nLab
Ehresmann connection

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 $G$ be a Lie group with Lie algebra $𝔤$ and $P\to X$ a $G$-principal bundle. Let

$$\rho :P\times G\to P$$\rho : P \times G \to P

be the action of $G$ on $P$ and

$${\rho }_{*}:𝔤\to \Gamma (TX)$$\rho_* : \mathfrak{g} \to \Gamma(T X)

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

For $v\in \Gamma (TX)$ and $\omega$ a differential form on $P$ write ${\iota }_v\omega$ for the contraction.

Definition

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

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

on $P$ satisfying two conditions

1. first Ehresmann condition

   for every $x\in 𝔤$ we have

   $${\iota }_{{\rho }_{*}(x)}A=x.$$\iota_{\rho_*(x)} A = x \,.

2. second Ehresmann condition

   for every $x\in 𝔤$ we have

   $${ℒ}_{{\rho }_{*}(x)}A={\mathrm{ad}}_{x}A,$$\mathcal{L}_{\rho_*(x)} A = ad_x A \,,

   where ${ℒ}_{{\rho }_{*}(x)}$ is the Lie derivative along ${\rho }_{*}(x)$ and where ${\mathrm{ad}}_x:𝔤\to 𝔤$ is the adjoint action of $𝔤$ on itself.

Proposition

This is equivalent to

1. first Ehresmann condition

   for every $x\in 𝔤$ we have

   $${\iota }_{{\rho }_{*}(x)}A=x.$$\iota_{\rho_*(x)} A = x \,.

2. second Ehresmann condition

   for every $x\in 𝔤$ we have

   $${\iota }_{{\rho }_{*}(x)}{F}_A=0,$$\iota_{\rho_*(x)} F_A = 0 \,,

   where $F_A\in {\Omega }^2(P,𝔤)$ is the curvature 2-form of $A$.

Proof

Using ${\iota }_{x}A=x$ we have by Cartan calculus

$$\begin{array}{rl}{\iota }_{{\rho }_{*}(x)}{F}_A& ={\iota }_{{\rho }_{*}(x)}{d}_{\mathrm{dR}}A+\frac{1}{2}[A\wedge A]\\ & ={ℒ}_{{\rho }_{*}(x)}A-d_{\mathrm{dR}}{\iota }_{{\rho }_{*}(x)}A+[x,A]\\ & ={ℒ}_{{\rho }_{*}(x)}A+[x,A].\end{array}$$\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 $\pi :E\to X$ with typical fiber $F$ (e.g. a smooth vectir bundle or a smooth principal $G$-bundle), there is a well defined vector subbundle $VE\subset TE$ over $E$ such that $V_p$ consists of the tangent vectors $v_p$ such that $(T_p\pi )(v_p)=0$. A smooth distribution (field) of horizontal subspaces is a choice of a vector subspace $H_{p}E\subset T_{p}E$ for every $p$ such that

E1. (complementarity) $T_{u}E=H_{u}E\oplus V_{u}E$

E2. $p\mapsto H_{p}E$ is smooth. That means that in the unique decomposition of any smooth vector field $X$ on $E$ into vector fields $X^H\in \Gamma (H_{u}E)$ and $X^V\in \Gamma (V_{u}E)$ such that $X=X^H+X^V$ the vector field $X^H$ is smooth (or equivalently $X^V$ is smooth, or equivalently both) as a section of $TE$ (there exist yet several other equivalent formulations of the smooothness criterium).

An Ehresmann connection describes a connection on a $G$-principal bundle $\pi :P\to X$ (for $G$ some Lie group) in terms of a distribution of horizontal subspaces $H\subset TP$ which is a subbundle of the tangent bundle of $P$ complementary at each point to the vertical tangent bundle to the fiber. More precisely, an Ehresmann connection on a principal $G$-bundle $\pi :P\to X$ is a smooth distribution of horizontal subspaces $p\mapsto H_{p}P$ which is equivariant:

E3. $H_{pg}P=(T_p{R}_g)H_{p}P$ for every $p\in P$ and $g\in G$.

This subbundle $H={\cup }_p{H}_p\subset TX$ over $X$ can be expressed as a field of subspaces $H_x=\mathrm{Ker}{A}_x=\mathrm{Ann}{A}_x\subset TP$ ($x\in P$) which are pointwise annihilators of a smooth Lie algebra-valued $1$-form $A\in {\Omega }^1(P,\mathrm{Lie}(G))$ on $P$ that satisfies two Ehresmann conditions from the previous subsection.

The Ehresmann connections on a principal $G$-bundle are in 1-1 correspondence with an appropriate notion of a connection on the associated bundle. Namely, if $T^{H}P\subset TP$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $G$-bundle $P$ over $X$, where $G$ is a Lie group and $F$ a smooth left $G$-space, then consider the total space $P{\times }_{G}F$ of the associated bundle with typical fiber $F$. Then, for a fixed $f\in F$ one defines a map ${\rho }_f:P\to P{\times }_{G}F$ assigning the class $[p,f]$ to $p\in P$. If $(T_p{\rho }_f)(T_p^{H}P)=:T_{[p,f]}^{H}P{\times }_{G}F$ defines the horizontal subspace $T_{[p,f]}^{H}P{\times }_{G}F\subset T_{[p,f]}P{\times }_{G}F$, the collection of such subspaces does not depend on the choice of $(p,f)$ in the class $[p,f]$, and the correspondence $p\mapsto T_{[p,f]}^{H}P{\times }_{G}F$ is a connection on the associated bundle $P{\times }_{G}F\to X$.

Properties

General

Definition

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

Proof

At each $p\in P$ take the horizontal subspace $H_{p}P$ to be the kernel of $A(p):T_{p}P\to 𝔤$

$$H_{p}P:=\ker A(p)$$H_p P := ker A(p)

Remark

This means we may think of $A$ as measuring how infinitesimal paths in $P$ fail to be horizontal or parallel to $X$ in the sense of parallel transport.

Curvature characteristic forms

Let $⟨-⟩\in W(𝔤)$ be an invariant polynomial on the Lie algebra. For $A\in {\mathrm{Omega}}^1(P,𝔤)$ an Ehresmann connection, write

$$⟨F_A⟩=⟨F_A\wedge F_A\wedge \cdots F_A⟩$$\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 $⟨F_A⟩\in {\Omega }^{2k}(P)$ are closed, descend along $p:P\to X$, in that they are pullbacks of forms along $p$, and their class in de Rham cohomology $H^{2k}(X)$ are independent of the choice of $A$ on $P$.

Proof

That the forms are closed follows from the Bianchi identity

$$d{F}_A=[A\wedge F_A]$$d F_A = [A\wedge F_A]

satisfied by the curvature 2-form and the defining as-invariance of $⟨-⟩$. More abstractly, the 1-form $A$ itself may be identified with a morphism of dg-algebras out of the Weil algebra $W(𝔤)$ (see there)

$${\Omega }^{•}(P)\leftarrow W(𝔤):A$$\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

$$W(𝔤)\leftarrow \mathrm{CE}(b^{2k-1}ℝ):⟨-⟩$$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 $x\in 𝔤$ we have

1. ${\iota }_{{\rho }_{*}(X)}⟨F_A⟩=0$

2. ${ℒ}_{{\rho }_{*}(x)}⟨F_A⟩=0$

The first follows from ${\iota }_{{\rho }_{*}(x)}{F}_A=0$. The second from this, the $d_{\mathrm{dR}}$-closure just discussed and Cartan's magic formula for the Lie derivative.

Remark

The form $⟨F_A⟩$ is called the curvature characteristic form of the connection $A$. The map

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

induced by $(P,A)$ 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

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis