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.
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.
Klein's Erlangen programme suggested that geometry could be regarded as a study of homogeneous spaces: in particular, it is the study of the many geometries of interest to geometers of 19th century. A Klein geometry consisted of a space, along with a law for motion within the space expressed as a Lie group of transformations. These generalized spaces turn out to be homogeneous smooth manifolds diffeomorphic to the quotient space of a Lie group by a Lie subgroup. The extra differential structure that these homogeneous spaces possess allows one to study and generalize their geometry using calculus.
The general approach of Cartan is to begin with such a smooth Klein geometry, given by a Lie group G and a Lie subgroup H, with associated Lie algebras and, respectively. Let P be the underlying principal homogeneous space of G. A Klein geometry is the homogeneous space given by the quotient P/H of P by the right action of H. There is a right H-action on the fibres of the canonical projection
given by Rhg = gh. Moreover, each fibre of π is a copy of H. P has the structure of a principal H-bundle over P/H.
A vector field X on P is vertical if dπ = 0. Any ξ ∈ 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 Maurer-Cartan form η of P is the -valued one-form on P which identifies each tangent space with the Lie algebra. It has the following properties:
  1. Ad Rh*η = η for all h in H
  2. η = ξ for all ξ in
  3. for all g∈P, η restricts a linear isomorphism of TgP with .
In addition to these properties, η satisfies the structure equation
Conversely, one can show that given a manifold M and a principal H-bundle P over M, and a 1-form η with these properties, then P is locally isomorphic as an H-bundle to the principal homogeneous bundle G→G/H. The structure equation is the integrability condition for the existence of such a local isomorphism.
A Cartan geometry is a generalization of a smooth Klein geometry, in which the structure equation is not assumed, but is instead used to define a notion of curvature. Thus the Klein geometries are said to be the flat models for Cartan geometries.

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 into 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 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 Up of p is given along with a mapping φp : Up → 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 hp ∈ H, parametrized by p, such that
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 pt. Then pt supplies a notion of transport of the model space along the curve. Let τt : G/H → G/H be the composite map
Intuitively, τt is the transport map. A pseudogroup structure requires that τt be a symmetry of the model space for each t: τt ∈ G. A Cartan connection requires only that the derivative of τt 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 = 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.

Formal definition

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. For example:
There are two main approaches to the definition. In both approaches, M is a smooth manifold of dimension n, H is a Lie group of dimension m, with Lie algebra, and G is a Lie group of dimension n+m, with Lie algebra, containing H as a subgroup.

Definition via gauge transitions

A Cartan connection consists of a coordinate atlas of open sets U in M, along with a -valued 1-form θU defined on each chart such that
  1. θU : TU →.
  2. θU mod : TuU → 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
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 ∈ U1 × H and ∈ 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 -valued 1-form η defined on P.

Definition via absolute parallelism

Let P be a principal H bundle over M. Then a Cartan connection is a -valued 1-form η on P such that
  1. for all h in H, AdRh*η = η
  2. for all ξ in, η = ξ
  3. for all p in P, the restriction of η defines a linear isomorphism from the tangent space TpP to.
The last condition is sometimes called the Cartan condition: it means that η defines an absolute parallelism on P. The second condition implies that η is already injective on vertical vectors and that the 1-form η mod, with values in, is horizontal. The vector space is a representation of H using the adjoint representation of H on, and the first condition implies that η mod is equivariant. Hence it defines a bundle homomorphism from TM to the associated bundle.
The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that η mod is a solder form.
The curvature of a Cartan connection is the -valued 2-form Ω defined by
Note that this definition of a Cartan connection looks very similar to that of a principal connection. There are several important differences, however. First, the 1-form η takes values in, but is only equivariant under the action of H. Indeed, it cannot be equivariant under the full group G because there is no G bundle and no G action. 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. Concretely, the existence of a solder form binds the Cartan connection to the underlying differential topology of the manifold.
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. Thus Cartan geometries are deformed analogues of Klein geometries. This deformation is roughly a prescription for attaching a copy of the model space G/H to each point of M and thinking of that model space as being tangent to the manifold at a point of contact. The fibre of the tautological bundle G → G/H of the Klein geometry at the point of contact is then identified with the fibre of the bundle P. Each such fibre carries a Maurer-Cartan form for G, and the Cartan connection is a way of assembling these Maurer-Cartan forms gathered from the points of contact into a coherent 1-form η defined on the whole bundle. The fact that only elements of H contribute to the Maurer-Cartan equation AdRh*η = η has the intuitive interpretation that any other elements of G would move the model space away from the point of contact, and so no longer be tangent to the manifold.
From the Cartan connection, defined in these terms, one can recover a Cartan connection as a system of 1-forms on the manifold by taking a collection of local trivializations of P given as sections sU : U → P and letting θU = s*η be the pullbacks of the Cartan connection along the sections.

As principal connections

Another way in which to define a Cartan connection is as a principal connection on a certain principal G-bundle. From this perspective, a Cartan connection consists of
such that the pullback η of α to P satisfies the Cartan condition.
The principal connection α on Q can be recovered from the form η by taking Q to be the associated bundle P ×H G. Conversely, the form η can be recovered from α by pulling back along the inclusion P ⊂ Q.
Since α is a principal connection, it induces a connection on any associated bundle to Q. In particular, the bundle Q ×G G/H of homogeneous spaces over M, whose fibers are copies of the model space G/H, has a connection. The reduction of structure group to H is equivalently given by a section s of E = Q ×G G/H. The fiber of over x in M may be viewed as the tangent space at s to the fiber of Q ×G G/H over x. Hence the Cartan condition has the intuitive interpretation that the model spaces are tangent to M along the section s. Since this identification of tangent spaces is induced by the connection, the marked points given by s always move under parallel transport.

Definition by an Ehresmann connection

Yet another way to define a Cartan connection is with an Ehresmann connection on the bundle E = Q ×G G/H of the preceding section. 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. The last condition, in particular, means that the tangent space of M at x is isomorphic to the tangent space of the model space at the point of contact. So the model spaces are, in this way, tangent to the manifold.
This definition also brings prominently into focus the idea of development. If xt is a curve in M, then the Ehresmann connection on E supplies an associated parallel transport map τt : Ext → Ex0 from the fibre over the endpoint of the curve to the fibre over the initial point. In particular, since E is equipped with a preferred section s, the points s transport back to the fibre over x0 and trace out a curve in Ex0. This curve is then called the development of the curve xt.
To show that this definition is equivalent to the others above, one must introduce a suitable notion of a moving frame for the bundle E. In general, this is possible for any G-connection on a fibre bundle with structure group G. See Ehresmann connection#Associated bundles for more details.

Special Cartan connections

Reductive Cartan connections

Let P be a principal H-bundle on M, equipped with a Cartan connection η : TP →. If is a reductive module for H, meaning that admits an Ad-invariant splitting of vector spaces, then the -component of η generalizes the solder form for an affine connection.
In detail, η splits into and components:
Note that the 1-form η is a principal H-connection on the original Cartan bundle P. Moreover, the 1-form η satisfies:
In other words, η is a solder form for the bundle P.
Hence, P equipped with the form η defines a H-structure on M. The form η defines a connection on the H-structure.

Parabolic Cartan connections

If is a semisimple Lie algebra with parabolic subalgebra and G and P are associated Lie groups, then a Cartan connection modelled on is called a parabolic Cartan geometry, or simply a parabolic geometry. A distinguishing feature of parabolic geometries is a Lie algebra structure on its cotangent spaces: this arises because the perpendicular subspace ⊥ of in with respect to the Killing form of is a subalgebra of, and the Killing form induces a natural duality between ⊥ and. Thus the bundle associated to ⊥ is isomorphic to the cotangent bundle.
Parabolic geometries include many of those of interest in research and applications of Cartan connections, such as the following examples:

Covariant differentiation

Suppose that M is a Cartan geometry modelled on G/H, and let be the principal G-bundle with connection, and 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 denotes the space of k-forms on M with values in V so that
is the space of sections of V and is the space of sections of
Hom. 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. Suppose instead that V is a -module: a representation of the group H with a compatible representation of the Lie algebra. 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. Let X be a vector field on M. Choose any right-invariant lift to the tangent bundle of P. Define
In order to show that ∇v is well defined, it must:
  1. be independent of the chosen lift
  2. be equivariant, so that it descends to a section of the bundle V.
For, the ambiguity in selecting a right-invariant lift of X is a transformation of the form where is the right-invariant vertical vector field induced from. So, calculating the covariant derivative in terms of the new lift, one has
since by taking the differential of the equivariance property at h equal to the identity element.
For, observe that since v is equivariant and is right-invariant, is equivariant. On the other hand, since η is also equivariant, it follows that is equivariant as well.

The fundamental or universal derivative

Suppose that V is only a representation of the subgroup H and not necessarily the larger group G. Let 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
where and.
For each k, the exterior derivative is a first order operator differential operator
and so, for k=0, it defines a differential operator
Because η is equivariant, if v is equivariant, so is Dv := φ. It follows that this composite descends to a first order differential operator D from sections of V=P×HV to sections of the bundle. This is called the fundamental or universal derivative, or fundamental D-operator.

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