Cartan connection
[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand.
Cartan connections induce covariant derivatives and other differential operators on certain associated bundles, hence a notion of parallel transport.
They have many applications in geometry and physics: see the method of moving frames, Cartan formalism and Einstein–Cartan theory for some examples.
In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space.
The flat Cartan geometries—those with zero curvature—are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
Another example is obtained by replacing the planes, as model surfaces, by spheres, which are homogeneous under the Möbius group of conformal transformations.
Differential geometers in the late 19th and early 20th centuries were very interested in using model families such as planes or spheres to describe the geometry of surfaces.
A key aspect of the Cartan connection point of view is to elaborate this notion in the context of principal bundles (which could be called the "general or abstract theory of frames").
: if h(t) is a 1-parameter subgroup with h(0)=e (the identity element) and h '(0)=ξ, then the corresponding vertical vector field is A principal H-connection on P is a 1-form
In general, suppose that M has dimension n and H acts on Rn (this could be any n-dimensional real vector space).
Frame bundles have a (canonical or tautological) solder form which sends a tangent vector X ∈ TpP to the coordinates of dπp(X) ∈ Tπ(p)M with respect to the frame p. The pair (ω, θ) (a principal connection and a solder form) defines a 1-form η on P, with values in the Lie algebra
The extra differential structure that these homogeneous spaces possess allows one to study and generalize their geometry using calculus.
gives rise to a canonical vertical vector field Xξ by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.
The process of attaching spaces to points, and the attendant symmetries, can be concretely realized by using special coordinate systems.
For a curve γ(t) = pt in M starting at p, we can associate the tangent vector X, as well as a transport map τtγ.
Taking the derivative determines a linear map 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 where ωH is the Maurer-Cartan form of H. A Cartan geometry modelled on a homogeneous space G/H can be viewed as a deformation of this geometry which allows for the presence of curvature.
-valued 1-form θU defined on each chart such that By analogy with the case when the θU came from coordinate systems, condition 3 means that φU is related to φV by h. The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by ΩU satisfy the compatibility condition: 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 Then P is a principal H-bundle on M, and the compatibility condition on the connection forms θU implies that they lift to a
Secondly, the 1-form is an absolute parallelism, which intuitively means that η yields information about the behavior of additional directions in the principal bundle (rather than simply being a projection operator onto the vertical space).
An intuitive interpretation of the Cartan connection in this form is that it determines a fracturing of the tautological principal bundle associated to a Klein geometry.
[10] A Cartan connection then consists of This definition makes rigorous the intuitive ideas presented in the introduction.
First, the preferred section s can be thought of as identifying a point of contact between the manifold and the tangent space.
In particular, since E is equipped with a preferred section s, the points s(xt) transport back to the fibre over x0 and trace out a curve in Ex0.
A distinguishing feature of parabolic geometries is a Lie algebra structure on its cotangent spaces: this arises because the perpendicular subspace
Parabolic geometries include many of those of interest in research and applications of Cartan connections, such as the following examples: Suppose that M is a Cartan geometry modelled on G/H, and let (Q,α) be the principal G-bundle with connection, and (P,η) the corresponding reduction to H with η equal to the pullback of α.
Let V a representation of G, and form the vector bundle V = Q ×G V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator where
For any section v of V, the contraction of the covariant derivative ∇v with a vector field X on M is denoted ∇Xv and satisfies the following Leibniz rule: for any smooth function f on M. The covariant derivative can also be constructed from the Cartan connection η on P. In fact, constructing it in this way is slightly more general in that V need not be a fully fledged representation of G.[14] Suppose instead that V is a (
Recall that a section v of the induced vector bundle V over M can be thought of as an H-equivariant map P → V. This is the point of view we shall adopt.
be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism given by
