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tubular neighbourhood

Contents

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& \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}{}& 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. .

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).

Equivariant version

These statements generalize to equivariant differential topology:

Proposition

(fixed loci of smooth proper actions are submanifolds)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

Then the GG-fixed locus X GXX^G \hookrightarrow X is a smooth submanifold.

If in addition XX is equipped with a Riemannian metric and GG acts by isometries then the submanifold X GX^G is a totally geodesic submanifold.

(e.g. Ziller 13, theorem 3.5.2, see also this MO discussion)

Proof

Let xX GXx \in X^G \subset X be any fixed point. Since this is in particular a closed invariant submanifold, Prop. applies and shows that an open neighbourhood of xx in XX is GG-equivariantly diffeomorphic to a linear representation VRO(G)V \in RO(G). The fixed locus V GVV^G \subset V of that is hence diffeomorphic to an open neighbourhood of xx in Σ\Sigma.

Remark

Without the assumption of proper action in Prop. the conclusion generally fails. See this MO comment for a counter-example.

Proposition/Definition

(GG-action on normal bundle to fixed locus)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

Then linearization of the GG-action aroujnd the fixed locus X GXX^G \subset X equips the normal bundle N X(X G)N_X\left( X^G\right) with smooth and fiber-wise linear GG-action.

(e.g. Crainic-Struchiner 13, Example 1.7)

Proposition

(existence of GG-invariant Riemannian metric)

Let XX be a smooth manifold, GG a compact Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

Then there exists a Riemannian metric on XX with its invariant with respect to the GG-action, hence such that all elements of GG act by isometries.

(Bredon 72,VI Theorem 2.1, see also Ziller 13, Theorem 3.0.2)

Definition

(GG-invariant tubular neighbourhood)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

For ΣX GX\Sigma \subset X^G \subset X a closed smooth submanifold inside the fixed locus, a GG-invariant tubular neighbourhood 𝒩(ΣX)\mathcal{N}(\Sigma \subset X) of Σ\Sigma in XX is

  1. a smooth vector bundle EΣE \to \Sigma equipped with a fiber-wise linear GG-action;

  2. an equivariant diffeomorphism EXE \overset{}{\longrightarrow} X onto an open neighbourhood of Σ\Sigma in XX which takes the zero section identically to Σ\Sigma.

Proposition

(existence of GG-invariant tubular neighbourhoods)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

If ΣιX\Sigma \overset{\iota}{\hookrightarrow} X is a closed smooth submanifold inside the GG-fixed locus

then

  1. Σ\Sigma admits a GG-invariant tubular neighbourhood ΣUX\Sigma \subset U \subset X (Def. );

  2. any two choices of such GG-invariant tubular neighbourhoods are GG-equivariantly isotopic;

  3. there always exists an GG-invariant tubular neighbourhood parametrized specifically by the normal bundle N(ΣX)N(\Sigma \subset X) of SigmaSigma in XX, equipped with its induced GG-action from Def. , and such that the GG-equivariant diffeomorphism is given by the exponential map

    exp ϵ:N(ΣX)𝒩(ΣX) \exp_\epsilon \;\colon\; N(\Sigma \subset X) \overset{\simeq}{\longrightarrow} \mathcal{N}(\Sigma \subset X)

with respect to a GG-invariant Riemannian metric (which exists according to Prop. ):

The existence of the GG-invariant tubular neighbourhoods is for instance in Bredon 72 VI Theorem 2.2, Kankaanrinta 07, theorem 4.4. The uniqueness up to equivariant isotopy is in Kankaanrinta 07, theorem 4.4, theorem 4.6. The fact that one may always use the normal bundle appears at the end of the proof of Bredon 72 VI Theorem 2.2, and as a special case of a more general statement about invariant tubular neighbourhoods in Lie groupoids it follows from Pflaum-Posthuma-Tang 11, Theorem 4.1 by applying the construction there to each point in Σ\Sigma for one and the same choice of background metric. See also for instance Pflaum-Wilkin 17, Example 2.5.

Pullbacks of tubular neighbourhoods

(…) propagating flow (…) (Godin).

References

Basics on tubular neighbourhoods are reviewed for instance in

Discussion in the generality of equivariant differential topology includes

  • Glen Bredon, Chapter VI.2 of Introduction to compact transformation groups, Academic Press 1972 (pdf)

  • Glen Bredon, Introduction to compact transformation groups, Academic Press 1972 (pdf)

  • Marja Kankaanrinta, Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions, Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 (euclid:agt/1513796653)

  • Markus Pflaum, Hessel Posthuma, X. Tang, Geometry of orbit spaces of proper Lie groupoids, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 (arXiv:1101.0180, doi:10.1515/crelle-2012-0092)

  • Wolfgang Ziller, Group actions, 2013 (pdf)

  • Marius Crainic, Ivan Struchiner, On the linearization theorem for proper Lie groupoids, Annales scientifiques de l’École Normale Supérieure, Série 4, Volume 46 (2013) no. 5, p. 723-746 (numdam:ASENS_2013_4_46_5_723_0 doi:10.24033/asens.2200)

  • Markus Pflaum, Graeme Wilkin, Equivariant control data and neighborhood deformation retractions, Methods and Applications of Analysis, 2019 (arXiv:1706.09539)

The homotopical uniqueness of tubular neighbourhoods is discussed in

For an analogue in homotopical algebraic geometry see

see also

Last revised on February 8, 2019 at 03:46:08. See the history of this page for a list of all contributions to it.