Cartan connection

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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 (and a way of formulating) his method of moving frames (repère mobile)^[1]. It operates with differential forms and so is computational in character. 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. For instance, 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 (pseudo) 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 (pseudo-)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 Method of moving frames, Cartan connection applications and Einstein-Cartan theory for some examples.

[edit] 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.

[edit] 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 (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 century 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(R²) 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, with G = O⁺(3,1) the (orthochronous) Lorentz group, and H the stabilizer of a null line in R^3,1.

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.

[edit] Klein geometries as model spaces

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 (and earlier). A Klein geometry consisted of a space, along with a law for motion within the space (analogous to the Euclidean transformations of classical Euclidean geometry) 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 $\mathfrak g$ and $\mathfrak h$ , 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 → P/H

given by R_hg = gh. Moreover, each fibre of π is a copy of H. P has the structure of a principal H-bundle over P/H.^[2]

A vector field X on P is vertical if dπ(X) = 0. Any ξ ∈ $\mathfrak h$ gives rise to a canonical vertical vector field X_ξ 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 $\mathfrak g$ -valued one-form on P which identifies each tangent space with the Lie algebra. It has the following properties:

Ad(h) R_h^*η = η for all h in H
η(X_ξ) = ξ for all ξ in $\mathfrak h$
for all g∈P, η restricts a linear isomorphism of T_gP with $\mathfrak g$ (η is an absolute parallelism on P).

In addition to these properties, η satisfies the structure (or structural) equation

$d\eta+\tfrac{1}{2}[\eta\wedge\eta]=0.$

Conversely, one can show that given a manifold M and a principal H-bundle P over M, and 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.^[3]

[edit] Cartan connections and 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 in to 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 (i.e., an element of the Lie algebra of 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.^[4] To each point p ∈ M, a neighborhood U_p of p is given along with a mapping φ_p : U_p → 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 h_p ∈ H, parametrized by p, such that

φ′_p = h_p φ_p^[5]

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 p_t. Then p_t supplies a notion of transport of the model space along the curve.^[6] Let τ_t : G/H → G/H be the (locally defined) composite map

τ_t = φ_{p_t} o φ_p₀^-1.

Intuitively, τ_t is the transport map. A pseudogroup structure requires that τ_t be a symmetry of the model space for each t: τ_t ∈ G. A Cartan connection requires only that the derivative of τ_t 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 (G-valued) 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(t) = p_t in M starting at p, we can associate the tangent vector X, as well as a transport map τ_t^γ. Taking the derivative determines a linear map

$X \mapsto \left.\frac{d}{dt}\tau_t^\gamma\right|_{t=0} = \theta(X) \in \mathfrak{g}.$

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

$\theta^\prime_p = Ad(h^{-1}_p)\theta_p + h^*_p\omega_H,$

where ω_H is the Maurer-Cartan form of H.

[edit] 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:

a Riemannian manifold can be seen as a deformation of Euclidean space
a Lorentzian manifold can be seen as a deformation of Minkowski space
a conformal manifold can be seen as a deformation of the conformal sphere
a manifold equipped with an affine connection can be seen as a deformation of an affine space.

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 $\mathfrak h$ , and G is a Lie group G of dimension n+m, with Lie algebra $\mathfrak g$ , containing H as a subgroup.

[edit] Definition via gauge transitions

A Cartan connection consists^[7]^[8] of a coordinate atlas of open sets U in M, along with a g-valued 1-form θ_U defined on each chart such that

θ_U : TU → g.
θ_U mod h : T_uU → g/h is a linear isomorphism for every u ∈ U.
For any pair of charts U and V in the atlas, there is a smooth mapping h : U ∩ V → H such that

$\theta_V = Ad(h^{-1})\theta_U + h^*\omega_H,\,$

where ω_H is the Maurer-Cartan form of H.

By analogy with the case when the θ_U came from coordinate systems, condition 3 means that φ_U is related to φ_V by h.

The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by

$\Omega_U = d\theta_U + \frac{1}{2}[\theta_U,\theta_U].$

Ω_U satisfy the compatibility condition:

If the forms θ_U and θ_V are related by a function h : U ∩ V → H, as above, then Ω_V = Ad(h^-1) Ω_U

The definition can be made independent of the coordinate systems by forming the quotient space

$P = (\coprod_U U\times H)/\sim$

of the disjoint union over all U in the atlas. The equivalence relation ~ is defined on pairs (x,h₁) ∈ U₁ × H and (x, h₂) ∈ U₂ × H, by

(x,h₁) ~ (x, h₂) if and only if x ∈ U₁ ∩ U₂, θ_U₁ is related to θ_U₂ by h, and h₂ = h(x)^-1 h₁.

Then P is a principal H-bundle on M, and the compatibility condition on the connection forms θ_U implies that they lift to a g-valued 1-form η defined on P (see below).

[edit] Definition via absolute parallelism

Let P be a principal H bundle over M. Then a Cartan connection^[9] is a $\mathfrak g$ -valued 1-form η on P such that

for all h in H, Ad(h)R_h^*η = η .
for all ξ in $\mathfrak h$ , η(X_ξ) = ξ
for all p in P, the restriction of η defines a linear isomorphism from the tangent space T_pP to $\mathfrak g$ .

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 $\mathfrak h$ , with values in $\mathfrak g/\mathfrak h$ , is horizontal. The vector space $\mathfrak g/\mathfrak h$ is a representation of H using the adjoint representation of H on $\mathfrak g$ , and the first condition implies that η mod $\mathfrak h$ is equivariant. Hence it defines a bundle homomorphism from TM to the associated bundle $P\times_H \mathfrak g/\mathfrak h$ . The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that η mod $\mathfrak h$ is a solder form.

The curvature of a Cartan connection is the $\mathfrak g$ -valued 2-form Ω defined by

$\Omega=d\eta+\tfrac{1}{2}[\eta\wedge\eta].$

Note that this definition of a Cartan connection looks very similar to that of a principal connection. There is one important difference however, in 1. above. The g-valued 1-form η is equivariant under the action of H (not G). Intuitively, this means that η yields information about the behavior of additional directions in the principal bundle (rather than simply being a projection operator onto the vertical space). Concretely, the existence of a solder form binds (or solders) 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 (and infinitesimally identical with) 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 (in G) 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 Ad(h)R_h^*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 (as in the gauge definition) by taking a collection of local trivializations of P given as sections s_U : U → P and letting θ_U = s^*η be the pullbacks of the Cartan connection along the sections.

[edit] Cartan connections 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 the following data

a principal G-bundle Q over M
a principal G-connection α on Q (the Cartan connection)
a principal H-subbundle P of Q (i.e., a reduction of structure group)

such that the pullback η of α to P satisfies the Cartan condition.

The principal connection α on Q can 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 $P\times_H \mathfrak g/\mathfrak h$ over x in M may be viewed as the tangent space at s(x) 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.

[edit] 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.^[10] A Cartan connection then consists of

A fibre bundle π : E → M with fibre G/H and vertical space VE ⊂ TE.
A section s : M → E.
A G-connection θ : TE → VE such that

s^*θ_x : T_xM → V_s(x)E is a linear isomorphism of vector spaces for all x ∈ M.

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.

Development of a curve into the model space at x₀

This definition also brings prominently into focus the idea of development. If x_t is a curve in M, then the Ehresmann connection on E supplies an associated parallel transport map τ_t : E_{x_t} → E_x₀ 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(x_t) transport back to the fibre over x₀ and trace out a curve in E_x₀. This curve is then called the development of the curve x_t.

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.

[edit] Special Cartan connections

[edit] Reductive Cartan connections

Let P be a principal H-bundle on M, equipped with a Cartan connection η : TP → g. If g is a reductive module for H, meaning that g admits an Ad(H)-invariant splitting of vector spaces g=h⊕p, then the p-component of η generalizes the solder form for an affine connection.^[11] In detail, η splits into h and p components:

η = η_h + η_p.

Note that the 1-form η_h is a principal H-connection on the original Cartan bundle P. Moreover, the 1-form η_p satisfies:

η_p(X) = 0 for every vertical vector X ∈ TP. (η_p is horizontal.)

R_h^*η_p = Ad(h^-1)η_p for every h ∈ H. (η_p is equivariant under the right H-action.)

In other words, η is a solder form for the bundle P.

Hence, P equipped with the form η_p defines a (first order) H-structure on M. The form η_h defines a connection on the H-structure.

[edit] Parabolic Cartan connections

If g is a semisimple Lie algebra equipped with a gradation g=g_-k ⊕ ... ⊕ g_k, then the non-negative part of the gradation h = g₀ ⊕ ... ⊕ g_k is the parabolic subalgebra. If G and H are associated Lie groups, then a Cartan connection modelled on (G,H,g,h) is called a parabolic Cartan geometry.

Parabolic Cartan geometries include many of those of interest in research and applications of Cartan connections. They include:

Conformal connections: Here G = SO(p+1,q+1), and H is the stabilizer of a null ray in Rⁿ⁺².
Projective connections: Here G = PGL(n+1) and H is the stabilizer of a point in RPⁿ.
CR structures: G = PSU(p+1,q+1), H = stabilizer of a point on the projective null hyperquadric.

[edit] Associated differential operators

[edit] Covariant differentiation

Suppose that M is a Cartan geometry modelled on G/H, and let (Q,α) be the principal G-bundle with connection, and (P,η) 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

$\nabla\colon \Omega^0_M(\mathbf V)\to \Omega^1_M(\mathbf V),$

where $\Omega^k_M(\mathbf V)$ denotes the space of k-forms on M with values in V so that $\Omega^0_M(\mathbf V)$ is the space of sections of V and $\Omega^1_M(\mathbf V)$ is the space of sections of Hom(TM,V). 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:

$\nabla_X(fv)=df(X)v+f \nabla_X v$

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.^[12] Suppose instead that that V is a ( $\mathfrak g$ , H)-module: a representation of the group H with a compatible representation of the Lie algebra $\mathfrak{g}$ . 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 $\bar{X}$ to the tangent bundle of P. Define

$\nabla_X v=dv(\bar{X})+\eta(\bar{X})\cdot v$ .

In order to show that ∇v is well defined, it must:

be independent of the chosen lift $\bar{X}$
be equivariant, so that it descends to a section of the bundle V.

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

$\nabla_X v=dv(\bar{X}+X_\xi)+\eta(\bar{X}+X_\xi))\cdot v$

$=dv(\bar{X}) +d v(X_\xi)+ \eta(\bar{X})\cdot v+ \xi\cdot v$

$=dv(\bar{X})+ \eta(\bar{X})\cdot v$

since $\xi\cdot v+dv(X_\xi)=0$ by taking the differential of the equivariance property $h\cdot R_{h}^*v=v$ at h equal to the identity element.

For (2), observe that since v is equivariant and $\bar{X}$ is right-invariant, $dv(\bar{X})$ is equivariant. On the other hand, since η is also equivariant, it follows that $\eta(\bar{X})\cdot v$ is equivariant as well.

[edit] The fundamental or universal derivative

Suppose that V is only a representation of the subgroup H and not necessarily the larger group G. Let $Ω k (P, V)$ be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism

$\varphi\colon \Omega^k(P,V)\cong \Omega^0(P,V\otimes\bigwedge\nolimits^k\mathfrak g^*)$

given by $\varphi(\beta)(\xi_1,\xi_2,\dots,\xi_k)=\beta(\eta^{-1}(\xi_1),\dots,\eta^{-1}(\xi_k))$ where $\beta \in \Omega^k(P,V)$ and $\xi_j \in \mathfrak g$ .

For each k, the exterior derivative is a first order operator differential operator

$d\colon \Omega^k(P,V)\rightarrow \Omega^{k+1}(P,V)\,$

and so, for k=0, it defines a differential operator

$\varphi\circ d\colon \Omega^0(P,V)\rightarrow \Omega^0(P,V\otimes \mathfrak g^*).\,$

Because η is equivariant, if v is equivariant, so is Dv := φ(dv). It follows that this composite descends to a first order differential operator D from sections of V=P×_HV to sections of the bundle $P\times_H (\mathbf V\otimes \mathfrak g^*)$ . This is called the fundamental or universal derivative, or fundamental D-operator.

[edit] Example: Affine connections

Main article: Affine connection

An affine connection on a manifold M is a Cartan connection modelled on the Klein geometry of affine space.^[13]

The affine group G = Aff(n) consists of all transformations of an n-dimensional vector space V, of the form

φ(x) = Ax + x₀,

where A is a general linear transformation. The affine group consists then of general linear transformations combined with translations. The stabilizer group of the origin in V is a copy H = GL(n) of the general linear group sitting inside G. We can also describe G and H as matrix groups by introducing a basis of V to identify it with Rⁿ. First embed Rⁿ in the natural way into Rⁿ⁺¹ as the hyperplane x_n+1 = 1. With these identifications

$G = \left\{ \left(\begin{matrix} A&x_0\\ 0&1 \end{matrix} \right); A\in GL(n), x_0\in\mathbb{R}^n\right\},\quad H = \left\{ \left(\begin{matrix} A&0\\ 0&1 \end{matrix} \right); A\in GL(n), x_0\in\mathbb{R}^n\right\}$

Likewise, the Lie algebras g = aff(n) and h = gl(n) are given as

$\mathfrak{g} = \left\{ \left(\begin{matrix} A&x_0\\ 0&0 \end{matrix} \right); A\in \mathfrak{gl}(n), x_0\in\mathbb{R}^n\right\},\quad \mathfrak{h} = \left\{ \left(\begin{matrix} A&0\\ 0&0 \end{matrix} \right); A\in \mathfrak{gl}(n), x_0\in\mathbb{R}^n\right\}$

Note that g admits an Ad(H)-invariant splitting

g = h ⊕ Rⁿ, (*)

and at the group level G is a semidirect product of H and Rⁿ.^[14]

An affine connection on a manifold M consists of a principal H-bundle P, along with a g-valued 1-form ω on P such that

ω is nondegenerate.
R_h^*ω = Ad(h^-1) ω for all h ∈ H.
ω(X_ξ) = ξ for all ξ ∈ h, generating the vertical vector field X_ξ.

To get a clearer picture of these three conditions, we decompose ω using the splitting (*). In this way, write ω = σ + θ, where σ is a 1-form with values in h and θ is a 1-form with values in Rⁿ. Stated in terms of σ and θ, properties 1. - 3. can be rewritten

1σ. σ is a surjective h-valued 1-form

2σ. R_h^*σ = Ad(h^-1) σ for all h ∈ H.

3σ. σ(X_ξ) = ξ for all ξ ∈ h.

and

1θ. θ is a surjective Rⁿ-valued 1-form

2θ. (R_h)^*θ(X) = h^-1θ(X) for any X ∈ TP.

3θ. (R_h)^*θ(X_ξ) = 0 for any vertical vector X_ξ

Note in particular that properties 1σ. - 3σ. correspond precisely to the definition of σ being a principal connection on P.^[15]

Properties 1θ. - 3θ. are what distinguish affine connections from more general principal connections. They bind the structure of the principal bundle P (and the affine connection) to the underlying differentiable structure of M, and θ is aptly named a solder form.

The key to seeing this is property 3θ. Suppose that x ∈ M and u ∈ P_x, the fibre over x. If X is a tangent vector to M at X, then we can unambiguously define θ_u(X), since θ_u is insensitive to the addition of vertical vectors. Incorporating now property 1θ:

$\theta_u: T_xM \stackrel{\approx}{\rightarrow} \mathbb{R}^n$ defines a linear isomorphism of vector spaces.

In other words, for each choice of u in the fibre, θ_u defines a frame of T_xM.

Finally, property 2θ. ensures that θ has the correct tensorial transformation law to identify the principal bundle P with the bundle of linear frames of M.

[edit] Prolongation and development

Associated with an affine connection (or much more generally, any Cartan connection: see below) is a notion of development. Development corresponds to the intuitive idea of rolling the tangent copies of the model space along curves in the manifold. Recall the example of the affine tangent plane rolling along a curve in a surface. Here, as the plane moves, the point of contact traces out a curve in the affine plane. So for each curve in the surface, there is a development of that curve into the model affine space. In a general setting, the development of a Cartan connection along a curve is a way of lifting the curve in a natural way to the model space (which may be thought of as the generalization of rolling the model space along the curve).^[16]

Developments are most naturally conceived using the technique of prolongation of a Cartan geometry. Later in the article, this will be discussed for arbitrary Cartan geometries. The way it works for affine connections is as follows.

From the preceding section, we have seen that once equipped with the data of an affine connection, the bundle P → M is realized as the bundle of linear frames of M. The prolongation of this Cartan geometry is a geometric structure defined on the bundle of affine frames of M.

Here we regard the tangent spaces of M as affine spaces rather than vector spaces. This defines the affine tangent bundle AM. An affine frame of M at a point x consists of a choice of base point (or origin) p ∈ A_xM and a choice of linear frame (X₁, ..., X_n) attached to the base point p.

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(n) rather than the orthogonal group O(n).
The group G need not be a semidirect product of H with Rⁿ.

[edit] Notes

^ Although Cartan only began formalizing this theory in particular cases in the 1920s (see Cartan, E. (1926)), he made much use of the general idea much earlier. In particular, the high point of his remarkable 1910 paper on Pfaffian systems in five variables is the construction of a Cartan connection modelled on a 5-dimensional homogeneous space for the exceptional Lie group G₂, which he an Engels had discovered independently in 1894.
^ Chevalley (1946), p. 110.
^ See Hermann, R., Appendix 1-3 in Cartan, E. (1983).
^ This appears to be Cartan's way of viewing the connection. Cf. with Cartan (1923) p. 362; (1924) p. 208, esp.: ..un repère définissant un système de coordonnées projectives...; Cartan (1983) p. 34. Though modern readers can arrive at various interpretations of these statements. Cf. with Hermann (in Cartan, 1983) pp. 384-385, 477.
^ More precisely, h_p is required to be in the isotropy group of φ_p(p), which is a group in G isomorphic to H.
^ In general, this is not the rolling map described in the motivation, although it is related.
^ Sharpe, R. (1997).
^ Lumiste, Ü. Conformal connection.
^ This is the standard definition of the connection. Cf. with Hermann (in Cartan 1983) Appendix 2; Kobayashi (1972) p. 127; Sharpe (1996); Slovák (1997).
^ Ehresmann (1950). Kobayashi (1957). Lumiste (2001), Connections on a manifold.
^ For a treatment of affine connections from this point of view, see Kobayashi and Nomizu, Volume I.
^ See, for instance, A. Čap and A.R. Gover (2002), Definition 2.4.
^ <The material in this section is standard, and essentially based on Kobayashi and Nomizu, Volume 1, with changes made to fit with the general Cartan connection concept. Kobayashi and Nomizu call an affine connection linear, and its prolongation affine. We shall not adopt this convention here.
^ Such geometries are called reductive and play a special role in the differential geometry of Cartan connections.
^ This is not, as it happens, a general feature of Cartan connections, but rather a consequence of the reductivity of aff(n).
^ In greater generality, we can also talk about developments of higher dimensional submanifolds or even the whole manifold itself, into the model space. However, such developments will not always exist, and depend on some integrability conditions of the curvature.

[edit] References

Čap, Andreas and Gover, A. Rod (2002). "Tractor calculi for parabolic geometries". Trans. Amer. Math. Soc. 354: 1511–1548. doi:10.1090/S0002-9947-01-02909-9.
Cartan, Elie (1910). "Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre". Annales scientifiques de l'École Normale Supérieure 27: 109-192.
Cartan, Elie (1923). "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)". Annales Scientifiques de l'École Normale Supérieure 40: 325-412.
Cartan, Elie (1924). "Sur les varietes a connexion projective". Bulletin de la Société Mathématique 52: 205-241.
Cartan, Elie (1926). "Espaces à connexion affine, projective et conforme". Acta Math. 48: 1-42.
Cartan, Elie (1983). Geometry of Riemannian Spaces. Math Sci Press, Massachusetts.
Chevalley, C.The Theory of Lie Groups, Princeton University Press, 1946. ISBN 0-691-08052-6.
Ehresmann, C. (1950). "Les connexions infinitésimals dans un espace fibré differential". Colloque de Topologie, Bruxelles: 29-55.
Griffiths, Phillip (1974). "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry". Duke Mathematics Journal 41 (4): 775-814.
Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 1 & 2. Wiley-Interscience. ISBN 0471157333.
Kobayashi, Shoshichi (1970 (First edition)). Transformation Groups in Differential Geometry. Springer. ISBN 3-540-05848-6.
Kobayashi, Shoshichi (1957). "Theory of Connections". Ann. Mat. Pura Appl. 43: 119-194.
Lumiste, Ü. (2001), “Conformal connection”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Lumiste, Ü. (2001), “Connections on a manifold”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Sharpe, R.W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, New York. ISBN 0-387-94732-9.
Slovak, J. Parabolic Geometries, Research Lecture Notes, Part of DrSc-dissertation, Masaryk University, 1997, 70pp.

[edit] External links

Ü. Lumiste (2001), “Affine connection”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

Categories: Differential geometry | Connection (mathematics)