Editing Holonomy (section)

==Ambrose–Singer theorem==
The Ambrose–Singer theorem (due to {{harvs|txt|first1=Warren|last1=Ambrose|authorlink1=Warren Ambrose|first2=Isadore M.| last2=Singer |authorlink2=Isadore Singer|year=1953}}) relates the holonomy of a [[connection (principal bundle)|connection in a principal bundle]] with the [[curvature form]] of the connection. To make this theorem plausible, consider the familiar case of an [[affine connection]] (or a connection in the tangent bundle{{snd}} the Levi-Civita connection, for example). The curvature arises when one travels around an infinitesimal parallelogram.

In detail, if σ: [0, 1] × [0, 1] → ''M'' is a surface in ''M'' parametrized by a pair of variables ''x'' and ''y'', then a vector ''V'' may be transported around the boundary of σ: first along (''x'', 0), then along (1, ''y''), followed by (''x'', 1) going in the negative direction, and then (0, ''y'') back to the point of origin. This is a special case of a holonomy loop: the vector ''V'' is acted upon by the holonomy group element corresponding to the lift of the boundary of σ. The curvature enters explicitly when the parallelogram is shrunk to zero, by traversing the boundary of smaller parallelograms over [0, ''x''] × [0, ''y'']. This corresponds to taking a derivative of the parallel transport maps at ''x'' = ''y'' = 0:

:<math>\frac{D}{dx}\frac{D}{dy}V - \frac{D}{dy}\frac{D}{dx}V = R\left(\frac{\partial\sigma}{\partial x}, \frac{\partial\sigma}{\partial y} \right)V</math>

where ''R'' is the [[Riemann curvature tensor|curvature tensor]].<ref>{{harvnb|Spivak|1999|p=241}}</ref> So, roughly speaking, the curvature gives the infinitesimal holonomy over a closed loop (the infinitesimal parallelogram). More formally, the curvature is the differential of the holonomy action at the identity of the holonomy group. In other words, ''R''(''X'', ''Y'') is an element of the [[Lie algebra]] of <math>\operatorname{Hol}_p(\omega).</math>

In general, consider the holonomy of a connection in a principal bundle ''P'' → ''M'' over ''P'' with structure group ''G''. Let '''g''' denote the Lie algebra of ''G'', the [[curvature form]] of the connection is a '''g'''-valued 2-form Ω on ''P''. The Ambrose–Singer theorem states:<ref>{{harvnb|Sternberg|1964|loc=Theorem VII.1.2}}</ref>

:The Lie algebra of <math>\operatorname{Hol}_p(\omega)</math> is spanned by all the elements of '''g''' of the form <math>\Omega_q(X,Y)</math> as ''q'' ranges over all points which can be joined to ''p'' by a horizontal curve (''q'' ~ ''p''), and ''X'' and ''Y'' are horizontal tangent vectors at ''q''.

Alternatively, the theorem can be restated in terms of the holonomy bundle:<ref>{{harvnb|Kobayashi|Nomizu|1963|loc=Volume I, §II.8}}</ref>

:The Lie algebra of <math>\operatorname{Hol}_p(\omega)</math> is the subspace of '''g''' spanned by elements of the form <math>\Omega_q(X, Y)</math> where ''q'' ∈ ''H''(''p'') and ''X'' and ''Y'' are horizontal vectors at ''q''.