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 ManiFold''s'' 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.  If anyone want to make some models, attach the sides of these (and remove the corners to puncture)

       *              *                  B                B
      v v            v ^              *>>>>>*          *>>>>>*
     v   v          v   ^             v     v          v     v
   A v   v A      A v   ^ A         A v     v A      A v     v A
     v   v          v   ^             v     v          v     v
      v v            v ^              *<<<<<*          *>>>>>*
       *              *                  B                B

   Sphere      Projective plane     Klein bottle        Torus
             (punct: MoebiusBand)                 (sphere w handle)