Smooth manifold

Note: It's worth looking at Motivation for smooth manifolds

Definition

A smooth manifold is a pair [ilmath](M,\mathcal{A})[/ilmath] where [ilmath]M[/ilmath] is a topological [ilmath]n[/ilmath]-manifold and [ilmath]\mathcal{A} [/ilmath] is a smooth structure on [ilmath]M[/ilmath]

We may now talk about "smooth manifolds"

Notes

• A topological manifold may have many different potential smooth structures it can be coupled with to create a smooth manifold.
• There do exist topological manifolds that admit no smooth structures at all
  • First example was a compact 10 dimensional manifold found in 1960 by Michel Kervaire

Specifying smooth atlases

Because of the huge number of charts that'd be in a smooth structure there's little point in even trying to explicitly define one, see:

• Smooth structure determined by an atlas

Other names

• Smooth manifold structure
• Differentiable manifold structure
• [ilmath]C^\infty[/ilmath] manifold structure

See also

• Manifold
• Smooth structure
• Smooth
• Smoothly compatible charts
• Motivation for smooth manifolds