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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). 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.

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 (with zero curvature) are homogeneous spaces, hence geometries in the sense of Klein.

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 thet 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, called the congruence of tangent planes. The tangent plane can be "rolled" long 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 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.

Affine connections

An affine connection on a manifold M is a connection (principal bundle) on the frame bundle of M (or equivalently, a connection (vector bundle) on the tangent bundle of M). A key aspect of the Cartan connection point of view is to elaborate this notion in the context of principal bundles (which could be called the "general or abstract theory of frames").

Let H be a Lie group. Then a principal H-bundle is 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(n)-bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O(n)-bundle.

Let R_h denote the (right) action of h ∈ H on P. The derivative of this action defines a vertical vector field on P for each element ξ of ${\mathfrak {h}}$: 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

$X_{\xi }={\frac {\mathrm {d} }{\mathrm {d} t}}R_{h(t)}{\biggr |}_{t=0}.\,$

A principal H-connection on P is a 1-form $\omega \colon TP\to {\mathfrak {h}}$ on P, with values in the Lie algebra ${\mathfrak {h}}$ of H, such that

1. ${\hbox{Ad}}(h)(R_{h}^{*}\omega )=\omega$
2. for any $\xi \in {\mathfrak {h}}$, ω(X_ξ) = 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 ${\mathfrak {h}}$.

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 (this could be any n-dimensional real vector space). 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 (canonical or tautological) solder form which sends a tangent vector X ∈ T_pP to the coordinates of dπ_p(X) ∈ T_π(p)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 ${\mathfrak {g}}$ of the semidirect product G of H with Rⁿ, which provides an isomorphism of each tangent space T_pP with ${\mathfrak {g}}$. 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.

• 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.

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) which 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. There is a right H-action on the fibres of the canonical projection

π: G → G/H

given by R_hg = gh. A vector field X on G is vertical if dπ(X) = 0. Any ξ ∈ ${\mathfrak {h}}$ gives rise to a canonical vertical vector field X_ξ by taking the differential of the right action.

The Maurer-Cartan form η of G is the ${\mathfrak {g}}$-valued one-form on G which identifies each tangent space with the Lie algebra. It has the following properties:

1. Ad(h) R_h^*η = η for all h in H
2. η(X_ξ) = ξ for all ξ in ${\mathfrak {h}}$
3. for all g∈G, η restricts a linear isomorphism of T_gG with ${\mathfrak {g}}$.

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.

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.

Definition via absolute parallelism

Let P be a principal H bundle over M. Then a Cartan connection is a ${\mathfrak {g}}$-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 ${\mathfrak {h}}$, 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 ${\mathfrak {g}}$.

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

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

Cartan connections as principal connections

A Cartan geometry is given by

• 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 Cartan connection α on Q can recovered from η by taking Q to be the associated bundle P ×_H G.

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 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.

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

$\nabla \colon \Omega _{M}^{0}(\mathbf {V} )\to \Omega _{M}^{1}(\mathbf {V} ),$

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

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

for any smooth function f on M.

The covariant derivative can also be constructed from the Cartan connection η on P. For this, recall that a section v of 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:

1. be independent of the chosen lift ${\bar {X}}$
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 $X\mapsto X+X_{\xi }$ where $X_{\xi }$ 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

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

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

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

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.

The fundamental or universal derivative

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

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

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.

Moving frames

In performing actual calculations with a Cartan connection, it is traditional to work in a particular gauge or moving frame, which is just a (local) section s of P. The pullback θ = s*η is sometimes called the gauge form or connection 1-form of the Cartan connection with respect to the moving frame. It is a 1-form on (an open subset of) M with values in ${\mathfrak {g}}$ with the property (the Cartan condition) that the quotient mapping

$\theta \mod {\mathfrak {h}}\colon T_{p}M\rightarrow {\mathfrak {g}}\rightarrow {\mathfrak {g}}/{\mathfrak {h}}$

is an isomorphism of vector spaces.

If two moving frames s and t are given, then they are pointwise related by the H-action on P, so s = th where h is an H-valued function on M. The induced connection 1-forms s*η and t*η are related by the equation

$s^{*}\eta =Ad(h^{-1})t^{*}\eta +h^{*}\eta _{H}\,$

where $\eta _{H}$ is the Maurer-Cartan form of H.

References

• 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. {{cite journal}}: External link in |title= (help)
• 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.
• Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 1 & 2. Wiley-Interscience. ISBN 0471157333. {{cite book}}: Check date values in: |year= (help)CS1 maint: multiple names: authors list (link) CS1 maint: year (link)
• Sharpe, R.W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, New York. ISBN 0-387-94732-9.

External links

• Ü. Lumiste (2001) [1994], "Affine connection", Encyclopedia of Mathematics, EMS Press
• Ü. Lumiste (2001) [1994], "Connections on a manifold", Encyclopedia of Mathematics, EMS Press