I want to find conditions under which one can pull-back vector fields (if it is at all possible).
Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. Let $Y$ be a vector field on $N$ (i.e. smooth section of the tangent bundle $TN$). Define: $T^\ast Y(p)(f)=Y(F(p))(f\circ F^{-1})$, where $f \in C^{\infty}(M,\mathbb R)$. We check that $T^\ast Y$ is a derivation, and this is true since:
$T^\ast Y(p)(fg)=Y(F(p)(fg \circ F^{-1})=Y(F(p)((f \circ F^{-1})(g \circ F^{-1}))$
My question: Is it enough for $F$ to be a local diffeomorphism for this to work?