ManiFold

A continuous n-ManiFold is a topological space where there is some open set around each point that is isomorphic to the ProductTopology R^n. That is, the space looks locally Euclidean. These isomorphisms are called charts, and the collection of all of them is called an atlas; by only including charts that relate smoothly to one another, we get an analytic manifold. All differentiable manifolds can be made analytic.

One of the most important kinds of ManiFold is a LieGroup. These can always be made differentiable.

The classification of all ManiFolds is an open problem. We know that every connected 1-D manifold is isomorphic either to R or the circle S. Connected, compact 2-Manifolds can be divided into three infinite series:

• Orientable with characteristic 2-2n (spheres with n handles)
• Non-orientable with characteristic 1-2n (projective planes with n handles)
• Non-orientable with characteristic -2n (klein bottles with n handles)

Non-compact connected 2-manifolds are just these with one or more punctures (missing points). A 2-manifold can be embedded in R^3 if it is orientable or if it has at least one puncture. All can be embedded in R^4.