nLab
tubular neighbourhood

Context

Manifolds and cobordisms

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

Definition

For i:XYi : X \hookrightarrow Y an embedding of manifolds, a tubular neighbourhood of XX in YY is

Remark

The derivative of i^\hat i provides an isomorphism of EE with the normal bundle ν X/Y\nu_{X/Y} of XX in YY.

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:XYi : X \to Y, write Tub(i)Tub(i) for the topological space whose underlying set is the set of tubular neighbourhoods of ii and whose topology is the subspace topology of Hom(N iX,Y)Hom(N_i X, Y) equipped with the C-infinity topology.

Proposition

If XX and YY are compact manifolds, then Tub(i)Tub(i) is contractible for all embeddings i:XYi : 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

see also

Revised on June 2, 2013 02:44:28 by Beren Sanders (76.171.102.144)