What is a... Manifold?

Transcription

1 What is a... Manifold? Steve Hurder Manifolds happens all the time! We just have to know them when we see them. Manifolds have dimension, just like Euclidean space: 1-dimension is the line, 2-dimension is the plane, and so on. Here are a few illustrations: Figure 1: Bottles But most manifolds do not exist in 3-space: Figure 2: Surfaces that dont fit in 3-space

2 Definition: A manifold is a set M which has a well-defined notion of neighborhoods, and each point has a neighborhood that looks like an open subset of Euclidean space. The most natural notion of the neighborhood of a point x M is to talk of an open ball of radius ɛ > 0 about x B(x, ɛ) = {y M d M (x, y) < ɛ} But this requires we have some notion of the distance d M (x, y) 0 between any two points in M. This is a property called a metric on M, and also means we know what is the shortest path between two points and more. These shortest paths are called geodesics or light-rays. What do we mean by a neighborhood that looks like an open subset of Euclidean space? Figure 3: 2-Sphere with some geodesics, and some not-geodesics Just that if we have an open neighborhood U = B(x, ɛ) say, of a point x M, then there is a homeomorphism (1-1, onto, and bi-continuous) into an open subset of some Euclidean space ϕ x : B(x, ɛ) R n Think of ϕ x as your local vision of M near to where you stand at x in the set M. When we look around the points nearby to where we stand at each point x of M, it seems to be just like Euclidean space. Polar coordinates give a chart near the south pole in this illustration:

3 Manifolds with 1 dimension Suppose you go for a long walk, walking straight ahead. And you start noticing, that every time you travel a distance T, the scenery looks exactly the same as before. Yet, you are certain you never stopped walking straight ahead? Where are you? You are stuck on a circle! Imagine Christopher Columbus, sailing along one of the great circles in Figure 3. This is the example that we all know, of how to get a circle. While you thought your path was given by x = c t + x 0 where c is your speed and x 0 was your starting position, it seems that your starting position x 0, and the position x 0 + T, are the same. Mathematically, a circle of circumference T is described by writing S 1 R = [0, R]/0 T. That is, we glue the points 0 and T together; or the points x 0 and x 0 + T together. It doesn t matter which two points, we just close up the line segment into a circle. This is not all that exciting (well, it was to Columbus, but not 500 years later.) In fact, the only interesting question is whether we care about the length of the circle. If we don t, then there is just one circle, and we pick a length for it that works for us. Like for radius 1, then the length is T = 2π. Or maybe some other length works better. If we do care about length, then it s clear how to get a circle of that length. Mathematically, here is the statement: Theorem: There are exactly two 1-dimensional manifolds without edges: The line R 1 and the circle S 1. If we wish to add length to our description, then there is still just one line R 1, but there are now circles S 1 T for each circumference T > 0. When we say there is just one circle, notice we don t care where it is: on the plane, in a sphere, in hyper-space, in some data set, or just in our minds. It is this concept, that there is an object that exists abstractly and need not be place in a special context, that powers up the idea of manifolds.

4 Manifolds with 2 dimensions - Surfaces Imagine now that you are standing on a 2-dimensional manifold, what is called a surface mathematically. Now pick a direction - you have 360 of possible starting directions. Suppose that you find that after traveling T units of distance, you are right back where you started. What sort of surface are you living on? It must be a 2-sphere, right? Like in Figure 3. Wrong! You might return back to where you started, going out in any direction, but you might return and find that your right and left hands have been switched! Left is right, and right is left. Then you are living on a non-orientable space. In this case, you are on the 2-dimensional Projective Space. The best we can do to picture what this space looks like in Euclidean space is a figure called Boy s Surface here is a picture of it: It is crinkled up because Projective Space cannot be embedded in Euclidean 3-space. But if you try imagining walking along the surface of this shape, when you go through a crease, what is outside the surface becomes inside, and vice-versa; and left become right. Now, if you take all your daily walked, and each time return to where you were after walking T distance, and your keep your orientation, then you are in fact living on the 2-sphere. This is a mathematical fact. Theorem: Let M be a 2-dimensional manifold without boundary, and suppose that every geodesic is closed with length T. Then either M is the round 2-sphere or radius R = 2π/T, or M is a Projective Space of radius π/t. The 2-sphere is Euclidean 3-space is described by the equation S 2 = {(x, y, z) x 2 + y 2 + z 2 = R 2 } The Projective Space is defined more abstractly. It is the space of all lines through the origin in R 3. A line through the origin is determined by the two points (0, 0, 0) and (x, y, z) except that the points (x, y, z) and ( x, y, z) define the same line through the origin: P 2 = {(x, y, z) x 2 + y 2 + z 2 = R 2 ; (x, y, z) ( x, y, z)} Here is another way to describe this space: take a Möbius band, which has a boundary circle, and glue onto this circle a disk, so there is no boundary anymore. What you get is P 2!

5 The examples of the 2-sphere S 2 and the Projective Space P 2 are only the beginning of the possible surfaces. Look at Figure 1 again these are all surfaces. What sets the illustrations in Figure 1 apart from the 2-sphere and the Projective Space, is that every surface in Figure 1 has at least one handle: The Klein Bottle has one handle, and the other two have 2-handles. The simplest surface after the 2-sphere is the two-torus T 2 If you have a surface M, you can always add another handle by smashing on a donut. This is called attaching an orientable 2-handle. Here is an illustration of gluing on a handle: You can also attach a Projective Space or a Klein Bottle to get a new surface, but these are impossible to draw in 3-Space. All 2-dimensional surfaces have a neat description: Theorem: Every 2-dimensional manifold without boundary is equivalent (up to topological equivalence - not metric equivalence!) to either the sphere S 2, or to n copies of the 2-torus attached together, or to k copies of the Projective Space attached together. That s it. Want to learn more about surfaces? check out wiki, Manifolds with higher dimensions You can continue to construct manifolds in all dimensions n beyond 2. The UIC Math Department is world-renown for its expertise in the study of manifolds of dimension 3. Check out the wiki, One of the big open questions in cosmology is: What is the shape of the universe? The universe we live in, is thought to be a 3-manifold, but which one? Is it a 3-sphere? (In physics this is the model that predicts the big crunch ); a flat Euclidean space R 3? (the steady-state model); or is it something more exotic, like some sort of hyperbolic space? Though, String Theory predicts we live in 10-dimensional space! Check out Jeff Weeks web site Geometry Games,