Editing Holonomy (section)

==Definitions==

===Holonomy of a connection in a vector bundle===
Let ''E'' be a rank-''k'' [[vector bundle]] over a [[smooth manifold]] ''M'', and let ∇ be a [[connection (vector bundle)|connection]] on ''E''. Given a [[piecewise]] smooth [[loop (topology)|loop]] ''γ'' : [0,1] → ''M'' based at ''x'' in ''M'', the connection defines a [[parallel transport]] map ''P''<sub>''γ''</sub> : ''E<sub>x</sub>'' → ''E<sub>x</sub>'' on the fiber of ''E'' at ''x''. This map is both linear and invertible, and so defines an element of the [[general linear group]] GL(''E<sub>x</sub>''). The '''holonomy group''' of ∇ based at ''x'' is defined as

:<math>\operatorname{Hol}_x(\nabla) = \{P_\gamma \in \mathrm{GL}(E_x) \mid \gamma \text{ is a loop based at } x\}.</math>

The '''restricted holonomy group''' based at ''x'' is the subgroup <math>\operatorname{Hol}^0_x(\nabla)</math> coming from [[contractible]] loops&nbsp;''γ''.

If ''M'' is [[connected space|connected]], then the holonomy group depends on the [[Pointed space|basepoint]] ''x'' only [[up to]] [[Conjugacy class|conjugation]] in GL(''k'', '''R'''). Explicitly, if ''γ'' is a path from ''x'' to ''y'' in ''M'', then

:<math>\operatorname{Hol}_y(\nabla) = P_\gamma \operatorname{Hol}_x(\nabla) P_\gamma^{-1}.</math>

Choosing different identifications of ''E<sub>x</sub>'' with '''R'''<sup>''k''</sup> also gives conjugate subgroups. Sometimes, particularly in general or informal discussions (such as below), one may drop reference to the basepoint, with the understanding that the definition is good up to conjugation.

Some important properties of the holonomy group include:
* <math>\operatorname{Hol}^0(\nabla)</math> is a connected [[Lie subgroup]] of GL(''k'', '''R''').
* <math>\operatorname{Hol}^0(\nabla)</math> is the [[identity component]] of <math>\operatorname{Hol}(\nabla).</math>
* There is a natural, [[surjective]] [[group homomorphism]] <math>\pi_1(M) \to \operatorname{Hol}(\nabla)/ \operatorname{Hol}^0(\nabla),</math> where <math>\pi_1(M)</math> is the [[fundamental group]] of ''M'', which sends the homotopy class <math>[\gamma]</math> to the [[coset]] <math>P_{\gamma}\cdot\operatorname{Hol}^0(\nabla).</math>
* If ''M'' is [[simply connected]], then <math>\operatorname{Hol}(\nabla) = \operatorname{Hol}^0(\nabla).</math>
* ∇ is flat (i.e. has vanishing curvature) [[if and only if]] <math>\operatorname{Hol}^0(\nabla)</math> is trivial.
<!--Find a better place for this, please:See also [[Wilson loop]].-->

===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.

===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''.

===Monodromy===
The holonomy bundle ''H''(''p'') is a principal bundle for <math>\operatorname{Hol}_p(\omega),</math> and so also admits an action of the restricted holonomy group <math>\operatorname{Hol}^0_p(\omega)</math> (which is a normal subgroup of the full holonomy group). The discrete group <math>\operatorname{Hol}_p(\omega)/\operatorname{Hol}^0_p(\omega)</math> is called the [[monodromy group]] of the connection; it acts on the quotient bundle <math>H(p)/ \operatorname{Hol}^0_p(\omega).</math> There is a surjective homomorphism <math>\varphi: \pi_1 \to \operatorname{Hol}_p(\omega)/\operatorname{Hol}^0_p(\omega),</math> so that <math>\varphi\left(\pi_1(M)\right)</math> acts on <math>H(p)/ \operatorname{Hol}^0_p(\omega).</math> This action of the fundamental group is a '''monodromy representation''' of the fundamental group.<ref>{{harvnb|Sharpe|1997|loc=§3.7}}</ref>

===Local and infinitesimal holonomy===
If π: ''P'' → ''M'' is a principal bundle, and ω is a connection in ''P'', then the holonomy of ω can be restricted to the fibre over an open subset of ''M''. Indeed, if ''U'' is a connected open subset of ''M'', then ω restricts to give a connection in the bundle π<sup>−1</sup>''U'' over ''U''. The holonomy (resp. restricted holonomy) of this bundle will be denoted by <math>\operatorname{Hol}_p(\omega, U)</math> (resp. <math>\operatorname{Hol}^0_p(\omega, U)</math>) for each ''p'' with π(''p'') ∈ ''U''.

If ''U'' ⊂ ''V'' are two open sets containing π(''p''), then there is an evident inclusion

:<math>\operatorname{Hol}_p^0(\omega, U)\subset\operatorname{Hol}_p^0(\omega, V).</math>

The '''local holonomy group''' at a point ''p'' is defined by

:<math>\operatorname{Hol}^*(\omega) = \bigcap_{k=1}^\infty \operatorname{Hol}^0(\omega,U_k)</math>

for any family of nested connected open sets ''U''<sub>''k''</sub> with <math>\bigcap_k U_k = \pi(p)</math>.

The local holonomy group has the following properties:

# It is a connected Lie subgroup of the restricted holonomy group <math>\operatorname{Hol}^0_p(\omega).</math>
# Every point ''p'' has a neighborhood ''V'' such that <math>\operatorname{Hol}^*_p(\omega) = \operatorname{Hol}^0_p(\omega, V).</math> In particular, the local holonomy group depends only on the point ''p'', and not the choice of sequence ''U''<sub>''k''</sub> used to define it.
# The local holonomy is equivariant with respect to translation by elements of the structure group ''G'' of ''P''; i.e., <math>\operatorname{Hol}^*_{pg}(\omega) = \operatorname{Ad} \left(g^{-1}\right) \operatorname{Hol}^*_p(\omega)</math> for all ''g'' ∈ ''G''. (Note that, by property 1, the local holonomy group is a connected Lie subgroup of ''G'', so the adjoint is well-defined.)

The local holonomy group is not well-behaved as a global object. In particular, its dimension may fail to be constant. However, the following theorem holds:

: If the dimension of the local holonomy group is constant, then the local and restricted holonomy agree: <math>\operatorname{Hol}^*_p(\omega) = \operatorname{Hol}^0_p(\omega).</math>