Editing Cartan connection (section)

===Definition via gauge transitions===
A '''Cartan connection''' consists<ref>{{Harvnb|Sharpe|1997}}.</ref><ref>{{Harvnb|Lumiste|2001a}}.</ref> of a [[atlas (topology)|coordinate atlas]] of open sets ''U'' in ''M'', along with a <math>\mathfrak g</math>-valued 1-form θ<sub>U</sub> defined on each chart such that
# θ<sub>U</sub> : T''U'' → <math>\mathfrak g</math>.
# θ<sub>U</sub> mod <math>\mathfrak h</math> : T<sub>u</sub>''U'' → <math>\mathfrak g/\mathfrak h</math> is a linear isomorphism for every ''u'' ∈ ''U''.
#For any pair of charts ''U'' and ''V'' in the atlas, there is a smooth mapping ''h'' : ''U'' ∩ ''V'' → ''H'' such that
::<math>\theta_V = Ad(h^{-1})\theta_U + h^*\omega_H,\,</math>
:where ω<sub>H</sub> is the [[Maurer-Cartan form]] of ''H''.
By analogy with the case when the θ<sub>U</sub> came from coordinate systems, condition 3 means that φ<sub>U</sub> is related to φ<sub>V</sub> by ''h''.

The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by
:<math>\Omega_U = d\theta_U + \tfrac{1}{2}[\theta_U,\theta_U].</math>
Ω<sub>U</sub> satisfy the compatibility condition:
:If the forms θ<sub>U</sub> and θ<sub>V</sub> are related by a function ''h'' : ''U'' &cap; ''V'' &rarr; ''H'', as above, then Ω<sub>V</sub> = Ad(''h''<sup>−1</sup>) Ω<sub>U</sub>

The definition can be made independent of the coordinate systems by forming the [[Quotient space (topology)|quotient space]] 
:<math>P = (\coprod_U U\times H)/\sim</math>
of the disjoint union over all ''U'' in the atlas.   The [[equivalence relation]] ~ is defined on pairs (''x'',''h''<sub>1</sub>) ∈ ''U''<sub>1</sub> &times; ''H'' and (''x'', ''h''<sub>2</sub>) ∈ ''U''<sub>2</sub> &times; ''H'', by
:(''x'',''h''<sub>1</sub>) ~ (''x'', ''h''<sub>2</sub>) if and only if ''x'' &isin; ''U''<sub>1</sub> &cap; ''U''<sub>2</sub>, θ<sub>''U''<sub>1</sub></sub> is related to θ<sub>''U''<sub>2</sub></sub> by ''h'', and ''h''<sub>2</sub> = ''h''(''x'')<sup>−1</sup> ''h''<sub>1</sub>.
Then ''P'' is a [[principal bundle|principal ''H''-bundle]] on ''M'', and the compatibility condition on the connection forms θ<sub>U</sub> implies that they lift to a <math>\mathfrak g</math>-valued 1-form η defined on ''P'' (see below).