.  Clearly, this is a huge improvement.

The way they did it is by using this obscure tool from the 80’s that is popular now because Russian recluse Grigori Perelman used it to prove the Poincaré conjecture.  It’s called the Ricci flow, and what it does is sort of “shrink the metric” so that its curvature was more uniform — bringing weirdly-shaped spheres down to standard spheres, for instance.  Brendle and Schoen rephrased the sectional curvature condition in terms of something they called isotropic curvature, which they showed forced the Ricci flow to converge to a standard sphere.

New and cool developments in math — pretty cool, eh?  The original Sphere Theorem is interesting to prove and nice because it generalizes in so many different ways — you can replace the curvature condition with one about the size of the manifold, for instance.  I checked out Brendle’s book on the differentiable version, but ended up concentrating on the original one and not reading very far into the book.  Since it uses such different methods, it presumably generalizes in very different ways, and I’ll be interested to see where future mathematicians take the method.