Cartan Connection

A Cartan connection consists of a coordinate atlas of open sets U in M, along with a g-valued 1-form θ_U defined on each chart such that

1. θ_U : TU → g.
2. θ_U mod h : T_uU → g/h is a linear isomorphism for every u ∈ U.
3. For any pair of charts U and V in the atlas, there is a smooth mapping h : U ∩ V → H such that

where ω_H is the Maurer-Cartan form of H.

By analogy with the case when the θ_U came from coordinate systems, condition 3 means that φ_U is related to φ_V by h.

The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by

Ω_U satisfy the compatibility condition:

If the forms θ_U and θ_V are related by a function h : U ∩ V → H, as above, then Ω_V = Ad(h-1) Ω_U

The definition can be made independent of the coordinate systems by forming the quotient space

of the disjoint union over all U in the atlas. The equivalence relation ~ is defined on pairs (x,h₁) ∈ U₁ × H and (x, h₂) ∈ U₂ × H, by

(x,h₁) ~ (x, h₂) if and only if x ∈ U₁ ∩ U₂, θ_U1 is related to θ_U2 by h, and h₂ = h(x)-1 h₁.

Then P is a principal H-bundle on M, and the compatibility condition on the connection forms θ_U implies that they lift to a g-valued 1-form η defined on P (see below).