Editing Holonomy (section)

===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>