Another definition of a smooth map between manifolds

Question

definition (1) of a smooth map is 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.

And here is another definition/claim: A map $f : X → Y$ between two smooth manifolds is smooth iff for any open $V ⊂ Y$ and any $φ ∈ C^∞(V)$ the composition $f^{-1}(V)\xrightarrow{f} V \xrightarrow{φ} R$ is in $C^∞(f^{-1}(V))$.

My question is how to show this claim?

For one side, if we consider the manifolds $X, Y$ as subsets of euclidean space. Then I can use the definition (1) above:

a function $φ∈C^∞(V)$ on an open subset $V⊂Y$ is a smooth function from $V$ to $R$.Therefore, we can take the charts $\phi:f^{−1}(V)→R^m$ and $ψ:V→R^n$ to be the identity maps, and we have: $ψ◦f◦{\phi}^{−1}: \phi(f^{-1}(V)=f^{-1}(V)→R^n$ which is a smooth map between Euclidean spaces. Composing this with $φ$ gives a smooth function on $f^{−1}(V) : φ◦(ψ◦f◦{\phi}^{−1}):f^{−1}(V)→R$ and this is the map $φ∘f$ as we chose $ψ,\phi$ to be the identity maps. Does this seem okay?

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

Answer 1

For maps $f : U \to V$ between open subsets $U \subset \mathbb R^m, V \subset \mathbb R^n$ of Euclidean spaces we have a standard definition of being smooth from multivariable calculus. For the moment let us call this property c-smooth. Your definition (1) explains when a map $f : X \to Y$ between smooth manifolds $X,Y$ is called smooth. For the moment let us call this property m-smooth.

Let us collect some well-known facts.

1. Each Euclidean space $\mathbb R^m$ has a canonical structure of a smooth manifold (with smooth atlas given by the identity $id: \mathbb R^m \to \mathbb R^m$).

2. Each open subset $U \subset X$ of a smooth manifold $X$ has a canonical structure of a smooth manifold. Its smooth charts are the smooth charts on $X$ whose domains are contained in $U$. Alternatively, if we have a smooth atlas $\mathcal A$ on $X$ (not necessarily the maximal smooth atlas on $X$), then we get a smooth atlas $\mathcal A \mid_U$ on $U$ by taking the restriction of each chart in $\mathcal A$ to the intersection of its domain with $U$.
   In particular, open subsets $U \subset \mathbb R^m$ have a canonical structure of a smooth manifold (with smooth atlas given by the inclusion $i: U \to \mathbb R^m$).

3. A map $f : X \to Y$ between smooth manifolds $X,Y$ is m-smooth if and only if for each $x \in X$ there exist charts $\phi : U \to \mathbb R^m$ on $X$ with $x \in U$ and $\psi : V \to \mathbb R^n$ on $Y$ with $f(x) \in V$ such that $\psi \circ f \circ \phi^{-1}$ is c-smooth.

This allows to prove easily various well-known corollaries:

4. A map $f : U \to V$ between open subsets of Euclidean spaces is c-smooth if and only if it is m-smooth. Thus we no longer need to use the phrases c-smooth and m-smooth to distinguish both concepts; we can use the word smooth for both cases.

5. The composition of smooth maps between smooth manifolds is smooth.

6. Let $f : X \to Y$ be a smooth map between smooth manifolds and $U \subset X, V \subset Y$ be open such that $f(U) \subset V$. Then $f_{U,V}: U \xrightarrow{f} V$ is smooth.

7. A map $f : X \to \mathbb R^n$ on a smooth manifold $X$ is smooth if and only if for each chart $\phi : U \to \mathbb R^m$ on $X$ the composition $f \circ \phi^{-1} : \phi(U) \to \mathbb R^n$ is smooth (which can be understood as c-smooth).

Let us use these facts to prove three lemmas.

Lemma 1. Each chart $\phi: U \to \mathbb R^m$ on a smooth manifold $X$ establishes a diffeomorphism $\phi_U : U \xrightarrow{\phi} \phi(U)$ between the smooth manifolds $U \subset X$ and $\phi(U) \subset \mathbb R^m$.

Proof. Take the charts $\phi$ on $U$ and $i: \phi(U) \hookrightarrow \mathbb R^m$ on $\phi(U)$ and apply 3. to see that $\phi_U$ and $\phi_U^{-1}$ are smooth.

Lemma 2. A map $f : X \to Y$ between smooth manifolds is smooth if and only if for each chart $\psi : V \to \mathbb R^n$ on $Y$ the composition $\psi \circ f : f^{-1}(V) \xrightarrow{f} V \xrightarrow{\psi} \mathbb R^n$ is smooth.

Proof. If $f$ is smooth, then clearly all $\psi \circ f$ are smooth. Conversely, let all $\psi \circ f$ be smooth. By 7. all $\psi \circ f \circ \phi^{-1}$ with charts $\phi: U \to \mathbb R^m$ on $X$ and $\psi: V \to \mathbb R^n$ on $Y$ are smooth. Thus $f$ is smooth.

Lemma 3. A map $f : X \to \mathbb R^n$ on a smooth manifold $X$ is smooth if and only all coordinate functions $f_i : X \to \mathbb R$ are smooth.

Proof. If $f$ is smooth, then all $f_i$ are smooth because $f_i = p_i \circ f$ with projections $p_i : \mathbb R^n \to \mathbb R$ which are smooth. Conversely, let all $f_i$ be smooth. By 7. it suffices to show that for each chart $\phi : U \to \mathbb R^m$ on $X$ the composition $f^\phi = f \circ \phi^{-1}$ is smooth. It is well-known from multivariable calculus that $f^\phi$ is smooth if and only if all coordinate functions $f_i^\phi$ are smooth. But we have $f_i^\phi = f_i \circ \phi^{-1}$ which are smooth maps by Lemma 1.

By $C^\infty(X)$ we denote the set of all smooth maps $f : X \to \mathbb R$ on a smooth manifold $X$. This notation applies automatically also to open $U \subset X$.

Let us prove your claim.

If $f$ is smooth, then trivially all $\varphi \circ f : f^{-1}(V) \to \mathbb R$ are smooth.

Conversely, let all $\varphi \circ f : f^{-1}(V) \to \mathbb R$ be smooth. Consider a chart $\psi : V \to \mathbb R^n$ on $Y$. Since all coordinate functions $\psi_i : V \to \mathbb R$ are smooth, all $\psi_i \circ f$ are smooth. Thus $\psi \circ f$ is smooth by Lemma 3. Now apply Lemma 2.