Why must subbundle be an embedded submanifold?

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:

1. It is a topological embedding.
2. 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?

• note that $i$ maps $D$ into $E$, not into $M$. (this is not an attempt to answer your question, but it may be a hint to a misconception you may have) – Thomas Sep 27 '18 at 9:10
• If it's not an embedding, then your trivialization maps may not restrict to homeomorphisms, since $D$ doesn't necessarily inherit the subspace topology of $E$. It may be still true that $D\to M$ is a smooth vector bundle, but it's not a subbundle of $E\to M$. – Matt Sep 27 '18 at 10:00
• ＠Thomas, thaanks that was a typo. @Matt is it possible that: D is a subbundle , D has subspace topology, but $D \rightarrow E$ is not an embedding? My problem is that I feel the condition of smooth immersion is "forced", if you get what I mean, but could not formalize it. – CL. Sep 27 '18 at 10:36
• @CL I don't understand what you're asking. If you don't have an immersion, then you can't think of $T_pD$ as a subspace of $T_pE$. So $D\subset E$, but the smooth structures aren't compatible in a nice way. This defeats the purpose of working in the smooth category. You could just have $D$ is a topological embedding into $E$ over some topological space $M$ and then have talking about the (topological) subbundle. If it's not an embedding, then you just have two smooth vector bundles $E\to M$ and $D\to M$ that have some overlapping (set) characteristics. – Matt Sep 28 '18 at 17:29
• @CL I think this would be similar to saying you have a smooth map $f:M\to N$ of smooth manifolds, and trying to say $T_pM\leq T_{f(p)}N$. You can certainly make sense of the pushforward, but you don't consider $M$ a submanifold of $N$. So why should you consider $D$ a subbundle of $E$? – Matt Sep 28 '18 at 17:31