Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where $M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where $N'$ is also an $n$-manifold.
Suppose there is fibre bundle $\xi': N'\longrightarrow E'\longrightarrow M'$.
Question. Could we impose any conditions on $M$, $M'$, $N$, $N'$ and $\xi'$ such that there exists a fibre bundle
$\xi: N\longrightarrow E\longrightarrow M$
satisfying $\xi'$ is a sub-bundle of $\xi\mid_{M'}$, where $\xi\mid_{M'}$ is the restriction of $\xi$ to $M'$?
Are there any references? Thanks.
