Cartan connection

In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, based on an understanding of the role of the affine group in the usual approach.

Aspects of the theory

It was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames. It operates with differential forms and so is computational in character, but has two other major aspects, both more geometric.

A general theory of frames

The first of these looks first to the theory of principal bundles (which one can call the general theory of frames). The ideal of a connection on a principal bundle for a Lie group G is relatively easy to formulate, because in the 'vertical direction' one can see that the required datum is given by translating all tangent vectors back to the identity element (into the Lie algebra), and the connection definition should simply add a 'horizontal' component, compatible with that. If G is a type of affine group with respect to another Lie group H - meaning that G is a semidirect product of H with a vector translation group T on which H acts, an H-bundle can be made into a G-bundle by the associated bundle construction. There is a T-bundle associated, too: a vector bundle, on which H acts by automorphisms that become inner automorphisms in G.

The first type of definition in this set-up is that a Cartan connection for H is a specific type of principal G-connection.

Identifying the tangent bundle

The second type of definition looks directly at the tangent bundle TM of the smooth manifold M assumed as the base. Here the datum is a certain type of identification of TM, as a bundle, as the 'vertical' tangent vectors in the T-bundle mentioned before (where M is natural identified as the zero section). This is called a soldering (sometimes welding): we now have TM within a richer setting, expressed by the H-valued transition data. A major point here, as with the previous discussion, is that it is not assumed that H acts faithfully on T. That immediately allows spinor bundles to take their place in the theory, with H a spin group rather than simply an orthogonal group.

Vierbeins, et cetera

The vierbein or tetrad theory is the special case of a four-dimensional manifold. It applies to metrics of any signature. In any dimension, for a pseudo Riemannian geometry (with metric signature (p,q)), this Cartan connection theory is an alternative method in differential geometry. In different contexts it has also been called the orthonormal frame, repère mobile, soldering form or orthonormal nonholonomic basis method.

This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, funfbein, elfbein etc. have been used. Vielbein covers all dimensions.

If you're looking for a basis-dependent index notation, see tetrad (index notation).

The basic ingredients

Suppose given differential manifold M of dimension n, and fixed natural numbers p and q with p+q = n. We suppose given a SO(p,q) principal bundle B over M (called the frame bundle), and a vector SO(p,q)-bundle V associated to B by means of with the natural n dimensional representation of SO(p,q).

Suppose given also a SO(p,q)-invariant metric η of signature (p,q) over V; and an invertible linear map between vector bundles over M, e:TM->V where TM is the tangent bundle of M.

Constructions

A (pseudo)Riemannian metric is defined over M as the push forward of η by e. To put it in other words, if we have two sections of TM, X and Y,

g(X,Y)=η(e(X),e(Y)).

A connection over V, A is defined as the unique connection satisfying these two conditions:

dη(a,b)=η(d_Aa,b)+η(a,d_Ab) for all differentiable sections a and b of V (i.e. d_Aη=0) where d_A is the covariant exterior derivative. (this basically states that A can be extended to a connection over the SO(p,q) principal bundle)
d_Ae=0. (this basically states that ∇ defined below is torsion-free)

Now that we've specified A, we can use it to define a connection over TM by the pullback (or is it push forward?) by e;

e(∇X)=d_Ae(X) for all differentiable sections X of TM.

Since what we now have here is a SO(p,q) gauge theory, the Riemann curvature F defined as ${\mathbf {F}}\equiv d{\mathbf {A}}+{\mathbf {A}}\wedge {\mathbf {A}}$ is pointwise gauge covariant. This is simply the Riemann tensor in a different guise.

Advantages

An advantage of the tetrad formalism, over the conventional metric tensor formulation, is in describing spinor bundles.

General theory

Cartan reformulated the differential geometry of (pseudo) Riemannian geometry; and not just those metric manifolds, but theories for an arbitrary manifold, including Lie group manifolds. This was in terms of "moving frames" (repère mobile) as an alternative reformulation of general relativity.

Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames. Although it is frame dependent, it is very well suited for computations. It can also be understood in terms of frame bundles, and it allows generalizations like the spinor bundle, and also the principal bundle.

The main idea is to develop expressions for connections and curvature using orthogonal frames.