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:
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)