Editing Connection (principal bundle) (section)

===Bundle of principal connections===
The group ''<math>G</math>'' acts on the [[tangent bundle]] ''<math>TP</math>'' by right translation.  The [[Quotient space (topology)|quotient space]] ''TP''/''G'' is also a manifold, and inherits the structure of a [[fibre bundle]] over ''TM'' which shall be denoted ''dπ'':''TP''/''G''→''TM''.  Let ρ:''TP''/''G''→''M'' be the projection onto ''M''.  The fibres of the bundle ''TP''/''G'' under the projection ρ carry an additive structure.

The bundle ''TP''/''G'' is called the '''bundle of principal connections''' {{harv|Kobayashi|1957}}.  A [[section (fiber bundle)|section]] Γ of dπ:''TP''/''G''→''TM'' such that Γ : ''TM'' → ''TP''/''G'' is a linear morphism of vector bundles over ''M'', can be identified with a principal connection in ''P''.  Conversely, a principal connection as defined above gives rise to such a section Γ of ''TP''/''G''.

Finally, let Γ be a principal connection in this sense.  Let ''q'':''TP''→''TP''/''G'' be the quotient map.  The horizontal distribution of the connection is the bundle

:<math>H = q^{-1}\Gamma(TM) \subset TP.</math> We see again the link to the horizontal bundle and thus Ehresmann connection.