A manifold is a powerful
topological object. In
physics, it is probably the most-used and least-appreciated
mathematical concept. You might like to think of it as an n-dimensional
generalization of a two-dimensional surface. Think about the surface of a
sphere. It can be parameterized by two coordinates:
lattitude and
longitude. Its
global structure differs from, say, a
torus, or a
flat plane, but
locally (in sufficiently small regions) these spaces all "look" the same, meaning they are parameterized by two
continuous coordinates. The n-dimensional
generalization follows quite naturally, so if things get too
abstract in what follows, a good way of getting a more concrete picture is to think back to the special case of surfaces, by simply setting n = 2. Since these are often realizable as an
embedding of
three-dimensional space, it is possible to literally picture what is going on mathematically.
The most common description of a manifold is that it is a topological space which locally "looks" like Rn. This is useful because it allows us to apply the mathematical machinery we have developed over the centuries for doing calculations on Rn in a mathematical setting which may seem completely abstract. Specifically, in the case of differential manifolds, we can do calculus, which is of great utility.
Examples* of Manifolds:
- Rn
- The n-dimensional sphere, Sn, which is a submanifold of Rn+1.
- The Torus T2 = S1 × S1 (the surface of a donut).
- The n-Torus Tn = S1 × S1 × ...
- Real Projective space RPn, the set of lines in Rn+1 passing through the origin (equivalent to the sphere Sn if you identify antipodal points with each other). RP2 is known as the projective plane.
- The set of n×n real matrices, GLnR, is an n2-dimensional manifold, whose coordinates could simply be given by the n2 matrix entries.
- The complex plane, C, can be considered a two-dimensional real manifold. There also exists the notion of a complex manifold, which instead locally "looks like" Cn, and therefore C is a one-dimensional complex manifold.
Before arrive at the proper definition of a manifold, we need to describe a few other concepts. First, we need to look at how coordinate systems are described in a manifold. This information is encoded in coordinate charts (or "patches") on the manifold (which we will now refer to as "M"). A coordinate chart is described by two objects: an open set U in the manifold M, and a homeomorphism from U to an open set in Rn. A homeomorphism is a bijective map which is continuous in the topological sense. It declares two spaces to be topologically equivalent, and gives us a "dictionary" between points in the two spaces, which explicitly shows how they are equivalent. So, a coordinate chart takes a piece of the manifold, and shows it to be topologically equivalent to Rn. This is basically what we want, but notice what a powerful statement that is. It means that each point in the patch U can be associated with n numbers ("coordinates") which categorize the point exactly. Not only that, but since U is topologically equivalent to its associated open set in Rn, we know that the notion of "nearness" between two points in the abstract space M is equivalent to the notion of "nearness" between the corresponding points in Rn, i.e. that the difference in coordinates is small.
A collection of charts which cover M (so that each point in M is contained in some coordinate patch) is called an "atlas" for M. We can now define a topological manifold (of dimension n) to be a topological space equipped with an atlas of n-dimensional coordinate charts. I'm leaving out a few points here; first the requirements that M be a Hausdorff space, and that M is second countable, which are caveats I'm not going to bother explaining, and another is that the atlas be maximal, i.e. that the atlas includes all possible consistent coordinate charts which could be defined on the manifold. This is done so that if I define an atlas on the manifold, and then my friend defines a completely different atlas on the manifold, both our manifolds should have the same maximal atlas, and therefore we won't consider them to be two different manifolds.
Now there is a more concrete realization of what it means when we say a map f: M → R is continuous. It can easily be shown that to check the continuity properties of f, we can just pull back the function to its coordinate representation. This is done by defining the map fU = f o φU -1, for each coordinate patch U in M, which is just the coordinate representation of f in the patch U. Since fU is just a map from Rn to R, we can study its continuity properties in the same way we would any function in analysis. Although M is just an abstract manifold, the coordinate chart provides a concrete representation for any function on M which is defined in terms of spaces we understand very well.
An important question arises when it comes to the overlap between coordinate patches. How do we describe the intuitive notion that our coordinate patches should somehow be "consistent" with each other in their overlap? Since a manifold is a topological space, what we need to know is that notions of continuity are the same when we go from one patch to another. Thus, we should say that two coordinate patches are consistent if they give us the same answer as to whether a function f on M is continuous, for any given f. Let's see how this plays out:
Say we are given a manifold M, and two coordinate patches U and V (subsets of M) with coordinate charts φU: U → Rn and φV: V → Rn, and a function f: M → R. Is f continuous in U ∩ V?
Since U is topologically equivalent to an open set in Rn, as is V, we can check f's continuity properties by finding out if it's continuous in its coordinate representation. This is given explicitly by the composition map fU = f o φU -1: Rn → R, and fV = f o φV -1: Rn → R.
Note that we can rewrite fU as f o φU -1 = f o φV -1 o φV o φU -1 = fV o (φV o φU -1). Thus, fU and fV are simply related by the function φV o φU -1, which is often called the transition function from U to V (Note that since the φ's are invertible, we could have played this same game in the other direction, from V to U). It is now easy to see that fU's continuity will always agree with fV's continuity, provided that the transition functions are all continuous. Thus, the transition functions are critical to defining the topological structure on M. Two patches are said to be compatible if their transition functions are continuous. Now, we can say precisely what a topological manifold is:
An n-dimensional topological manifold is a second countable Hausdorff topological space M, with a maximal atlas of compatible coordinate charts {Ui, φUi:Ui → Rn}.
This is how you should think about a topological manifold: imgagine you just start with a collection of patches, i.e. a bunch of abstract spaces {U1, U2, U3...} which are equivalent to Rn, but so far are unrelated to one another. Now, take these patches and "glue" them together, via the transition functions, φUi o φUj -1. The patches themselves have no interesting structure; they are just pieces of Rn. What makes the manifold unique is the precise way in which these patches are glued together, which is encoded in the transition functions.
Now that we understand how to treat the question of continuity on a manifold, let's generalize to ask about differentiability on the manifold. That is, given a map f: M → R, can we say anything meaningful about its differentiability ("meaningful" in the sense of coordinate-independence)?
To make a long story slightly less long, the answer turns out to be dependent, once again, upon the transition functions. Since we have not yet said anything about the differentiability of the transition functions, we cannot yet say anything meaningful about f's differentiability. So, we define a differentiable manifold, or differential manifold, to be a topological manifold whose transition functions are all differentiable. Now, if M1 and M2 are both differential manifolds, and there is a map f:M1 → M2, the question of f's differentiability will not depend on which patches we use. A differential manifold is often simply called a smooth manifold.
Examples of coordinate charts and transition functions:
1. Consider M = Rn (using the standard topology). For this example, let's make our lives easy, and just use one patch to cover M: U = Rn, where φU: Rn → Rn is just the identity map. Since there is only one coordinate chart, there are no transition functions, so we are already done.
2. Consider the two-dimensional sphere S2. Normally we think of this object as being embedded in three-dimensional space (x,y,z), using the restriction x2 + y2 + z2 = 1. Now, (x,y,z) will not make good local coordinates on the sphere. If for no other reason, there are three of them, and there should only be two coordinates on any given patch of the sphere. If we just chose two, say x and y, this patch will not be one-to-one; there are two points on the sphere corresponding to each x and y (except at the equator). Thus, we can only use this patch to cover the upper hemisphere or the lower hemisphere, which gives us two patches, and we still haven't properly covered the equator (remember, we need to use open sets). All told, we'd need to use six patches with this method, which gives us 30 transition functions to calculate, so it might be a good idea to try something else.
We can do a little bit better by using azimuthal and axial angles θ and φ (commonly known as lattitude and longitude**). We can effectively cover the sphere with these coordinates. Unfortunately, this "coordinate chart" is not one-to-one. The point θ = 0 on the sphere (the north pole) gets mapped to a line in R2, because at θ = 0, φ can take on any value from zero to 2π, and it will correspond to the same point. This same issue occurs at θ = π, the south pole. The bottom line is, there are problems with this coordinate chart. It is valid at some points, but not all.
The standard approach turns out to require only two patches. These patches are found via stereographic projection. For a given point on the sphere (x2 + y2 + z2 = 0), draw a line between that point and the north pole. Choose coordinates x and y to be the x and y coordinates where this line crosses the plane z = 0. Since the plane z = 0 is just R2, this is a map from the sphere into R2. Note that for this construction, you can choose any point other than the north pole. Thus, the sphere minus the north pole is topologically equivalent to R2, a flat plane. The second patch can be generated by drawing lines through the south pole instead of the north pole. You can check using basic euclidean geometry that the transition function between the two patches is continuous and smooth, except at the poles (where there is no overlap). It is impossible to cover the sphere with just one patch, which is good news, since we want the sphere to look globally different from R2, even though locally it looks the same.
3. Consider the n-dimensional sphere Sn, which can be thought of as a submanifold of Rn+1 with coordinates (x1, x2...xn+1), and the restriction Σ xk2 = 1. Once again these coordinates are one-too-many, but we could generalize stereographic projection in the same way we did for R2. Thus, Sn can always be covered by two patches.
Some ways of constructing differentiable manifolds:
1. Instead of defining S2 via stereographic projection, we could have defined it as a submanifold of the ambient space, R3, using the six patches we mentioned before. This turns out to be a special case of a very general method of defining a manifold, i.e. giving an ambient space with some restricting conditions on the coordinates (like x2 + y2 + z2 = 1). By the implicit function theorem, if we have k independent equations, we can locally write k of the coordinates in terms of the others (in a proof that I'm not going to get into). For example, z = (1 - x2 + y2)1/2, in the upper-hemisphere. Then, in the patch where this holds true, we can uniquely map the coordinates to Rn-k, by simply dropping extraneous coordinates (the z-coordinate), knowing that they are given implicitly by the other coordinates. This is the basic idea; you can check to make sure that the axioms still hold. In particular, curves through a manifold M are themselves manifolds, of dimension one.
2. Product Manifolds are generated by taking a list of smooth manifolds as input, and taking their product as topological spaces, producing a new smooth manifold as output. For example, R × R = R2. The coordinate charts are obtained via Uij = Vi × Vj, and φij = (φi, φj). Since these charts map M x N to Rm × Rn = Rm+n, the product of an m-dimensional manifold with an n-dimensional manifold is an m+n-dimensional manifold. Other examples include S1 x S1 = T2 (torus) and S1 x R = a Cylinder.
We know that smooth manifolds are defined by a set of coordinate charts whose transition functions are differentiable. Is it possible to define two different differential structures on a manifold? In other words, is it possible to define two separate atlases on a manifold which are differentially consistent with themselves, but inconsistent with each other? The answer turns out to be yes, but not always. Two examples are S7, on which there are exactly 28 possible differential structures, and R4, on which there are infinitely many. Actually describing these differential structures is well beyond the scope of this writeup.
Examples of Manifolds in Physics
For some universal reason, manifolds seem to be the natural language in which to talk about physics. I'm personally trying my hardest to think of a single physics problem that doesn't use a manifold in one way or another, and I can't come up with anything reasonable. Why, the very setting of most physics problems, that of spacetime, is itself a manifold. Certainly, we can perform physics calculations in spacetime without having any idea what a manifold is (and a great many physicists do), but this is merely an example of the fact that physicists aren't mathematicians. They don't always have a full appreciation for the math that they are using. Consider this: for calculations in spacetime, we implicitly use the metric topology. However, the metric of spacetime is not a bona fide distance function! Try defining open sets with the interval Δx2 - Δt2, and see how far this gets you. How do physicists typically deal with this problem? They ignore it, which seems to work pretty well. For an example of a physics problem which actually requires a complete understanding of manifolds, read about the magnetic monopole.
The most obvious example of a manifold in physics is simply three-dimensional physical space. Physical space is a submanifold of physical spacetime, a four-dimensional manifold. This manifold is not simply R4, although it might be topologically and differentially equivalent; there is curvature, which is commonly known as gravity, and the global structure of the universe is still unknown. Four dimensional spacetime, in turn, is believed to be a submanifold of a larger-dimensional (maybe 11-dimensional?) spacetime, which string theorists have yet to nail down. It is believed, however, that this spacetime is the product manifold of four-dimensional spacetime with a Calabi-Yau space.
A more subtle example is configuration space, i.e. the set of all possible physical states for a system. For example, the physical system of a simple pendulum exhibits the configuration space of all possible angles at which the pendulum could be found. Thus, the configuration space of a simple pendulum is just S1, a circle. For a double-pendulum (one pendulum attached to the bottom of another pendulum), the configuration space is S1 × S1 = a torus.
Another interesting example could be found in a nematic liquid crystal, in which case the physical state is completely specified by the orientation of the crystal's axis. Since this can just be given by a line in three dimensions, the configuration space here is in fact RP2, the projective plane.
Configuration space is a subspace of phase space, the space of all positions and conjugate momenta of a physical system. Specifically, phase space is identified with the tangent bundle of configuration space.
Clearly, manifolds have great mathematical and physical significance. Essentially, any time you want to use a continuous parameter to describe something, you're implicitly defining at least one manifold. There are many other structures which can be defined on a manifold, like tangent spaces, curvature, distances between points, and the like, but these are not necessary to defining the space, defining functions on the space, and even doing calculus on the space (assuming it's smooth).
*If you have some ideas for additional examples, please let me know.
**I'm using the physicist's definition of θ and &phi, where φ is the axial angle, and θ goes from 0 to π, instead of going from -π/2 to π/2. I'm also pronouncing it "phi", as in "rhymes with fly". Mathematicians everywhere wince.
Thanks to jrn for giving me seven thousand comments on how I could improve this writeup. I welcome more.