Smooth Maps

Smoothness

Smooth Map

Let be a smooth -manifold with dimension , an atlas representing the differentiable structure on . Let be another smooth -manifold, and differentiable atlas . Let be a map from to . Then is smooth if it is continuous, and for all , and every chart in with and in with , the map is smooth on .

The collection of all such smooth maps are denoted as , and we usually write for .

Remark

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, because our atlases are smooth, so transition maps are all smooth!
Moreover, to also check continuity in a streamlined way, we simply need to show that for each there exists a chart at and a chart at such that and is smooth.

e.g.

• Every constant map is smooth;
• The identity map is smooth;
• The inclusion map is smooth for every open subset ;
• The composition of smooth maps is smooth;
• The inclusion map is smooth. We can take
• The quotient map is smooth. Note that
  

Proposition

A map is smooth if and only if each component function is smooth.

Proof If is smooth, then is smooth for every chart . Conversely, if each is smooth, then is smooth.

Proposition

A map between smooth manifolds and is smooth if and only if is a smooth function on whenever is a smooth function on .

Proof Clearly if is smooth, then is smooth. Conversely, if is smooth on , then for every chart in the atlas of , the composition is smooth on . Hence is smooth by definition.

Proposition

A map between smooth manifolds is smooth if and only if for every smooth function , the precomposition is a smooth function on . Equivalently, the pullback satisfies .

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 an open 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.

Regular Value Theorem