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...