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 $Y$.
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).
Basics on tubular neighbourhoods are for instance in section 3 of
The homotopical uniqueness of tubular neighbourhoods is discussed in
For an analogue in homotopical algebraic geometry see
see also