I try generalising the following "classical" result to the abstract fibre bundle (Rudolph-Schmidt, Differential Geometry and Mathematical Physics part I)
I can state it as follows.
Global result Let $p\colon E\to B$ be a fibre bundle and $N$ a submanifold of $E$. Let us assume that $N$ intersects transversally the fibres of $p$ at most once. If $N$ shares the dimension of $B$, then $U=p(N)$ is open in $B$, and $N=s(U)$ for some $s\in \Gamma_U(p)$.
Now, I'm considering what happens if I drop the dimensional assumption on $N$. I conjecture something like that (but I need to figure out how to make it more formal).
Global result Let $p\colon E\to B$ be a fibre bundle and $N$ a submanifold of $E$. Let us assume that $N$ intersects transversally the fibres of $p$ at most once. Then, there is a local section $s$ of $p$ such that $N$ contains the image of $s$ as a submanifold.
Some ideas The transversality condition reads for any $a\in N$ $$T_a N+\ker p_a'=T_a E\,.$$ In particular, by the Grassman formula, we have: $$\dim(T_a N\cap \ker p_a')=\max(\dim N-\dim B,0)\,.$$ If $\dim N=\dim B$, $p_a'$ is an immersion for any $a\in N$. Thus, $p_{\restriction N}$ is an injective immersion. Moreover, $U$ is open for dimensional reasons. Therefore, $p_{\restriction N}$ is a diffeomorphism onto $U$. Its inverse defines the section of $p$ we are looking for.
I cannot replicate the above arguments if $\dim N>\dim B$. Probably, I have to study the distribution $T N\cap \ker p'$.
Would you happen to have any suggestions? Thanks for considering my request.