Differentiable Manifold

[JohnEB]

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. Note that a differentiable manifold as it stands does not have any metric structure or any notion of orthogonality. The addition of metric (or pseudo-metric) structure corresponds to the linear space mentioned above actually being Euclidean space (or pseudo-Euclidean space).

http://en.wikipedia.org/wiki/Differentiable_manifold

[JohnEB]

INTRO to Manifolds Video Collection

http://www.youtube.com/watch?v=AaYXO8wifSU&feature=related