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.

Associated with every point on an analytic manifold is a TangentSpace? and its dual, the CotangentSpace?. The former consists of the possible directional derivatives, and the latter the differentials, which can be thought of as infinitesimal elements of the manifold. These spaces always have the same dimension as the manifold does.

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

The classification of all manifold 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)