nLab
Ehresmann connection

Context

\infty-Chern-Weil theory

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The notion of Ehresmann connection is one of the various equivalent definitions of connection on a bundle.

Definition

In terms of differential forms

Let GG be a Lie group with Lie algebra 𝔤\mathfrak{g} and PXP \to X a GG-principal bundle. Let

ρ:P×GP \rho : P \times G \to P

be the action of GG on PP and

ρ *:𝔤Γ(TX) \rho_* : \mathfrak{g} \to \Gamma(T X)

its derivative, sending each element x𝔤x \in \mathfrak{g} to the vector field on PP that at pPp \in P is the push-forward ρ(p,) *(x)\rho(p,-)_*(x).

For vΓ(TX)v \in \Gamma(T X) and ω\omega a differential form on PP write ι vω\iota_v \omega for the contraction.

Definition

A Cartan-Ehresmann connection on PP is a Lie algebra-valued 1-form

AΩ 1(P,𝔤) A\in \Omega^1(P, \mathfrak{g})

on PP satisfying two conditions

  1. first Ehresmann condition

    for every x𝔤x \in \mathfrak{g} we have

    ι ρ *(x)A=x. \iota_{\rho_*(x)} A = x \,.
  2. second Ehresmann condition

    for every x𝔤x \in \mathfrak{g} we have

    ρ *(x)A=ad xA, \mathcal{L}_{\rho_*(x)} A = ad_x A \,,

    where ρ *(x)\mathcal{L}_{\rho_*(x)} is the Lie derivative along ρ *(x)\rho_*(x) and where ad x:𝔤𝔤ad_x : \mathfrak{g} \to \mathfrak{g} is the adjoint action of 𝔤\mathfrak{g} on itself.

Proposition

This is equivalent to

  1. first Ehresmann condition

    for every x𝔤x \in \mathfrak{g} we have

    ι ρ *(x)A=x. \iota_{\rho_*(x)} A = x \,.
  2. second Ehresmann condition

    for every x𝔤x \in \mathfrak{g} we have

    ι ρ *(x)F A=0, \iota_{\rho_*(x)} F_A = 0 \,,

    where F AΩ 2(P,𝔤)F_A \in \Omega^2(P, \mathfrak{g}) is the curvature 2-form of AA.

Proof

Using ι xA=x\iota_x A = x we have by Cartan calculus

ι ρ *(x)F A =ι ρ *(x)d dRA+12[AA] = ρ *(x)Ad dRι ρ *(x)A+[x,A] = ρ *(x)A+[x,A]. \begin{aligned} \iota_{\rho_*(x)} F_A &= \iota_{\rho_*(x)} d_{dR} A + \frac{1}{2}[A\wedge A] \\ & = \mathcal{L}_{\rho_*(x)} A - d_{dR} \iota_{\rho_*(x)} A + [x,A] \\ & = \mathcal{L}_{\rho_*(x)} A + [x,A] \,. \end{aligned}

In terms of distributions

Given a smooth bundle π:EX\pi: E\to X with typical fiber FF (e.g. a smooth vectir bundle or a smooth principal GG-bundle), there is a well defined vector subbundle VETEV E\subset T E over EE such that V pV_p consists of the tangent vectors v pv_p such that (T pπ)(v p)=0(T_p \pi)(v_p) = 0. A smooth distribution (field) of horizontal subspaces is a choice of a vector subspace H pET pEH_p E\subset T_p E for every pp such that

E1. (complementarity) T uE=H uEV uET_u E \: = \: H_u E \oplus V_u E

E2. pH pEp\mapsto H_p E is smooth. That means that in the unique decomposition of any smooth vector field XX on EE into vector fields X HΓ(H uE)X^H \in \Gamma(H_u E) and X VΓ(V uE)X^V \in \Gamma(V_u E) such that X=X H+X VX = X^H + X^V the vector field X HX^H is smooth (or equivalently X VX^V is smooth, or equivalently both) as a section of TET E (there exist yet several other equivalent formulations of the smooothness criterium).

An Ehresmann connection describes a connection on a GG-principal bundle π:PX\pi : P \to X (for GG some Lie group) in terms of a distribution of horizontal subspaces HTPH \subset T P which is a subbundle of the tangent bundle of PP complementary at each point to the vertical tangent bundle to the fiber. More precisely, an Ehresmann connection on a principal GG-bundle π:PX\pi:P\to X is a smooth distribution of horizontal subspaces pH pPp\mapsto H_p P which is equivariant:

E3. H pgP=(T pR g)H pPH_{p g}P = (T_p R_g) H_p P for every pPp \in P and gGg \in G.

This subbundle H= pH pTXH = \cup_p H_p\subset T X over XX can be expressed as a field of subspaces H x=KerA x=AnnA xTPH_x = Ker A_x = Ann A_x\subset T P (xPx\in P) which are pointwise annihilators of a smooth Lie algebra-valued 11-form AΩ 1(P,Lie(G))A \in \Omega^1(P,Lie(G)) on PP that satisfies two Ehresmann conditions from the previous subsection.

The Ehresmann connections on a principal GG-bundle are in 1-1 correspondence with an appropriate notion of a connection on the associated bundle. Namely, if T HPTPT^H P\subset T P is the smooth horizontal distrubution of subspaces defining the principal connection on a principal GG-bundle PP over XX, where GG is a Lie group and FF a smooth left GG-space, then consider the total space P× GFP\times_G F of the associated bundle with typical fiber FF. Then, for a fixed fFf\in F one defines a map ρ f:PP× GF\rho_f : P\to P\times_G F assigning the class [p,f][p,f] to pPp\in P. If (T pρ f)(T p HP)=:T [p,f] HP× GF(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F defines the horizontal subspace T [p,f] HP× GFT [p,f]P× GFT_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F, the collection of such subspaces does not depend on the choice of (p,f)(p,f) in the class [p,f][p,f], and the correspondence pT [p,f] HP× GFp\mapsto T_{[p,f]}^H P\times_G F is a connection on the associated bundle P× GFXP\times_G F\to X.

In terms of cohesive homotopy type theory

One may also describe(flat) Ehresmann connections in cohesive homotopy type theory.

The general abstract discussion is here. The discussion of how in smooth infinity-groupoids this reduces to the traditional notion is here.

Properties

General

Definition

The two definitions in terms of 1-forms and in terms of horizontal distributions are equivalent.

Proof

At each pPp \in P take the horizontal subspace H pPH_p P to be the kernel of A(p):T pP𝔤A(p) : T_p P \to \mathfrak{g}

H pP:=kerA(p) H_p P := ker A(p)
Remark

This means we may think of AA as measuring how infinitesimal paths in PP fail to be horizontal or parallel to XX in the sense of parallel transport.

Curvature characteristic forms

Let W(𝔤)\langle - \rangle \in W(\mathfrak{g}) be an invariant polynomial on the Lie algebra. For AOmega 1(P,𝔤)A \in Omega^1(P,\mathfrak{g}) an Ehresmann connection, write

F A=F AF AF A \langle F_A \rangle = \langle F_A \wedge F_A \wedge \cdots F_A\rangle

for the curvature characteristic form obtained by evaluating this on wedge powers of the curvature 2-form.

Proposition

The forms F AΩ 2k(P)\langle F_A \rangle \in \Omega^{2k}(P) are closed, descend along p:PXp : P \to X, in that they are pullbacks of forms along pp, and their class in de Rham cohomology H 2k(X)H^{2k}(X) are independent of the choice of AA on PP.

Proof

That the forms are closed follows from the Bianchi identity

dF A=[AF A] d F_A = [A\wedge F_A]

satisfied by the curvature 2-form and the defining as-invariance of \langle-\rangle. More abstractly, the 1-form AA itself may be identified with a morphism of dg-algebras out of the Weil algebra W(𝔤)W(\mathfrak{g}) (see there)

Ω (P)W(𝔤):A \Omega^\bullet(P) \leftarrow W(\mathfrak{g}) : A

and the evaluation of the curvature in the invariant polynomials corresponds to the precomposition with the morphism

W(𝔤)CE(b 2k1): W(\mathfrak{g}) \leftarrow CE(b^{2k-1}\mathbb{R} ) : \langle - \rangle

described at ∞-Lie algebra cohomology.

to show that these forms descend, it is sufficient to show that for all x𝔤x \in \mathfrak{g} we have

  1. ι ρ *(X)F A=0\iota_{\rho_*(X)} \langle F_A \rangle = 0

  2. ρ *(x)F A=0\mathcal{L}_{\rho_*(x)} \langle F_A \rangle = 0

The first follows from ι ρ *(x)F A=0\iota_{\rho_*(x)} F_A = 0. The second from this, the d dRd_{dR}-closure just discussed and Cartan's magic formula for the Lie derivative.

Remark

The form F A\langle F_A \rangle is called the curvature characteristic form of the connection AA. The map

inv(𝔤)Ω (X) \inv(\mathfrak{g}) \to \Omega^\bullet(X)

induced by (P,A)(P,A) as above is the Chern-Weil homomorphism.

Note on terminology

The terminology for the various incarnations of the single notion of connection on a bundle varies throughout the literature. What we here call an Ehresmann connection is sometimes, but not always, called principal connection (as it is defined for principal bundles).

References

The original definition is due to

  • Charles Ehresmann, Les connexions infinitésimale dans une espace fibré différentiable, Colloque de Topologie, Bruxelles (1950) 29-55, MR0042768

See also

  • Nakahara, Mikio: Geometry, topology and physics (ZMATH entry)

A formulation and discussion of Ehresmann connections using language and tools from synthetic differential geometry is in section 6 of

Revised on June 28, 2012 14:01:02 by Urs Schreiber (82.172.178.200)