Connections Between Vector Fields and Topology: as Motivated by Non-Linear Dynamics.

In the field of Non-Linear Dynamics one runs across an interesting phenomenon. For well-behaved systems in two dimensions, we find that they all have a rather interesting property in common. If we plot the motion for a system, we find that we end up with four dimensions to plot. One dimension is for each spatial dimension (say x, and y), and one for each dimension of momentum for the particle. However, physical properties like energy conservation restrict the degrees of freedom that we have for plotting, and thus lower the dimensionality that the system is free to move about in. The result of this is that the motion gets constrained (when it’s not chaotic), to be on some sort of surface. This surface isn’t quite as simple as something like a sphere, it turns out that the shape this carves out is the torus.

From the perspective of a physicist, an in depth discussion of why this must be true is years out of the scope of this paper. An intuitive idea is that associated to every point must be a vector that uniquely describes the motion we are looking at. Thus if we are at one point, then there is one and only one vector at that point that sends us towards where we are going to go next. When we have a vector field like this on a surface that doesn’t vanish at any point and varies continuously we say that the surface is “combable”. What does this really have to do with the torus or the sphere though?

Since we must have a continuous field of vectors that describes the motion, this is something that we need to be able to place onto whatever surface we want to look at. The simple fact though is that this cannot be done with the sphere, or any other surface than the torus (of the same dimension).

There are really nice ways to study this idea of putting vector fields on surfaces using the tools of topology. To this end we want to look at something called fibre bundles, and then at a rather intriguing theorem related to the Euler characteristic. For now we put all this off in favor of considering a related theorem about spheres.

Theorem (1): No nonvanishing continuous tangent vector field can be placed on the sphere Sⁿ if n is even.

proof: Suppose n is even, and let v denote a continuous tangent vector field. We want to assume towards a contradiction that for every . Since we are assuming that v is never zero, we can normalize it so we are dealing with vectors of unit length. We can then define a map F of into by . From this we can find that

, for all x and t. What this means is that F is really a mapping from Sⁿ into Sⁿ itself. We also have that and . Since F is continuous, we have that F is a homotopy on Sⁿ from the identity to the antipodal map. Of course this can’t be as the identity map on Sⁿ isn’t homotopic to the antipodal map, and here we have our contradiction (Naber 161). One way to see this last bit is through the idea of degrees. A map f from Sⁿ to Sⁿ will induce a mapping between the corresponding homology groups, f_*, of H_n(Sⁿ) into H_n(Sⁿ). Yet both of these are basically just the integers. So the mapping basically returns an integer multiple of what it acted on, this is the degree of the map. A relatively simple thing to show is that if two maps are homotopic then they must have the same degree. It is also relatively straight-forward to compute the degree of the antipodal map for Sⁿ, though the end result is that it has degree (-1)ⁿ⁺¹. Then we simply note that the identity map has degree one, and since, for n even, the antipodal map has degree one, the two maps don’t have the same degree and thus can’t be homotopic.

So we have from this that if we place a continuous vector field on the sphere of an even dimension that it must vanish somewhere (v(x)=0 for some x in Sⁿ). Getting back to our situation in Non-Linear Dynamics however, it turns out that we really want to look at a special case of this, namely the every day sphere, S².

Now that we have that S² isn’t combable, it’s time to turn to more advanced topological ideas that relate to this whole situation, the study of fibre bundles. Intuitively, a fibre bundle is an example of a topological structure that is locally, but not always globally, a product of two spaces. To make this notion a bit easier to understand, consider the cases of the cylinder and the Möbius band. Both of these are identification spaces formed from a simple rectangle by identifying the ends in some way. Locally, i.e. in some small region, the cylinder looks like the rectangle it sprang from, thus we and have . From the global view we see that the cylinder is no more than . The Möbius band on the other hand does not share this later property. Globally, the Möbius band isn’t a product of spaces, though locally it is exactly the same as the cylinder. When a bundle is globally a product of spaces (it’s isomorphic to a product) then we say that it is trivial.

Now we are ready to give the definition of a fibre bundle.

Definition: A fibre bundle is a collection of items consisting of:

i. A topological space E, called the total, or bundle space.

ii. A topological space X, called the base space.

iii. A continuous, onto projection P:E®X.

iv. A topological space F, called the fibre such that F is homeomorphic to for all x in X.

As an example of this, we return to the Möbius band with the picture. Here the total space is the Möbius band, and the base space is simply the circle. The spaces denoted by F in the picture are two fibres, one for x, and one for y as . F is said to be the fibre over x or the fibre over y. The somewhat shaded region on the Möbius band is there to help visualize the idea that locally it is a product. Clearly if one is just looking at that section, it appears to be nothing more than a square. The way we get that is by taking some neighborhood about a point in the base space, say U_z about z. Then the shaded region is , and is homeomorphic to .

One fascinating aspect of this is that each fibre is a topological space associated to a point in the base space (as in figure 1 we see that the fibres are actually the inverse of the projection of the point). It is not the usual thing to have that a function acting on a single point gives not another point, but rather a whole topological space. This is how we obtain these fibres however, in that the fibre over a point really is a whole space that depends on that point (though all are homeomorphic).

While there are other, more complicated and useful, ways to define a fibre bundle, this is the basic definition and should be sufficient for what we need. The term fibre bundle however really is just the most general name for several different types of bundles. Of the more specific types of bundles (such as tensor, cotangent, or tangent), the one in particular that we are going to want to be concerned with is the tangent bundle of our base space. For a tangent bundle T(M), the base space is the manifold (a space where every point has a neighborhood homeomorphic to Rⁿ, where n is the dimension of the manifold) in question, say M (for us: S² and the torus), the fibre over a point p of the manifold is the tangent space at that point: T_p(M). The projection P takes vectors in T_p(M) and returns the point p that they are based at.

It is at this point that we have to introduce the big difference between working with fibre bundles and just looking at the surface. In the latter case, looking at the idea of combing is enough. That is, putting one vector field on the surface that doesn’t vanish anywhere. It turns out that this really isn’t enough when it comes to a vector bundle where all our fibres are full-fledged vector spaces, isomorphic to Rⁿ. What we want there ends up being more of a generalization of combing. Instead of putting one vector field on the surface, we want to put n independent vector fields on our bundle (where n is the dimension of the bundle). This process we will call framing the bundle, where these n vector fields represent an n-frame.

In general we have a base space that is a manifold, say M, and V will be our total space, which is an n-dimensional vector bundle. Here P is again the projection function, but we want to create new functions f_n(x) from M into V. These functions are called cross-sections. A cross-section, say f, is a continuous map such that P (f (x)) = x. In the case of tangent bundles of manifolds, these cross-sections just end up representing the vector fields that we want. The indexing n in f_n(x) is not chosen at random however, it is the same as the dimension of F (the fibre), and thus we want to have these cross-sections form a basis for F.

The part of figure 2 on the right is really an important result that can be understood fairly easily. If we can construct these n vector fields, what this induces is really a mapping from the manifold M into where the analogous. Here x is just the same point it always was, and e_n represents the standard basis vector in Rⁿ. If this works for every x in M, then what we have is an isomorphism between our bundle V, and the Cartesian product of M and Rⁿ, thus satisfying the definition of what it means for a bundle to be trivial. Thus we have:

Theorem (2): We can construct a global frame for a manifold, M, if and only if the corresponding tangent bundle, TM, is trivial.

proof:

We give here an outline of the proof of one direction of this theorem. Starting with the assumption that we can construct a global frame for the manifold, the task is then to exhibit the existence of an isomorphism between the tangent bundle and the product . Start with a function between the two, say f. We define f so that it takes the basis for T_x(M) to the normal basis e₁, …,e_n in Rⁿ, eg: . We note then that linear functions are completely determined by what they do to a basis. The end result is that we can get that f is linear on each fibre. One-to-one will follow from the fact that the dimension of both the TM and Rⁿ is n, as this implies that the kernel of f must be empty. We can get onto similarly from the fact that the dimensions are the same. The result will be that f is an isomorphism and that TM is trivial.

The task now is to consider this process of constructing independent vector fields on the torus and on S² and see what comes of this. The case of S² is somewhat easy in that we’ve already shown (Theorem 1) that the 2-sphere cannot be combed. This means that we can’t even put one vector field onto the sphere. Therefore we surely can’t put 2 independent ones on the sphere, and so we can’t build up this 2-frame for S². Since this theorem is an if and only if, this implies that the tangent bundle for the sphere is not trivial.

We now want to turn our attention from the sphere to the case of the torus. As mentioned, the torus is a surface that can be combed, but then there is the question of whether or not it can be framed, and if the associated tangent bundle is trivial or not. There are a couple different ways to answer the latter of these questions, but we proceed by constructing a pair of independent vector fields on the torus. Intuitively, if we start with one field that goes around the hole, to the right, as indicated in figure 3 (only one arrow shown to make things clearer), this is a continuous, and non-vanishing vector field. At every point though we can define the orthogonal field that loops about the torus in the other way. Since the two fields will be independent at every point of the torus, we have our f₁(x) and f₂(x), as needed for our 2-frame for the torus. The tangent space at a point for the torus looks like a plane, and it is fairly immediate that with two independent such fields we will indeed have a basis for this space.

This of course gives us another interesting difference between the torus and the 2-sphere. Since we can build the 2-frame for the torus, this implies that the tangent bundle for it is trivial. Utilizing figure 2 again, we find that the tangent bundle is equivalent to as the torus is the Cartesian product of two circles.

A quick and interesting thing to take note of is that while the normal torus, and the 2-sphere are different in that one can be framed, and the other can’t …this is not the case for all the spheres. Steenrod gives an argument using quaternions that the tangent bundle for S³ is trivial, and he says that using the analogous argument with the Cayley numbers one can obtain the same result for S⁷. Later on Steenrod lays out a theorem saying that for Sⁿ with , the tangent bundle is not trivial. He goes on to say that G. W. Whitehead proved that if , and then it also is not trivial (this takes care of all the other cases than 1, 3, and 7 for n, for which the sphere’s tangent bundle is trivial). An interesting question that apparently has yet to be settled is the higher dimensional motion one. One-dimensional systems can be set out on S¹, but the question to ask is if the same can be done for S³ and S⁷ with their respective dimensions.

There is another interesting connection in topology to this concept of placing a vector field on a surface. There is a very intriguing theorem that relates this idea with the very useful notion of the Euler characteristic (often denoted by c). The Poincaré-Hopf Index Theorem gives us an easy way to compute the characteristic in certain situations. The theorem states: “If is a smooth vector field on the compact, oriented manifold X with only finitely many zeros, then the global sum of the indices of equals the Euler characteristic of X.” This notion is easiest to understand in the two-dimensional case, where it is easy for us to understand what is meant by the indices of the vector field. In this case the index of a point (where the field is zero), is a measure of the number of times that the field’s direction rotates completely around as we move counterclockwise in a circle about the point. The direction that the field rotates here matters, as a complete rotation in the counterclockwise direction for the field adds –1 to the total, whereas the opposite direction would add +1.

As an example of this and how it all works, consider the two pictures A and B. If the dots are the places where these two fields have zeros, then all we have to do is go around each one and count how many times, and in what direction, the vector field changes. For A, we travel around counterclockwise and (depending on starting point) the field goes from left to up to right to down to left, so it has made one rotation clockwise, so the index of this is –1. The case of B is a little bit more important for us in that this is one particular way that we can lay out a vector field on S². If we travel around this one, the field has made one complete rotation by the time we are halfway around, and then another one by the time it’s done. The field changes in a counterclockwise sense, so the index associated to this is +2. If we do put this field onto a sphere, then the point indicated is the only place where the field goes to zero. Thus, according to the Poincaré-Hopf Index Theorem, the Euler characteristic of the 2-sphere is simply the sum of this one index, and we have c(S²) = +2, as expected.

There are many connections between vector fields for a surface and various aspects of Topology. When we consider the case of the 2-sphere, it is plain that we can’t comb the sphere by setting a continuous, non-vanishing vector field on it. As a consequence of this, the representation of motion certainly cannot lie on a sphere. The torus is a different surface in many ways however. It can be combed, but more than that, the torus can be framed. In general a much stronger condition, this requires us to build a basis for the tangent bundle consisting of independent vector fields. There are certain higher dimensional spheres that share this property, when n equals three and seven, though n equals one is also acceptable.

Other interesting math can come out of looking at placing vector fields on manifolds though. In particular we have the example of the Poincaré-Hopf Index Theorem. This draws together these ideas of vectors and the important Euler characteristic. It provides a rather easy way to calculate the characteristic in some situations as well. By simply considering the zeros of whatever field we end up laying down on the manifold, in the two dimensional case we simply have to watch how the field changes as we move about the zeros.

From the motivation of considering physical systems and how they can be represented in space, we get the notion of putting vector fields onto manifolds. Considering a stronger condition than just putting one vector field onto a manifold and we look into the amazing topological structure of fibre bundles. Further connections are found with a relatively easy way to calculate the Euler characteristic for certain types of surfaces. All together we end up with a pair of excellent topological ideas. In his classic text “The Topology of Fibre Bundles,” Winston Steenrod remarks that the study of fibre bundles contains “some of the finest applications of topology to other fields, and gives promise of many more.” An interesting study in its own right, here we’ve seen Steenrod’s words truly bear fruit.
Bibliography

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Guillemin, Victor and Pollack, Alan. Differential Topology. Prentice-Hall. New Jersey, 1974.

Naber, Gregory L. Topological Methods in Euclidean Spaces. Dover. New York, 2000.

Nash, Charles and Sen, Siddhartha. Topology and Geometry for Physicists. Academic Press. New York. 1983.

Steenrod, Norman. The Topology of Fibre Bundles. Princeton UP. Princeton, New Jersey, 1960.