nLab tubular neighbourhood

Context

Manifolds and cobordisms

manifolds and cobordisms

Theorem

Differential geometry

differential geometry

synthetic differential geometry

Contents

Definition

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 $Y$.

Remark

The derivative of $\hat i$ provides an isomorphism of $E$ with the normal bundle $\nu_{X/Y}$ of $X$ in $Y$.

Properties

General

Proposition

(tubular neighbourhood theorem)

Every embedding does admit a tubular neighbourhood.

For instance (DaSilva, theorem 3.1).

Moreover, tubular neighbourhoods are unique up to homotopy in a suitable sense:

Definition

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.

Proposition

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).

Pullbacks of tubular neighbourhoods

(…) propagating flow (…) (Godin).

References

Basics on tubular neighbourhoods are for instance in section 3 of

• Ana Cannas da Silva, Prerequisites from differential geometry (pdf)

The homotopical uniqueness of tubular neighbourhoods is discussed in

For an analogue in homotopical algebraic geometry see