nLab
Ehresmann connection

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 Lie algebra-valued $1$-form $A\in {\Omega }^1(P,\mathrm{Lie}(G))$ on $P$ that satisfies two conditions.

(Or rather, in the original formulation, it describes it equivalently in terms of the horizontal subbundle $H:=\ker A\subset TP$ of the tangent bundle of $P$ of vectors on which $A$ vanishes, see 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

$$\cdots \to P{\times }_{X}P\simeq P\times G\overrightarrow{\to }P$$\cdots \to P \times_X P \simeq P \times G \stackrel{\to}{\to} P

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 $G$

$$i_x^{*}A=\theta$$i_x^* A = \theta

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

$$\exists c_I\in {\Omega }^{2k}(X):I(F_A)=p^{*}{c}_I.$$\exists c_I \in \Omega^{2k}(X) : I(F_A) = p^* c_I \,.

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 Toplogie, Bruxelles (1950) 29-55.

A fair idea of what exactly that original article defined can be gained from its MathScinet review

A useful statement of the defintion 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

• Moerdijk-Reyes, Models for Smooth Infinitesimal Analysis