Smoothness
Smooth Map
Let
be a differentiable manifold with dimension , an atlas representing the differentiable structure on . A function is smooth if for every chart in , the map is smooth on . Let be another differentiable manifold with dimension , and differentiable atlas . Let be a map from to . Then is smooth if for all , and every chart in with and in with , the map is smooth on . 
e.g.
- The inclusion map
is smooth. We can take - The quotient map
is smooth. Note that
Proposition
Although the definition requires that the map
is smooth for every chart in the atlas, it is enough to check the smoothness for a single chart around each point. Similar argument applies to check that a map between manifolds is smooth.
Proof If
Proposition
A map
is smooth if and only if each component function is smooth.
Proof If
Proposition
A map
between differentiable manifolds and is smooth if and only if is a smooth function on whenever is a smooth function on .
Proof Clearly if
Further Classification of Maps
Diffeomorphism
A map
between smooth manifolds is a diffeomorphism if it is smooth, and has a smooth inverse. is a local diffeomorphism about if for every there is a neighborhood of such that is a diffeomorphism to an open subset of .
Immersion
is an immersion if for every in there exist charts for and for with and , such that the map has derivative which is injective at .
Submersion
is a submersion if for each there are charts and such that the derivative of at is surjective.
Smooth Embedding
A smooth map
is an embedding if is a homeomorphism onto its image with the subspace topology, and for any charts and for and respectively, has derivative of full rank.
Remark
A smooth embedding is a topological embedding that is also an immersion.