Cartan connection


In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.
The theory of Cartan connections was developed by Élie Cartan, as part of his method of moving frames. 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. In relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
Cartan reformulated the differential geometry of Riemannian geometry, as well as the differential geometry of manifolds equipped with some non-metric structure, including Lie groups and homogeneous spaces. The term 'Cartan connection' most often refers to Cartan's formulation of a Riemannian, affine, projective, or conformal connection. Although these are the most commonly used Cartan connections, they are special cases of a more general concept.
Cartan's approach seems at first to be coordinate dependent because of the choice of frames it involves. However, it is not, and the notion can be described precisely using the language of principal bundles. 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.

Introduction

At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries—those with zero curvature—are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
A Klein geometry consists of a Lie group G together with a Lie subgroup H of G. Together G and H determine a homogeneous space G/''H, on which the group G'' acts by left-translation. Klein's aim was then to study objects living on the homogeneous space which were congruent by the action of G. A Cartan geometry extends the notion of a Klein geometry by attaching to each point of a manifold a copy of a Klein geometry, and to regard this copy as tangent to the manifold. Thus the geometry of the manifold is infinitesimally identical to that of the Klein geometry, but globally can be quite different. In particular, Cartan geometries no longer have a well-defined action of G on them. However, a Cartan connection supplies a way of connecting the infinitesimal model spaces within the manifold by means of parallel transport.

Motivation

Consider a smooth surface S in 3-dimensional Euclidean space R3. Near to any point, S can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. The affine subspaces are model surfaces—they are the simplest surfaces in R3, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme. Every smooth surface S has a unique affine plane tangent to it at each point. The family of all such planes in R3, one attached to each point of S, is called the congruence of tangent planes. A tangent plane can be "rolled" along S, and as it does so the point of contact traces out a curve on S. Conversely, given a curve on S, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve by affine transformations, and is an example of a Cartan connection called an affine connection.
Another example is obtained by replacing the planes, as model surfaces, by spheres, which are homogeneous under the Möbius group of conformal transformations. There is no longer a unique sphere tangent to a smooth surface S at each point, since the radius of the sphere is undetermined. This can be fixed by supposing that the sphere has the same mean curvature as S at the point of contact. Such spheres can again be rolled along curves on S, and this equips S with another type of Cartan connection called a conformal connection.
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 family of model spaces attached to each point of a surface S is called a congruence: in the previous examples there is a canonical choice of such a congruence. A Cartan connection provides an identification between the model spaces in the congruence along any curve in S. An important feature of these identifications is that the point of contact of the model space with S ''always moves with the curve. This generic condition is characteristic of Cartan connections.
In the modern treatment of affine connections, the point of contact is viewed as the origin in the tangent plane, and the movement of the origin is corrected by a translation, and so Cartan connections are not needed. However, there is no canonical way to do this in general: in particular for the conformal connection of a sphere congruence, it is not possible to separate the motion of the point of contact from the rest of the motion in a natural way.
In both of these examples the model space is a homogeneous space G''/H.
  • In the first case, G/''H is the affine plane, with G'' = Aff the affine group of the plane, and H = GL the corresponding general linear group.
  • In the second case, G/''H is the conformal sphere, with G'' = O+ the Lorentz group, and H the stabilizer of a null line in R3,1.
The Cartan geometry of S consists of a copy of the model space G/''H at each point of S'' together with a notion of "parallel transport" along curves which identifies these copies using elements of G. This notion of parallel transport is generic in the intuitive sense that the point of contact always moves along the curve.
In general, let G be a group with a subgroup H, and M a manifold of the same dimension as G/''H. Then, roughly speaking, a Cartan connection on M'' is a G-connection which is generic with respect to a reduction to H.

Affine connections

An affine connection on a manifold M is a connection on the frame bundle of M. A key aspect of the Cartan connection point of view is to elaborate this notion in the context of principal bundles.
Let H be a Lie group, its Lie algebra. Then a principal H-bundle is a fiber bundle P over M with a smooth action of H on P which is free and transitive on the fibers. Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL-bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O-bundle.
Let Rh denote the action of h ∈ H on P. The derivative of this action defines a vertical vector field on P for each element ξ of : if h is a 1-parameter subgroup with h=''e and h'' '=ξ, then the corresponding vertical vector field is
A principal H-connection on P is a 1-form on P,
with values in the Lie algebra of H, such that
  2. for any, ω = ξ.
The intuitive idea is that ω provides a vertical component of X, using the isomorphism of the fibers of π with H to identify vertical vectors with elements of.
Frame bundles have additional structure called the solder form, which can be used to extend a principal connection on P to a trivialization of the tangent bundle of P called an absolute parallelism.
In general, suppose that M has dimension n and H acts on Rn. A solder form on a principal H-bundle P over M is an Rn-valued 1-form θ: TP → Rn which is horizontal and equivariant so that it induces a bundle homomorphism from TM to the associated bundle P ×H Rn. This is furthermore required to be a bundle isomorphism. Frame bundles have a solder form which sends a tangent vector X ∈ TpP to the coordinates of dπp ∈ TπM with respect to the frame p.
The pair defines a 1-form η on P, with values in the Lie algebra of the semidirect product G of H with Rn, which provides an isomorphism of each tangent space TpP with. It induces a principal connection α on the associated principal G-bundle P ×H G. This is a Cartan connection.
Cartan connections generalize affine connections in two ways.
  • The action of H on Rn need not be effective. This allows, for example, the theory to include spin connections, in which H is the spin group Spin rather than the orthogonal group O.
  • The group G need not be a semidirect product of H with Rn.