User:Silly rabbit/Sandbox/Cartan connection

Cartan connections specialize the notion of a principal connection. Roughly speaking, a Cartan connection can be thought of as a G-connection on a principal G-bundle on M, together with an additional piece of information: an analog of the solder form to identify the tangent spaces of M with ${\mathfrak {g}}/{\mathfrak {h}}$ in an H-equivariant manner.

Cartan connections and pseudogroups

Cartan connections are closely related to pseudogroup structures on a manifold. Each is thought of as modelled on a Klein geometry G/H, in a manner similar to the way in which Riemannian geometry is modelled on Euclidean space. On a manifold M, one imagines attaching to each point of M a copy of the model space G/H. The symmetry of the model space is then built in to the Cartan geometry or pseudogroup structure by positing that the model spaces of nearby points are related by a transformation in G. The fundamental difference between a Cartan geometry and pseudogroup geometry is that the symmetry for a Cartan geometry relates infinitesimally close points by an infinitesimal transformation in G (i.e., an element of the Lie algebra of G) and the analogous notion of symmetry for a pseudogroup structure applies for points that are physically separated within the manifold.

The process of attaching spaces to points, and the attendant symmetries can be concretely realized by using special coordinate systems. To each point p ∈ M, a neighborhood U_p of p is given along with a mapping φ_p : U_p → G/H. In this way, the model space is attached to each point of M by realizing M locally at each point as an open subset of G/H. We think of this as a family of coordinate systems on M, parametrized by the points of M. Two such parametrized coordinate systems φ and φ′ are H-related if there is an element h_p ∈ H, parametrized by p, such that

φ′_p = h_p φ_p

This freedom corresponds roughly to the physicists' notion of a gauge.

Nearby points are related by joining them with a curve. Suppose that p and p′ are two points in M joined by a curve p_t. Then p_t supplies a notion of transport (or rolling) the model space along the curve. Let τ_t : G/H → G/H be the (locally defined) composite map

τ_t = φ_pt o φ_p0^-1.

Intuitively, τ_t is the transport map. A pseudogroup structure requires that τ_t be a symmetry of the model space for each t: τ_t ∈ G. A Cartan connection requires only that the derivative of τ_t be a symmetry of the model space: τ′₀ ∈ g, the Lie algebra of G.

Typical of Cartan, one motivation for introducing the notion of a Cartan connection was to study the properties of pseudogroups from an infinitesimal point of view. A Cartan connection defines a pseudogroup precisely when the derivative of the transport map τ′ can be integrated, thus recovering a true transport map between the coordinate systems. There is thus an integrability condition at work, and Cartan's method for realizing integrability conditions was to introduce a differential form.

In this case, τ′₀ defines a differential form at the point p as follows. For a curve γ(t) = p_t in M starting at p, we can associate the tangent vector X, as well as a transport map τ_t^γ. Taking the derivative determines a linear map

$X\mapsto \left.{\frac {d}{dt}}\tau _{t}^{\gamma }\right|_{t=0}=\theta (X)\in {\mathfrak {g}}.$

So θ defines a g-valued differential 1-form on M.

This form, however, is dependent on the choice of parametrized coordinate system. If h : U → H is an H-relation between two parametrized coordinate systems φ and φ′, then the corresponding values of θ are also related by

$\theta _{p}^{\prime }=Ad(h_{p}^{-1})\theta _{p}+h_{p}^{*}\omega _{H},$

where ω_H is the Maurer-Cartan form of H.