nLab
tubular neighbourhood

Context

Manifolds and cobordisms

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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: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 of smooth manifolds does admit a tubular neighbourhood, def. 1.

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 reviewed for instance in

The homotopical uniqueness of tubular neighbourhoods is discussed in

For an analogue in homotopical algebraic geometry see

see also

Revised on May 30, 2016 13:44:14 by Urs Schreiber (131.220.184.222)