Cartan connection

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In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. See Method of moving frames, Cartan connection applications and Einstein-Cartan theory for some applications of the method.

Contents

Conceptual 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. 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 in particular of general relativity.

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

Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames. Although it is frame-dependent in its most basic form, 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.

A general theory of frames

The first aspect of the theory looks first to the theory of principal bundles (which one can call the general theory of frames). The idea 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.

General theory in formal terms

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. Of course, a flat Cartan geometry should be a geometry without curvature. Beginning then with the flat case, we describe what is meant by a Cartan Geometry in general formal mathematical terms.

The flat case

Motivation

The Erlangen program focuses upon the study of homogeneous spaces of topological groups, and in particular, most geometries of interest (at least during the 19th century and early 20th century) turn out to be homogeneous differential manifolds isomorphic to the quotient space of a Lie group by a Lie subgroup. It is precisely the differential structure which is inherited from the differential structure of the Lie group which endows these homogeneous spaces with more structure (of a differential kind) than homogeneous spaces in general.

Mathematical details

The general approach to Cartan is to begin with a Lie group G and a Lie subgroup H with associated Lie algebras <math>\mathfrak g<math> and <math> \mathfrak h<math>, respectively. There is a right H-action on the fibres of the canonical homomorphism

<math>\pi:G \rightarrow G/H <math>

given by <math>R_hg=gh<math>. A vector field is vertical if <math>\pi_*X=0 <math>. Any <math>X\in\mathfrak h <math> gives rise to a canonical vertical vector field <math>X^+<math> by taking the differential of the right action. So for instance, if h(t) is a 1-parameter subgroup with tangent vector at the identity h'(e)=X, then the vertical vector field is

<math>X^+=\frac{d}{dt}|_{t=e}R_{h(t)}.<math>

The Maurer-Cartan form w for G can be reinterpreted in terms of such principal bundles over homogeneous spaces axiomatically as follows:

  1. w is a g-valued one-form on G, which is a linear isomorphism of the tangent space of G.
  2. <math>(R_h)_*w=Ad(h^{-1})w<math> for all h in H.
  3. <math>w(X^+)=X<math> for all X in <math>\mathfrak h<math>.
  4. <math> dw+\frac{1}{2}[w,w]=0 <math> (the structural equation)

Conversely, one can show that given a manifold M and a principal H-bundle over M, if a form w obeying these properties is given on the bundle, then that principal bundle is locally isomorphic as an H-bundle to the principal homogeneous bundle <math>G\rightarrow G/H<math>. Property 4 of the Maurer-Cartan form is tantamount to an integrability condition for the problem of establishing such an isomorphism. A Cartan geometry is a fracturing of the integrability condition in this picture, allowing for the presence of curvature.

The curved case

Starting with the basic data for a homogeneous space <math>(G,H,{\mathfrak g},{\mathfrak h},w)<math> as above, we are now prepared to define a Cartan geometry as a certain deformation of this structure, allowing for the presence of curvature.

Motivation

Riemannian geometry can be seen as a "deformation" of Euclidean geometry, a pseudo-Riemannian manifold as a deformation of Minkowski space, a differential manifold equipped with a conformal structure (a Weyl manifold) can be seen as a deformation of a conformal geometry, a differential manifold equipped with an affine connection (but no Riemannian metric) can be seen as a deformation of an affine geometry, etc.

Other examples are abundant. In particular, G need not be an affine group over H. For an example from physics, if M is a four-dimensional manifold and H the spin Lorentz group Spin(3,1), then G could be

R4Image:rtimes2.pngSpin(3,1),

or Spin(4,1) or Spin(3,2). This corresponds to a choice between Minkowski space, de Sitter space and anti de Sitter space respectively. The curved analogs of these structures are of importance in general relativity. (The choice of which group to use depends upon the sign of the cosmological constant.)

For another example, G could be SO(n+1,1) acting on n+2 dimensional Minkowski space, and H could be the isotropy group of a ray through the origin. The resulting geometric structure is isomorphic to group of conformal motions of the n-sphere. Curved analogs of these data correspond to the prescription of a conformal structure on the manifold.

Mathematical details

A Cartan geometry consists of the following. A smooth manifold M of dimension n, a Lie group H of dimension r having Lie algebra <math>\mathfrak h<math>, a principal H-bundle P on M, and Lie group G of dimension n+r with Lie algebra <math>\mathfrak g<math> containing H as a subgroup. A Cartan connection is a <math>\mathfrak g<math>-valued 1-form on P satisfying

  1. w is a linear isomorphism of the tangent space of P.
  2. <math>(R_h)_*w=Ad(h^{-1})w<math> for all h in H.
  3. <math>w(X^+)=X<math> for all X in <math>\mathfrak h<math>.

The curvature of a Cartan connection is the <math>\mathfrak g<math>-valued 2-form

<math>\Omega=dw+\frac{1}{2}[w,w]<math>.

If M is equipped with a Cartan geometry, the tangent space of M carries a canonical H-representation. Indeed, the projection <math>\pi:P\rightarrow M<math> has differential <math>\pi_*:TP\rightarrow TM<math>. The kernel of <math>\pi_*<math> consists of the subbundle of vertical vectors, which the Cartan connection trivializes to <math>\mathfrak h<math>. Thus the tangent bundle of M is isomorphic to the fibre product

<math>TM\cong P\times_H \mathfrak g/\mathfrak h<math>

where <math>\mathfrak g/\mathfrak h<math> is acted upon by the adjoint representation of H.

Gauges for a Cartan connection

In performing actual calculations with a Cartan connection, it is traditional to work in a particular gauge. A gauge on M is just a <math>\mathfrak g<math>-valued 1-form <math>\theta<math> on (an open subset of) M such that the quotient mapping <math>\theta:T_p M\rightarrow{\mathfrak g}\rightarrow {\mathfrak g}/{\mathfrak h}<math> is an isomorphism of vector spaces.

In terms of the connection w, a gauge can be determined by choosing a section <math>s:M\rightarrow P<math>, and setting <math>\theta=s^*w<math>. Such a section of the bundle is called a moving frame. If a pair of sections s and t are given, then they are related by the H-action, so <math>s=kt<math> where k is an H-valued function on M. The induced gauges <math>s^*w<math> and <math>t^*w<math> are related by the equation

<math> s^*w=Ad(k^{-1})t^*w+k^*\omega_H<math>

where <math>\omega_H<math> is the Maurer-Cartan form of H.

The fundamental D operator

Let V be a real or complex representation of H, with the action of H denoted by <math>\rho<math>. Let <math>A^0(P,V)<math> be the space of equivariant V-valued functions on P, so that

<math> R_h^*f=\rho(h^{-1})f <math> for all <math>f\in A^0(P,V)<math>.

Or equivalently

<math> A^0(P,V)=P\times_H V <math>.

Let <math>A^q(P,V)<math> be the space of equivariant V-valued q-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism

<math>\phi:A^q(P,V)\cong A^0(P,\bigwedge^q\mathfrak g^*)<math>

given by <math>\alpha(X_1,X_2,\dots,X_q)=\phi(\alpha)(w^{-1}X_1,\dots,w^{-1}X_q)<math>

The de Rham operator preserves equivariance and so descends to give a first order differential operator

<math>d:A^q(P,V)\rightarrow A^{q+1}(P,V)<math>.

The fundamental D operator is then the composite operator

<math>D=\phi^{-1}\circ d:A^0(P,V)\rightarrow A^0(P,V\otimes \mathfrak g^*)<math>.

Acting on functions in <math>A^0(P,V)<math>, one has

<math>D_X f=w^{-1}(X)f.<math>

Covariant differentiation

The covariant derivative is a first order differential operator which can be defined in a wide class of Cartan geometries. As in the previous section, let the data <math>(P,H,\mathfrak g,\mathfrak h, w)<math> specify a Cartan geometry, and let <math>(V,\rho)<math> be a representation of H, and form the vector bundle <math>\mathbb V=A^0(P,V)<math> over M. The covariant derivative is a first-order differential operator

<math>\nabla_X:\Gamma(\mathbb V)\rightarrow \Gamma(\mathbb V)<math>

for each <math>X\in TM<math> satisfying the usual axioms: If v and w are sections of <math>\mathbb V<math>, k is a function on M, and X and Y are sections of TM, then

  • <math> \nabla_X(v+w)=\nabla_Xv+\nabla_Xw<math>
  • <math> \nabla_{X+Y}v=\nabla_Xv+\nabla_Yv<math>
  • <math> \nabla_{kX}v=k\nabla_Xv<math>
  • <math> \nabla_X(kv)=X(k)v+k\nabla_Xv.<math>

To construct the covariant derivative, let v be any section of <math>\mathbb V<math>. Recall that v may be thought of as an H-equivariant map <math>P\rightarrow V<math>. This is the point of view we shall adopt. Let X be a section of the tangent bundle of M. Choose any right-invariant lift <math>\bar{X}<math> to the tangent bundle of P. Define

<math>\nabla_X v=\bar{X}(v)+\rho(\omega(\bar{X}))(v)<math>.

In order to show that <math>\nabla_X<math> has the required properties, it must: (1) be independent of the chosen lift <math>\bar{X}<math>, (2) be equivariant, so that it descends to a section of the bundle <math>\mathbb V<math>.

For (1), the ambiguity in selecting a right-invariant lift of X is a transformation of the form <math>X\mapsto X+\zeta^+<math> where <math>\zeta^+<math> is the right-invariant vertical vector field induced from <math>\zeta\in\mathfrak h<math>. So, calculating the covariant derivative in terms of the new lift <math>\bar{X}+\zeta^+<math>, one has

<math>\nabla_Xv=\bar{X}(v)+\zeta^+(v)+\rho(\omega(\bar{X}+\zeta^+))(v)<math>
<math>=\bar{X}(v)+\zeta^+(v)+\rho(\omega(\bar{X}))(v)+\rho(\zeta)(v)<math>
<math>=\bar{X}(v)+\rho(\omega(\bar{X}))(v)<math>

since <math>\rho(\zeta)(v)=-\zeta^+(v)<math> by taking the differential of the equivariance property <math>R_{h}^*v=\rho(h^{-1})v<math>.

For (2), since <math>\bar{X}<math> is right-invariant,

<math> R_h^*(\bar{X}(v))=\bar{X}(R_h^*v)=\bar{X}(\rho(h^{-1})(v))=\rho(h^{-1})(\bar{X}(v))<math>

and furthermore

<math> R_h^*[\rho(\omega(\bar{X}))(v)]=\rho(Ad(h^{-1})\omega(\bar{X}))(\rho(h^{-1})v)<math>
<math> =\rho(h^{-1})\rho(\omega(\bar{X}))(v)<math>

so <math>R_h^*(\nabla_X v)=\rho(h^{-1})\nabla_X v<math> as required.

See also: Riemannian geometry, General relativity

Further reading

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