Manifold

Manifold

A manifold is a curve, surface, or higher-dimensional space that has, at every point in that space, a small neighborhood around each point that looks like a ball in the corresponding Euclidean space of the appropriate dimension. More precisely, the neighborhood should be topologically equivalent to a ball--in other words, it should be possible to stretch and distort the neighborhood, without tearing or gluing it, so that it is a ball.

The easiest manifolds to visualize are the 2-dimensional manifolds. In dimension 2, a 'ball' in Euclidean space is simply a filled-in circle, or a disk. So a 2-dimensional manifold (or '2-manifold') is an object for which every point has a small neighborhood that looks like a (distorted) disk--these are what we commonly think of as surfaces. One of the simplest examples of a 2-manifold is the surface of the earth, a sphere; every point on the surface of the earth has a neighborhood that looks like a distorted disk. Another example is a torus, the surface of a donut. Manifolds do not have to be orientable--the Möbius strip, a non-orientable surface formed by gluing the opposite ends of a strip of paper with a 180-degree twist, is a manifold, since each point has a neighborhood that looks like a flat disk. Thus, manifolds can look quite different from the Euclidean space on which they are modeled; it is only locally that they must resemble Euclidean space.

In dimension 1, the Euclidean space is a line, and a 'ball' is simply a line segment (this is the proper analogue, since we think of a ball as the locus of points within a certain fixed distance from a given point). Thus, a 1-dimensional manifold is any 'space' that locally looks like a distorted line segment. A circle is a 1-dimensional manifold, but the letter 'X' is not; at almost every point of the 'X', there is a neighborhood that looks like a line segment, but at the center point, where four line segments come together, there is no neighborhood that looks like a single piece of line segment.

The correspondence between a small neighborhood and a piece of Euclidean space is often called a 'map'. This terminology arises from maps of the surface of the earth, which create correspondences between pieces of the sphere and pieces of the plane. A fruitful way of thinking of a manifold is in terms of its maps, which give concrete ways of visualizing small pieces of the manifold. A given point in the manifold can be contained in many different maps, in just the same way that a point on the surface of the earth can appear in many different maps, since there are many useful ways of mapping the earth. We can think of a manifold as being built from its maps, by gluing together pieces of Euclidean space whenever they correspond to the same piece of the manifold. In this way, a manifold (which often is very difficult to visualize) can be described by a collection of pieces of Euclidean space, together with gluing rules. These rules must satisfy some technical 'compatibility' requirements in order for the result of the gluing to be a manifold; roughly speaking, these conditions ensure that maps are always glued together over ball-shaped pieces, and that they are glued without any folding or tearing of the pieces.

One of the most important questions in the study of manifolds is, How many different manifolds are there in each dimension? In dimension 1, the only kinds of manifolds are curves (possibly infinitely long) and circles (possibly distorted). In dimension 2, there is a complete classification of the 'closed' manifolds, surfaces that have no boundary curves and that fit in a finite region. The orientable, compact 2-manifolds are the sphere, the torus, the double torus (a torus with 2 holes instead of 1), the triple torus, and so on. The non-orientable compact 2-manifolds are the projective plane (a manifold formed from a flat circular paper by gluing opposite points on the boundary circle), the Klein bottle, which is obtained from the projective plane by attaching a handle, and surfaces formed from the projective plane by adding 2 handles, 3 handles, and so forth. The Möbius strip is not included in this list since it is a surface with boundary, hence is not closed.

In dimensions higher than 2, it becomes much more difficult to classify, or even visualize, manifolds. We are all familiar with one 3-dimensional manifold: the space around us. We generally assume, based on our local picture, that the universe is shaped like 3-dimensional Euclidean space, extending infinitely far in all dimensions. However, physicists entertain the possibility (which at present cannot be tested conclusively) that our universe might be a more complicated 3-manifold. This seems counterintuitive, but it stems from the fact that we can only see a small portion of the universe, and a small piece is not enough to give information about global properties. For example, an ant that lived on the Möbius strip but could only see a very tiny part of it might never realize that it was not the Euclidean plane. In the same way, the universe might have some 3-dimensional 'twists' in it that are simply too far away for us to see.

Still higher-dimensional manifolds have proven to be important in describing the physical world. Einstein's theory of relativity is based on the idea of thinking of the universe as a 4-dimensional manifold, in which time is one of the dimensions, and then considering geometric structures on that 4-manifold. More recently, with the advent of string theory, has come the idea that the universe might be an even higher-dimensional manifold, several of whose dimensions are too small for us to perceive. Another important manifold that arises in the study of Hamiltonian dynamics is the phase space, a high-dimensional manifold whose points represent the different possible configurations (positions, velocities, and so forth) of a given collection of physical objects.