Editing Holonomy (section)

===Holonomy of a connection in a principal bundle===
The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let ''G'' be a [[Lie group]] and ''P'' a [[principal bundle|principal ''G''-bundle]] over a [[smooth manifold]] ''M'' which is [[paracompact]]. Let ω be a [[connection form|connection]] on ''P''. Given a piecewise smooth [[loop (topology)|loop]] ''γ'' : [0,1] → ''M'' based at ''x'' in ''M'' and a point ''p'' in the fiber over ''x'', the connection defines a unique ''horizontal lift'' <math>\tilde\gamma : [0,1] \to P</math> such that <math>\tilde\gamma(0) = p.</math> The end point of the horizontal lift, <math>\tilde\gamma(1)</math>, will not generally be ''p'' but rather some other point ''p''·''g'' in the fiber over ''x''. Define an [[equivalence relation]] ~ on ''P'' by saying that ''p'' ~ ''q'' if they can be joined by a piecewise smooth horizontal path in ''P''.

The '''holonomy group''' of ω based at ''p'' is then defined as

:<math>\operatorname{Hol}_p(\omega) = \{ g \in G \mid p \sim p \cdot g\}.</math>

The '''restricted holonomy group''' based at ''p'' is the subgroup <math>\operatorname{Hol}^0_p(\omega)</math> coming from horizontal lifts of [[contractible]] loops&nbsp;''γ''.

If ''M'' and ''P'' are [[connected space|connected]] then the holonomy group depends on the [[Pointed space|basepoint]] ''p'' only up to [[Glossary of Riemannian and metric geometry#C|conjugation]] in ''G''. Explicitly, if ''q'' is any other chosen basepoint for the holonomy, then there exists a unique ''g'' ∈ ''G'' such that ''q'' ~ ''p''·''g''. With this value of ''g'',

:<math>\operatorname{Hol}_q(\omega) = g^{-1} \operatorname{Hol}_p(\omega) g.</math>

In particular,

:<math>\operatorname{Hol}_{p\cdot g}(\omega) = g^{-1} \operatorname{Hol}_p(\omega) g,</math>

Moreover, if ''p'' ~ ''q'' then <math>\operatorname{Hol}_p(\omega) = \operatorname{Hol}_q(\omega).</math> As above, sometimes one drops reference to the basepoint of the holonomy group, with the understanding that the definition is good up to conjugation.

Some important properties of the holonomy and restricted holonomy groups include:

*<math>\operatorname{Hol}^0_p(\omega)</math> is a connected [[Lie subgroup]] of ''G''.
*<math>\operatorname{Hol}^0_p(\omega)</math> is the [[identity component]] of <math>\operatorname{Hol}_p(\omega).</math>
*There is a natural, surjective [[group homomorphism]] <math>\pi_1 \to \operatorname{Hol}_p(\omega)/\operatorname{Hol}^0_p(\omega).</math>
*If ''M'' is [[simply connected]] then <math>\operatorname{Hol}_p(\omega) = \operatorname{Hol}^0_p(\omega).</math>
*ω is flat (i.e. has vanishing curvature) if and only if <math>\operatorname{Hol}^0_p(\omega)</math> is trivial.