Editing Connection (principal bundle) (section)

===Relation to Ehresmann connections===
A principal ''<math>G</math>''-connection <math>\omega</math> on <math>P</math> determines an [[Ehresmann connection]] on <math>P</math> in the following way. First note that the fundamental vector fields generating the <math>G</math> action on <math>P</math> provide a bundle isomorphism (covering the identity of <math>P</math>) from the [[Fiber bundle|bundle]] <math>V</math> to <math>P\times\mathfrak g</math>, where <math>V=\ker(d\pi)</math> is the kernel of the [[Pushforward (differential)|tangent mapping]] <math>{\mathrm d}\pi\colon TP\to TM</math> which is called the [[vertical bundle]] of <math>P</math>. It follows that <math>\omega</math> determines uniquely a bundle map <math>v:TP\rightarrow V</math> which is the identity on <math>V</math>. Such a projection <math>v</math> is uniquely determined by its kernel, which is a smooth subbundle <math>H</math> of <math>TP</math> (called the [[horizontal bundle]]) such that <math>TP=V\oplus H</math>. This is an Ehresmann connection.

Conversely, an Ehresmann connection <math>H\subset TP</math> (or <math>v:TP\rightarrow V</math>) on <math>P</math> defines a principal <math>G</math>-connection <math>\omega</math> if and only if it is <math>G</math>-equivariant in the sense that <math>H_{pg}=\mathrm d(R_g)_p(H_{p})</math>.