October 1, 2019
Now that we have established the concept of local coordinates of a manifold M , we can define smooth maps between manifolds that preserve properties we have learned from advanced calculus.
1 Smooth Maps
Just like smooth functions, the smoothness of a map F : M → N at a point p ∈ M is defined to be the smoothness of the coordinate representation of F at p and F (p):
Definition 1.1. Given smooth manifolds N ,M of dimensions n, m respectively, a map F : N → M is said to be smooth at p, if there are charts (U, ϕ), (V, ψ) at p and F (p) so that F (U ) ⊂ V and
ψ ◦ F ◦ ϕ−1: ϕ(U )(⊂ Rn) → ψ(V )(⊂ Rm) is a smooth map in the ordinary sense.
It is now clear that compatibilities of charts on N and M make the definition above independent of choice of coordinate:
The following definitions can now be defined on manifolds:
Definition 1.2. A map F : N → M is smooth if it is smooth at every point.
Definition 1.3. A map F : N toM is called a diffeomorphism if it is smooth, bijective, and F−1 is smooth.
Constant maps from M to N , identity maps from M to itself, and inclusion maps of an open subset U to M , should all be smooth. Indeed, their coordinate representations are constant maps in Euclidean spaces. Some basic properties should hold true for smooth maps:
Theorem 1.4. Smooth maps are continuous.
Theorem 1.5. Composition of smooth maps is smooth.
Proposition 1.6. Let N and M be smooth manifolds and F : N → M be a map.
• If every p ∈ M has an open neighborhood U such that F |U is smooth, then F is smooth.
• If F is smooth, then its restriction to every open subset is smooth.
The proposition immediately implies
Corollary 1.7. Let M and N be smooth manifolds, and {Uα}αbe an open cover of M . Suppose that for each α, there exists smooth functions Fα: Uα → N so that for all α, β,
Fα|Uα∩Uβ = Fβ|Uα∩Uβ.
Then, there exists a smooth function F : M → N so that FUα= Fα.
Let’s study some example of smooth maps and diffeomorphisms.
Diffeomorphisms are basically the ”identifiers” of smooth manifolds without concerns of metrics. Though a manifold may have many distinct smooth struc- tures, they may be related to each other via diffeomorphisms. It is a fascinating (and very deep) subject to classify smooth type of a topological manifold, i.e.
smooth structures up to diffeomorphisms.
It is not difficult to show the un-surprising fact that dimensions and bound- aries are invariants under diffeomorphisms:
Theorem 1.8. An m-dimensional smooth manifolds can not be diffeomorphic to an n-dimensional manifolds unless n = m.
Theorem 1.9. Suppose M and N are smooth manifolds with a diffeomorphism F : M → N . Then F (∂M ) = ∂N and F restricts to a diffeomorphism from IntM to IntN .
