For $i : X \hookrightarrow Y$ an embedding of manifolds, a tubular neighbourhood of $X$ in $Y$ is
a real vector bundle $E \to X$;
an extension of $i$ to an isomorphism
with an open neighbourhood of $X$ in $Y$.
The derivative of $\hat i$ provides an isomorphism of $E$ with the normal bundle $\nu_{X/Y}$ of $X$ in $Y$.
(tubular neighbourhood theorem)
Every embedding of smooth manifolds does admit a tubular neighbourhood, def. 1.
For instance (DaSilva, theorem 3.1).
Moreover, tubular neighbourhoods are unique up to homotopy in a suitable sense:
For an embedding $i : X \to Y$, write $Tub(i)$ for the topological space whose underlying set is the set of tubular neighbourhoods of $i$ and whose topology is the subspace topology of $Hom(N_i X, Y)$ equipped with the C-infinity topology.
If $X$ and $Y$ are compact manifolds, then $Tub(i)$ is contractible for all embeddings $i : X \to Y$.
This appears as (Godin, prop. 31).
(…) propagating flow (…) (Godin).
key application: Pontrjagin-Thom collapse map
Basics on tubular neighbourhoods are reviewed for instance in
Ana Cannas da Silva, section 3 of Prerequisites from differential geometry (pdf)
Stanley Kochmann, section 1.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
The homotopical uniqueness of tubular neighbourhoods is discussed in
For an analogue in homotopical algebraic geometry see
see also