Definition
A smooth structure on a manifold M is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold M is an atlas for M such that each transition function is a smooth map, and two smooth atlases for M are smoothly equivalent provided their union is again a smooth atlas for M. This gives a natural equivalence relation on the set of smooth atlases.
A smooth manifold is a topological manifold M together with a smooth structure on M.
Read more about this topic: Smooth Structure
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