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?
Best Answer
To pull back vector fields on $N$ it suffices for $F : M \to N$ to be a local diffeomorphism (at each point of $M$):
If $Y$ is a smooth vector field on $\operatorname{Im} F$ (i.e. the image of $F$, which is an open subset of $N$ since $F : M \to N$ is an open map) then you can define a vector field $X$ on $M$ by $m \in M \mapsto \left( \operatorname{T}_m F \right)^{-1}(Y_{F(m)})$, which is well-defined since each $\operatorname{T}_m F : \operatorname{T}_m M \to \operatorname{T}_{F(m)} N$ is a bijection. $X$ will be smooth since for any $m \in M$ we can select an open (in $M$) neighbourhood $U$ of $m$ such that $F\big\vert_U : U \to F(U)$ is a diffeomorphism onto the open subset $F(U)$ of $N$. Note that $X\big\vert_U$ is equal to the pushforward $X\big\vert_U = \left( F\big\vert_U^{-1} \right)_{*}\left( Y\big\vert_{F(U)} \right) \circ F\big\vert_U$, which is smooth since it is a composition of the smooth maps $$U \xrightarrow{F\vert_U} F(U) \xrightarrow{Y\vert_{F(U)}} \operatorname{T}_{F(U)} N \xrightarrow{\left( F\vert_U^{-1} \, \right)_{*}} \operatorname{T}_{U} M$$ so that $X$ is smooth around each point of $M$.
Edit: Using the above result, by going into coordinates it is then straightforward to show that if $F : M \to N$ is an immersion and $Y_{F(m)} \in \operatorname{Im} \left( \operatorname{T}_m F \right)$ for all $m \in M$ then the map $X : M \to \operatorname{T} M$ defined by $m \mapsto \left( \operatorname{T}_m F \right)^{-1}(Y_{F(m)})$ is a well-defined smooth vector field on $M$ that can be pushed-forward to $Y\big\vert_{\operatorname{Im} F}$. Note that if $\operatorname{Im} F$ is dense in $N$ then the continuous map $Y$ is completely determined by $Y\big\vert_{\operatorname{Im} F}$ so that in this sense, $X$ could be viewed as "essentially" being the pullback of the entire vector field $Y$.