Editing Cartan connection (section)

==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&mdash;those with zero curvature&mdash;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 '''R'''<sup>3</sup>. 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&mdash;they are the simplest surfaces in '''R'''<sup>3</sup>, 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 '''R'''<sup>3</sup>, 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 (in fact Euclidean) 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 (which is then a vector space), 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('''R'''<sup>2</sup>) the [[affine group]] of the plane, and ''H'' = GL(2) the corresponding general linear group.
* In the second case, ''G''/''H'' is the conformal (or [[celestial sphere|celestial]]) sphere, with ''G'' = O<sup>''+''</sup>(3,1) the [[Lorentz group|(orthochronous) Lorentz group]], and ''H'' the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of a null line in '''R'''<sup>3,1</sup>.
The Cartan geometry of ''S'' consists of a copy of the model space ''G''/''H'' at each point of ''S'' (with a marked point of contact) 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===
{{main|Affine connection}}
An '''[[affine connection]]''' on a manifold ''M'' is a [[Connection (principal bundle)|connection]] on the [[Frame bundle|frame bundle (principal bundle)]] of ''M'' (or equivalently, a [[Connection (vector bundle)|connection]]  on the [[Tangent bundle|tangent bundle (vector bundle)]] of ''M''). A key aspect of the Cartan connection point of view is to elaborate this notion in the context of [[principal bundle]]s (which could be called the "general or abstract theory of frames").

Let ''H'' be a [[Lie group]], <math>\mathfrak h</math> its [[Lie algebra]]. Then a '''principal ''H''-bundle''' is a [[fiber bundle]] ''P'' over ''M'' with a smooth [[Group action (mathematics)|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'' &times; ''H'' → ''M''. The frame bundle of ''M'' is a principal GL(''n'')-bundle, while if ''M'' is a [[Riemannian manifold]], then the [[orthonormal frame bundle]] is a principal O(''n'')-bundle.

Let ''R''<sub>''h''</sub> denote the (right) action of ''h'' ∈ H on ''P''. The derivative of this action defines a '''[[vertical bundle|vertical vector]] field''' on ''P'' for each element ''ξ'' of <math>\mathfrak h</math>: 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
:<math>X_\xi=\frac{\mathrm d}{\mathrm dt}R_{h(t)}\biggr|_{t=0}.\,</math>

A '''principal ''H''-connection''' on ''P'' is a [[differential 1-form|1-form]] <math>\omega\colon TP\to \mathfrak h</math>  on ''P'',
with values in the [[Lie algebra]] <math>\mathfrak h</math> of ''H'', such that
# <math>\hbox{Ad}(h)(R_h^*\omega)=\omega</math>
# for any <math>\xi\in \mathfrak h</math>, ''ω''(''X''<sub>''ξ''</sub>) = ''ξ'' (identically on ''P'').

The intuitive idea is that ''ω''(''X'') provides a ''vertical component'' of ''X'', using the isomorphism of the fibers of ''π'' with ''H'' to identify vertical vectors with elements of <math>\mathfrak h</math>.

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 '''R'''<sup>''n''</sup> (this could be any ''n''-dimensional real vector space). A '''solder form''' on a principal ''H''-bundle ''P'' over ''M'' is an '''R'''<sup>''n''</sup>-valued 1-form ''θ'': T''P'' → '''R'''<sup>''n''</sup> which is horizontal and equivariant so that it induces a [[bundle homomorphism]] from T''M'' to the [[associated bundle]] ''P'' &times;<sub>''H''</sub> '''R'''<sup>''n''</sup>. This is furthermore required to be a bundle isomorphism. Frame bundles have a (canonical or tautological) solder form which sends a tangent vector ''X'' ∈ T<sub>''p''</sub>''P'' to the coordinates of d''π''<sub>''p''</sub>(''X'') ∈ T<sub>''π''(''p'')</sub>''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 <math>\mathfrak g</math> of the [[semidirect product]] ''G'' of ''H'' with '''R'''<sup>''n''</sup>, which provides an isomorphism of each tangent space T<sub>''p''</sub>''P'' with <math>\mathfrak g</math>. It induces a principal connection ''α'' on the associated principal ''G''-bundle ''P'' &times;<sub>''H''</sub> ''G''. This is a Cartan connection.

Cartan connections generalize affine connections in two ways.
* The action of ''H'' on '''R'''<sup>''n''</sup> need not be effective. This allows, for example, the theory to include spin connections, in which ''H'' is the [[spin group]] Spin(''n'') rather than the [[orthogonal group]] O(''n'').
* The group ''G'' need not be a semidirect product of ''H'' with '''R'''<sup>''n''</sup>.

===Klein geometries as model spaces===

Klein's [[Erlangen programme]] suggested that geometry could be regarded as a study of [[homogeneous space]]s: in particular, it is the study of the many geometries of interest to geometers of 19th century (and earlier).   A Klein geometry consisted of a space, along with a law for motion within the space (analogous to the [[Euclidean transformation]]s of classical [[Euclidean geometry]]) expressed as a [[Lie group]] of [[transformation group|transformations]].  These generalized spaces turn out to be homogeneous [[smooth manifold]]s diffeomorphic to the [[Quotient space (topology)|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 <math>\mathfrak g</math> and <math>\mathfrak h</math>, 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
:''π'': ''P'' &rarr; ''P''/''H''
given by ''R''<sub>''h''</sub>''g'' = ''gh''.  Moreover, each [[fibre bundle|fibre]] of ''π'' is a copy of ''H''. ''P'' has the structure of a [[principal bundle|principal ''H''-bundle]] over ''P''/''H''.<ref>{{Harvnb|Chevalley|1946| p=110}}.</ref>

A vector field ''X'' on ''P'' is ''vertical'' if d''π''(''X'') = 0.  Any ''ξ'' ∈ <math>\mathfrak h </math> gives rise to a canonical vertical vector field ''X''<sub>''ξ''</sub> 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 <math>\mathfrak g</math>[[Lie algebra-valued form|-valued one-form]] on ''P'' which identifies each tangent space with the Lie algebra. It has the following properties:

# Ad(''h'') ''R''<sub>''h''</sub><sup>*</sup>''η'' = ''η'' for all ''h'' in ''H''
# ''η''(''X''<sub>''ξ''</sub>) = ''ξ'' for all ''ξ'' in <math>\mathfrak h</math>
# for all ''g''∈''P'', ''η'' restricts a linear isomorphism of T<sub>''g''</sub>''P'' with <math>\mathfrak g</math>  (η is an '''absolute parallelism''' on ''P'').

In addition to these properties, ''η'' satisfies the '''structure''' (or '''structural''') '''equation'''
: <math> d\eta+\tfrac{1}{2}[\eta,\eta]=0. </math>

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.<ref>See R. Hermann (1983), Appendix 1&ndash;3 to {{Harvtxt|Cartan|1951}}.</ref>