Ehresmann connection

A choice of horizontal subspaces complementary to the vertical tangent spaces of a fibered manifold.

Ehresmann connection

Let $\pi:E\to M$ be a surjective submersion between smooth manifolds (a fibered manifold). The map $\pi$ is a smooth map , so it has a differential $d\pi:TE\to TM$ between the tangent bundles .

Define the vertical subbundle

VE:=\ker(d\pi)\subset TE.

Definition. An Ehresmann connection on $\pi:E\to M$ is a choice of a horizontal subbundle $HE\subset TE$ such that, as vector bundles over $E$ ,

TE = HE \oplus VE.

Equivalently, it is a smooth splitting of the short exact sequence

0 \longrightarrow VE \longrightarrow TE \xrightarrow{d\pi} \pi^*TM \longrightarrow 0,

or, equivalently again, a smooth projection $K:TE\to VE$ with $K|_{VE}=\mathrm{id}$ (so that $HE=\ker K$ ).

An Ehresmann connection determines horizontal lifts and hence parallel transport along curves. On a principal G-bundle , every principal connection induces an Ehresmann connection whose horizontal spaces are $G$ -equivariant.

Examples

Product bundle. For $E=M\times F$ with $\pi(x,f)=x$ , the vertical bundle is $0\oplus TF$ , and the choice $HE:=TM\oplus 0$ defines an Ehresmann connection (the “product connection”).
Connection on a vector bundle induces horizontals on the total space. A linear connection on a vector bundle $E\to M$ canonically defines horizontal subspaces in $TE$ (viewed as “directions of infinitesimal parallel translation of vectors in the fiber”).
From principal connections. If $P\to M$ is a principal $G$ -bundle with principal connection, the horizontal subspaces are the kernels of the connection 1-form at each point, giving an Ehresmann connection on $P\to M$ .