Cartan connection

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 (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. 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.

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. Let τ_t : G/H → G/H be the (locally defined) composite map

τ_t = φ_pt 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

${\displaystyle X\mapsto {\frac{d}{dt}{\tau }_t^{\gamma }|}_{t=0}=\theta (X)\in \mathfrak{g}.}$

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

${\displaystyle {\theta }_p^{\mathrm{\prime }}=Ad(h_p^{-1}){\theta }_p+h_p^{\ast }{\omega }_H,}$

where ω_H 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:

• 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 ${\displaystyle \mathfrak{h}}$, and G is a Lie group of dimension n+m, with Lie algebra ${\displaystyle \mathfrak{g}}$, 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 ${\displaystyle \mathfrak{g}}$-valued 1-form θ_U defined on each chart such that

1. θ_U : TU → ${\displaystyle \mathfrak{g}}$.
2. θ_U mod ${\displaystyle \mathfrak{h}}$ : T_uU → ${\displaystyle \mathfrak{g}/\mathfrak{h}}$ 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

${\displaystyle {\theta }_V=Ad(h^{-1}){\theta }_U+h^{\ast }{\omega }_H,}$

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

${\displaystyle {\mathrm{\Omega }}_U=d{\theta }_U+\frac{1}{2}[{\theta }_U,{\theta }_U].}$

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

${\displaystyle P=(\coprod _{U}U\times H)/\sim }$

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 ${\displaystyle \mathfrak{g}}$-valued 1-form η defined on P (see below).

Definition via absolute parallelism

Let P be a principal H bundle over M. Then a Cartan connection is a ${\displaystyle \mathfrak{g}}$-valued 1-form η on P such that

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

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

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

${\displaystyle \mathrm{\Omega }=d\eta +\frac{1}{2}[\eta \wedge \eta ].}$

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 ${\displaystyle \mathfrak{g}}$, 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 (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.

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

• 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 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 ${\displaystyle P{\times }_H\mathfrak{g}/\mathfrak{h}}$ 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.

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

• A fiber 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.

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_xt → 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.

Special Cartan connections

Reductive Cartan connections

Let P be a principal H-bundle on M, equipped with a Cartan connection η : TP → ${\displaystyle \mathfrak{g}}$. If ${\displaystyle \mathfrak{g}}$ is a reductive module for H, meaning that ${\displaystyle \mathfrak{g}}$ admits an Ad(H)-invariant splitting of vector spaces ${\displaystyle \mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}}$, then the ${\displaystyle \mathfrak{m}}$-component of η generalizes the solder form for an affine connection. In detail, η splits into ${\displaystyle \mathfrak{h}}$ and ${\displaystyle \mathfrak{m}}$ components:

η = η_{${\displaystyle \mathfrak{h}}$} + η_{${\displaystyle \mathfrak{m}}$}.

Note that the 1-form η_{${\displaystyle \mathfrak{h}}$} is a principal H-connection on the original Cartan bundle P. Moreover, the 1-form η_{${\displaystyle \mathfrak{m}}$} satisfies:

η_{${\displaystyle \mathfrak{m}}$}(X) = 0 for every vertical vector X ∈ TP. (η_{${\displaystyle \mathfrak{m}}$} is horizontal.)

R_h^*η_{${\displaystyle \mathfrak{m}}$} = Ad(h⁻¹)η_{${\displaystyle \mathfrak{m}}$} for every h ∈ H. (η_{${\displaystyle \mathfrak{m}}$} 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_{${\displaystyle \mathfrak{m}}$} defines a (first order) H-structure on M. The form η_{${\displaystyle \mathfrak{h}}$} defines a connection on the H-structure.

Parabolic Cartan connections

If ${\displaystyle \mathfrak{g}}$ is a semisimple Lie algebra with parabolic subalgebra ${\displaystyle \mathfrak{p}}$ (i.e., ${\displaystyle \mathfrak{p}}$ contains a maximal solvable subalgebra of ${\displaystyle \mathfrak{g}}$) and G and P are associated Lie groups, then a Cartan connection modelled on (G,P,${\displaystyle \mathfrak{g}}$,${\displaystyle \mathfrak{p}}$) 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 ${\displaystyle \mathfrak{p}}$^⊥ of ${\displaystyle \mathfrak{p}}$ in ${\displaystyle \mathfrak{g}}$ with respect to the Killing form of ${\displaystyle \mathfrak{g}}$ is a subalgebra of ${\displaystyle \mathfrak{p}}$, and the Killing form induces a natural duality between ${\displaystyle \mathfrak{p}}$^⊥ and ${\displaystyle \mathfrak{g}/\mathfrak{p}}$. Thus the bundle associated to ${\displaystyle \mathfrak{p}}$^⊥ 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:

• Conformal connections: Here G = SO(p+1,q+1), and P is the stabilizer of a null ray in Rⁿ⁺².
• Projective connections: Here G = PGL(n+1) and P is the stabilizer of a point in RPⁿ.
• CR structures and Cartan-Chern-Tanaka connections: G = PSU(p+1,q+1), P = stabilizer of a point on the projective null hyperquadric.
• Contact projective connections: Here G = SP(2n+2) and P is the stabilizer of the ray generated by the first standard basis vector in Rⁿ⁺².
• Generic rank 2 distributions on 5-manifolds: Here G = Aut(O_s) is the automorphism group of the algebra O_s of split octonions, a closed subgroup of SO(3,4), and P is the intersection of G with the stabilizer of the isotropic line spanned by the first standard basis vector in R⁷ viewed as the purely imaginary split octonions (orthogonal complement of the unit element in O_s).

Associated differential operators

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

${\displaystyle \mathrm{\nabla }:{\mathrm{\Omega }}_M^0(\mathbf{V})\to {\mathrm{\Omega }}_M^1(\mathbf{V}),}$

where ${\displaystyle {\mathrm{\Omega }}_M^k(\mathbf{V})}$ denotes the space of k-forms on M with values in V so that ${\displaystyle {\mathrm{\Omega }}_M^0(\mathbf{V})}$ is the space of sections of V and ${\displaystyle {\mathrm{\Omega }}_M^1(\mathbf{V})}$ 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:

${\displaystyle {\mathrm{\nabla }}_X(fv)=df(X)v+f{\mathrm{\nabla }}_{X}v}$

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

${\displaystyle {\mathrm{\nabla }}_{X}v=dv(\overline{X})+\eta (\overline{X})\cdot v}$.

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

1. be independent of the chosen lift ${\displaystyle \overline{X}}$
2. 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 ${\displaystyle X\mapsto X+X_{\xi }}$ where ${\displaystyle X_{\xi }}$ is the right-invariant vertical vector field induced from ${\displaystyle \xi \in \mathfrak{h}}$. So, calculating the covariant derivative in terms of the new lift ${\displaystyle \overline{X}+X_{\xi }}$, one has

${\displaystyle {\mathrm{\nabla }}_{X}v=dv(\overline{X}+X_{\xi })+\eta (\overline{X}+X_{\xi }))\cdot v}$

${\displaystyle =dv(\overline{X})+dv(X_{\xi })+\eta (\overline{X})\cdot v+\xi \cdot v}$

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

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

For (2), observe that since v is equivariant and ${\displaystyle \overline{X}}$ is right-invariant, ${\displaystyle dv(\overline{X})}$ is equivariant. On the other hand, since η is also equivariant, it follows that ${\displaystyle \eta (\overline{X})\cdot v}$ 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 ${\displaystyle {\mathrm{\Omega }}^k(P,V)}$ be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism

${\displaystyle \phi :{\mathrm{\Omega }}^k(P,V)\cong {\mathrm{\Omega }}^0(P,V\otimes {\bigwedge }^k{\mathfrak{g}}^{\ast })}$

given by ${\displaystyle \phi (\beta )({\xi }_1,{\xi }_2,\dots ,{\xi }_k)=\beta ({\eta }^{-1}({\xi }_1),\dots ,{\eta }^{-1}({\xi }_k))}$ where ${\displaystyle \beta \in {\mathrm{\Omega }}^k(P,V)}$ and ${\displaystyle {\xi }_j\in \mathfrak{g}}$.

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

${\displaystyle d:{\mathrm{\Omega }}^k(P,V)\to {\mathrm{\Omega }}^{k+1}(P,V)}$

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

${\displaystyle \phi \circ d:{\mathrm{\Omega }}^0(P,V)\to {\mathrm{\Omega }}^0(P,V\otimes {\mathfrak{g}}^{\ast }).}$

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 ${\displaystyle P{\times }_H(\mathbf{V}\otimes {\mathfrak{g}}^{\ast })}$. This is called the fundamental or universal derivative, or fundamental D-operator.

Books

• Kobayashi, Shoshichi (1972), Transformation Groups in Differential Geometry (Classics in Mathematics 1995 ed.), Springer-Verlag, Berlin, ISBN 978-3-540-58659-3.

The section 3. Cartan Connections [pages 127–130] treats conformal and projective connections in a unified manner.