I have the following problem:
Let $\pi:M\to N$ a surjective local diffeomorphism. Let $X\in\mathfrak{X}(M), Y\in\mathfrak{X}(M)$ be $\pi$-related, i.e. $(d\pi)_p X_p=Y_{\pi(p)}$ for all $p\in M$. Suppose that $Y$ is complete (it means all its integral curves are defined for $t\in\mathbb{R}$), does this imply that $X$ is complete?
I don't know if its true or false, I tried to think about examples where $N$ is compact so every vector field is complete, but I've stuck because I found it difficult to find whether exist $X$ a $\pi$-related vector field with $Y$.
Thank you in advance!
I think I've encountered a counterexample (It would be helpful if someone could check this)
Let $F:(-\varepsilon, 2\pi)\to S^1$ be given by $F(t)=(\cos t, \sin t)$. It is a surjective local diffeomorphism.
The vector field $X=\frac{d}{dt}$ is not complete on $(-\varepsilon, 2\pi)$ but it is $F$ related to the vector field $Y=-y \frac{\partial}{\partial x}+x \frac{\partial}{\partial y}$, which is tangent to the sphere, then there exists $Z\in\mathfrak{X}(S^1)$ such that $Y_{\iota(p)}=(d\iota)_p Z_p$ for all $p\in S^1$, but if $t\in(-\varepsilon, 2\pi)$, we have $Y_{\iota(F(t))}=(d\iota)_{F(t)} Z_{F(t)}$ and by the other hand we have $$Y_{\iota(F(t))}=Y_{F(t)}=(dF)_t X_t=(d\iota)_{F(t)} (dF)_t X_t$$ so $(dF)_t X_t=Z_{F(t)}$. Therefore, $X$ is $F$-related with $Z$, $Z$ is complete ($S^1$ is compact) but $X$ not.