Normal and conormal bundles of an embedded submanifold

Let $S\subseteq M$ be an embedded submanifold.

Equip $S$ with the smooth structure supplied by Slice-chart restrictions form a smooth atlas. The inclusion $i:S\hookrightarrow M$ is then a smooth embedding by The inclusion of an embedded submanifold is a smooth embedding, so its differential is defined. Identify $T_{p}S$ with the linear subspace $d{i}_p(T_{p}S)\subseteq T_{p}M$. Here the notation $TM{\mathrm{\mid }}_S:={\coprod }_{p\in S}{T}_{p}M,T^{\ast }M{\mathrm{\mid }}_S:={\coprod }_{p\in S}{T}_p^{\ast }M$ means the fibrewise restriction of these disjoint unions to base points in $S$; it does not invoke restriction to an open subset.

The normal-bundle set of $S$ in $M$ is the fibrewise quotient

$\nu (S):={\coprod }_{p\in S}{T}_{p}M\mathrm{/}{T}_{p}S\mathrm{.}$

The conormal-bundle set of $S$ in $M$ is the fibrewise annihilator

$N^{\ast }S:=\{\xi \in T^{\ast }M{\mathrm{\mid }}_S:\xi {\mathrm{\mid }}_{T_{p}S}=0\text{ for the base point }p\}\mathrm{.}$

Both are intrinsic constructions attached to the embedding $S\hookrightarrow M$. The next proposition supplies their smooth vector-bundle structures.