nLab tubular neighborhood

Contents

Definition

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

$\hat i : E \to U_{i(X)}$

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 does admit a tubular neighbourhood.

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).

Ana Cannas da Silva, §3 of: Prerequisites from differential geometry [pdf]
Ana Cannas da Silva, §6.2 in: Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer (2008) [doi:10.1007/978-3-540-45330-7, pdf]

The homotopical uniqueness of tubular neighbourhoods is discussed in

For an analogue in homotopical algebraic geometry see