In John Lee's smooth manifold, pg 264, or 199 in this version, he defines
Let $\pi:E \rightarrow M$ be a vector smooth bundle. A subbundle $\pi|_D:D \rightarrow M$ is called a smooth subbundle if it is smooth vector bundle and an embedded submanifold with or without boundary.
Being embedded in Lee's definition has two conditions:
- It is a topological embedding.
- It is a smooth immersion. That is $di_p :T_pD \rightarrow T_pE$ is injective for all $p \in D$.
$D$ can be given subspace topology. 1. is then satisfied.
But why do we require in the definition that it is a smooth immersion? Is this condition additonal? Or could actually be deduced?