nLab tubular neighborhood

Contents

Context

Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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

The homotopical uniqueness of tubular neighbourhoods is discussed in

For an analogue in homotopical algebraic geometry see