Differentiable Manifolds/Diffeomorphisms and related vector fields

Diffeomorphisms
We shall now define the notions of homeomorphisms and diffeomorphisms for mappings between manifolds.

The rank of the differential
The dimension of $$\text{Im } d \psi_p$$ is well-defined since $$d \psi_p$$ is a linear function, which is why it's image is a vector space; further, it is a vector subspace of $$T_{\psi(p)} N$$, which is a $$b$$-dimensional vector space, which is why it has finite dimension.

Related vector fields
Proof:

1. We show that $$\mathbf V$$ and $$\mathbf W$$ are $$\psi$$-related.

Let $$p \in M$$ be arbitrary. Then we have:
 * $$\mathbf W(\psi(p)) = d \psi_{\psi^{-1}(\psi(p))}(\mathbf V(\psi^{-1}(\psi(p)))) = d\psi_p (\mathbf V(p))$$

2. We show that there are no other vector fields besides $$\mathbf W$$ which are $$\psi$$-related to $$\mathbf V$$.

Let $$\mathbf Z$$ be also contained in $$\mathfrak X(N)$$ such that $$\mathbf V$$ and $$\mathbf Z$$ are $$\psi$$-related. We show that $$\mathbf Z = \mathbf W$$, thereby excluding the possibility of a different to $$\mathcal X$$ $$\psi$$-related vector field.

Indeed, for every $$q \in N$$ we have:

Due to the bijectivity of $$\psi$$, there exists a unique $$p \in M$$ such that $$\psi(p) = q$$, and we have $$p = \psi^{-1}(q)$$. Therefore, and since $$\mathbf Z$$ was required to be $$\psi$$-related to $$\mathbf V$$:
 * $$\mathbf Z(q) = \mathbf Z(\psi(p)) = d \psi_p (\mathbf V(p)) = d \psi_{\psi^{-1}(q)} (\mathbf V(\psi^{-1}(q))) = \mathbf W(q)$$

Proof:

Let $$\vartheta \in \mathcal C^m(N)$$. Inserting a few definitions from chapter 2, we obtain
 * $$\begin{align}

\mathbf W \vartheta (p) & = (d \psi_{\psi^{-1}(p)} (\mathbf V (\psi^{-1}(p))))(\vartheta) \\ & = (\mathbf V (\psi^{-1}(p)) \circ \psi^*)(\vartheta) \\ & = \mathbf V (\psi^{-1}(p)) (\vartheta \circ \psi) \\ & = \mathbf V (\vartheta \circ \psi) (\psi^{-1}(p)) \\ & = \left( \left( \psi^{-1} \right)^* \mathbf V (\vartheta \circ \psi) \right)(p) \\ \end{align}$$ , and therefore
 * $$\mathbf W \vartheta = \left( \psi^{-1} \right)^* \mathbf V (\vartheta \circ \psi)$$

Since $$\psi$$ is differentiable of class $$\mathcal C^j$$ and $$j \ge m$$, $$\psi$$ is also differentiable of class $$\mathcal C^m$$. Further, the function
 * $$\mathbf V (\vartheta \circ \psi)$$

is differentiable of class $$\mathcal C^m$$, because $$\mathbf V$$ is differentiable of class $$\mathcal C^m$$. Due to lemma 2.17, it follows that also
 * $$\mathbf W \vartheta = \left( \psi^{-1} \right)^* \mathbf V (\vartheta \circ \psi)$$

is differentiable of class $$\mathcal C^m$$, and therefore, due to the definition of differentiability of vector fields, so is $$\mathbf W$$.