Contents

Idea

The notion of Ehresmann connection describes a connection on a $G$-principal bundle $p:P\to X$ (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. This subbundle can be expressed as 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 conditions spelled out below.

This can be understood as the special case of nonabelian differential G-cocycle – namely a cocycle with values in the groupoid of Lie-algebra valued forms $\overline{B}G$ – in Čech cohomology using the “canonical” Čech cover

that comes from the total space surjection $p:P\to X$ of the bundle itself.

By the general mechanism of nonabelian Čech cohomology this means that a $\overline{B}G$-valued cocycle with respect to this cover is

• a morphism $A:P\to \overline{B}G$ : this is precisely given by the 1-form $A\in {\Omega }^1(P,\mathrm{Lie}(G))$;

• a tranformation $g:p_1^{*}A\to p_2^{*}A$ that restricts to the $BG$-cocycle of the underlying $G$-bundle.

Since $P{\times }_{X}P\simeq P\times G$ one may differentiate this transformation $g$ at the identity element of $G$. It is an exercise to check that this differential version of the Čech cocycle condition yields the following two conditions on $A$

1. first Ehresmann condition – restricted to the fibers, i.e. pulled back along $\begin{array}{ccc}G& \stackrel{i_x}{\to }& P\\ \downarrow & & \downarrow \\ *& \stackrel{x}{\to }& X\end{array}$ it becomes the canonical left-invariant $\mathrm{Lie}(G)$-valued 1-form $\theta \in {\Omega }^1(G,\mathrm{Lie}(G))$ on $G$

1. second Ehresmann condition – The form $A$ is equivariant with respect to the $G$-action on $P$ in some sense.

The crucial implication of the second property is that all characteristic form?s of $A$ in ${\Omega }^{•}(P)$ are pullbacks of forms on $X$: for $I$ any degree $k$ invariant polynomial on $\mathrm{Lie}(G)$ and for $F_A\in {\Omega }^2(P,\mathrm{Lie}(G))$ the curvature 2-form, we have

definition via subspaces

As already mentioned a connection can equally well be defined by a consistent separation of every tangent space $T_{u}P$ into the vertical subspace $V_{u}P$ and the horizontal subspace $H_{u}P$ such that

1. $T_{u}P=H_{u}P\oplus V_{u}P$

2. Every smooth vector field X on P is separated into smooth vector fields $X^H\in H_{u}P$ and $X^V\in V_{u}P$ such that $X=X^H+X^V$

3. $H_{\mathrm{ug}}P=R_{g*}{H}_{u}P$ for every $u\in P$ and $g\in G$.

The condition 3. states that horinzontal subspaces $H_{u}P$ and $H_{\mathrm{ug}}P$ on the same fibre are related by a linear map $R_{g*}$ induced by the right action of the gauge group.

theorem: equivalence of definitions: Every connection one-form $A$ defines a separation of tangent spaces as defined above, the horizontal subspaces are given by the kernel of $A$. Conversly, given a separation of tangent spaces it is possible to construct a connection one-form.

References

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

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

A useful statement of the definition in terms of a 1-form on the total space is for instance on p. 13 of

• Derek Wise, MacDowell-Mansouri gravity and Cartan geometry (arXiv)

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