- #1

aalma

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- TL;DR Summary
- Trying to prove a claim using the definition of a smooth map.

Given the definition of a smooth map as follows:

A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition

$$\psi ◦ f ◦ \phi^{-1}$$ is smooth.

Claim: A map ##f : X → Y## between two smooth manifolds is smooth iff for any open ##U ⊂ Y## and any ##φ ∈ C^∞(U)## the composition ##f^{-1}(U)→ U → R \in C^∞(f^{-1}(U))##.

My question is how to show this claim?

For one side, I can use the definition above:

a function ##φ∈C^∞(U)## on an open subset ##U⊂Y##

is a smooth function from ##U## to ##R##.Therefore, we can take the charts ##\phi:f^{−1}(U)→R^m##

and ##ψ:U→R^n## to be the identity maps, and we have: $$ψ◦f◦{\phi}^{−1}:R^m→R^n$$

which is a smooth map between Euclidean spaces. Composing this with ##φ## gives a smooth function on $$f^{−1}(U):

φ◦(ψ◦f◦{\phi}^{−1}):f^{−1}(U)→R$$

and this is the map ##φ∘f##

as we choose ##ψ,\phi## to be the identity maps.

Does this seem fine?

For the other direction, we suppose that for any open ##U ⊂ Y## and any ##φ ∈ C^∞(U)## the composition ##f^{-1}(U)→ U → R \in C^∞(f^{-1}(U))##. And we want to see that ##f## is smooth globaly (the definiton above). How to show this with charts..?

A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition

$$\psi ◦ f ◦ \phi^{-1}$$ is smooth.

Claim: A map ##f : X → Y## between two smooth manifolds is smooth iff for any open ##U ⊂ Y## and any ##φ ∈ C^∞(U)## the composition ##f^{-1}(U)→ U → R \in C^∞(f^{-1}(U))##.

My question is how to show this claim?

For one side, I can use the definition above:

a function ##φ∈C^∞(U)## on an open subset ##U⊂Y##

is a smooth function from ##U## to ##R##.Therefore, we can take the charts ##\phi:f^{−1}(U)→R^m##

and ##ψ:U→R^n## to be the identity maps, and we have: $$ψ◦f◦{\phi}^{−1}:R^m→R^n$$

which is a smooth map between Euclidean spaces. Composing this with ##φ## gives a smooth function on $$f^{−1}(U):

φ◦(ψ◦f◦{\phi}^{−1}):f^{−1}(U)→R$$

and this is the map ##φ∘f##

as we choose ##ψ,\phi## to be the identity maps.

Does this seem fine?

For the other direction, we suppose that for any open ##U ⊂ Y## and any ##φ ∈ C^∞(U)## the composition ##f^{-1}(U)→ U → R \in C^∞(f^{-1}(U))##. And we want to see that ##f## is smooth globaly (the definiton above). How to show this with charts..?

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