Editing Holonomy (section)

===Holonomy bundles===
Let ''M'' be a connected paracompact smooth manifold and ''P'' a principal ''G''-bundle with connection ω, as above. Let ''p'' ∈ ''P'' be an arbitrary point of the principal bundle. Let ''H''(''p'') be the set of points in ''P'' which can be joined to ''p'' by a horizontal curve. Then it can be shown that ''H''(''p''), with the evident projection map, is a principal bundle over ''M'' with structure group <math>\operatorname{Hol}_p(\omega).</math> This principal bundle is called the '''holonomy bundle''' (through ''p'') of the connection. The connection ω restricts to a connection on ''H''(''p''), since its parallel transport maps preserve ''H''(''p''). Thus ''H''(''p'') is a reduced bundle for the connection. Furthermore, since no subbundle of ''H''(''p'') is preserved by parallel transport, it is the minimal such reduction.<ref>{{harvnb|Kobayashi|Nomizu|1963|loc=§II.7}}</ref>

As with the holonomy groups, the holonomy bundle also transforms equivariantly within the ambient principal bundle ''P''. In detail, if ''q'' ∈ ''P'' is another chosen basepoint for the holonomy, then there exists a unique ''g'' ∈ ''G'' such that ''q'' ~ ''p'' ''g'' (since, by assumption, ''M'' is path-connected). Hence ''H''(''q'') = ''H''(''p'') ''g''. As a consequence, the induced connections on holonomy bundles corresponding to different choices of basepoint are compatible with one another: their parallel transport maps will differ by precisely the same element ''g''.