Do Lie algebroids pull back (along submersions)?
Regarding the linked question, I am interested in pulling back only the anchor map, ignoring the bracket. Formally, let
- $p_E:E\rightarrow M$ and $p_F:F\rightarrow N$ be vector bundles,
- $a:E\rightarrow TM$ an anchor map (i.e. a vector bundle morphism over $M$),
- and $\varphi:F\rightarrow E$ a vector bundle morphism covering a smooth map $f:N\rightarrow M$ (so $f\circ p_F = p_E\circ \varphi$).
Clearly, the composition $a\circ\varphi:F\rightarrow TM$ is also a vector bundle morphism.
From here, I would like to construct a new anchor map $a^\varphi:F\rightarrow TN$. However, it seems that this is not what people usually mean by the pullback anchor map (see the references in the linked question for more details).
Here is my attempt (with some conditions on $F$ and $f$): suppose that
- $f$ is an injective immersion with derivative $T(f):TN\rightarrow TM$,
- and $\mathrm{Im}(a\circ\varphi)\subseteq \mathrm{Im}(T(f))$.
Then, for each $q\in N$ and $v\in F_q$, there exists a unique $w\in T_qN$ such that
$$(a\circ\varphi)(v) = T(f)(w).$$
This defines a map $a^\varphi:F\rightarrow TN$.
My questions are:
- Is $a^\varphi$ a smooth vector bundle morphism?
- Why is this not the standard definition of the pullback anchor map? Am I overlooking something by ignoring the bracket?
Thanks in advance.
Note for experts (not me): the example I have in mind does not have a surjective anchor map (is not transitive).
