Proposition 8.19 Suppose $M$ and $N$ are smooth manifolds with or without boundary, and $F:M\to N$ is a diffeomorphism. For every $X\in\mathfrak{X}(M)$, there is a unique smooth vector field on $N$ that is $F$-related to $X$.
Problem 8-9. Show by finding a counterexample that Proposition 8.19 is false if we replace the assumption that $F$ is a smooth diffeomorphism by the weaker assumption that it is smooth and bijective.
My solution:
I started by thinking of a smooth bijection that is not a diffeomorphism. Let $M=\mathbb{R}=N$. Then $F(x):=x^3$ is such a map, because its inverse is not smooth at $0$. Let $X=d/dx$. Then
$$dF_x(X_x)f=\frac{d}{dx}(f\circ F)=\frac{d}{dx}(f(x^3))=3x^2\frac{df}{dx}\,.$$
Let $Y=\alpha(x)\frac{d}{dx}$. Then
$$Y_{F(x)}=\alpha(x^3)\frac{d}{dx}\,,$$
so for $Y$ to be $F$-related to $X$, we need $\alpha(x^3)=3x^2$, which implies that $\alpha(x)=3x^{2/3}$, so $\alpha$ is not smooth, and thus $Y\notin\mathfrak{X}(\mathbb{R})$.
Is this correct?
Best Answer
I think your example is fine. I have one that I think this one works also, but it is more complicated (not as good as) than yours, and I am posting it as an answer, but it's really the same question as yours. That is, is it correct?
Let $p:[0,1)\to \mathbb S^1$ be the restriction of the usual covering map of $\mathbb S^1$. Then, if $X_s=(1-2s)\left(\frac{d}{dt}\right)_s$, we have
$dp(X_s)f=(1-2s)\left(\frac{df(e^{2\pi it})}{dt}\right)_s=2\pi i(1-2s)e^{2\pi i s}f'(e^{2\pi i s})$
so, using the notation of your example, we would need $2\pi i(1-2s)e^{2\pi i s}=\alpha(e^{2\pi is})$.
Now, if $s=0$, then $\alpha(1)=2\pi i$. On the other hand, if we let $s\to 1^-$, then continuity of the composition implies that the RHS is $\alpha(s)\to \alpha(1)$ while the LHS is $-2\pi i$ so there is no way to define $\alpha$ at $(1,0)\in \mathbb S^1$