Cartan connection
Encyclopedia
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 space
s.
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 form
s and so is local in character. The main idea is to develop a suitable notion of the connection form
s and curvature
using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frame
s 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 manifold
s equipped with some non-metric structure, including Lie group
s and homogeneous space
s. 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 the method of moving frames, Cartan connection applications
and Einstein–Cartan theory
for some examples.
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
.
'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.
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.
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 bundle
s (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 Rh 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
: 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
A principal H-connection on P is a 1-form
on P,
with values in the Lie algebra
of H, such that
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
.
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 ∈ TpP to the coordinates of dπp(X) ∈ Tπ(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
of the semidirect product
G of H with Rn, which provides an isomorphism of each tangent space TpP with
. 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.
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 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
and 
, 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
given by Rhg = gh. Moreover, each fibre of π is a copy of H. P has the structure of a principal H-bundle
over P/H.
A vector field X on P is vertical if dπ(X) = 0. Any ξ ∈
gives rise to a canonical vertical vector field Xξ 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
-valued one-form on P which identifies each tangent space with the Lie algebra. It has the following properties:
In addition to these properties, η satisfies the structure (or structural) equation
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.
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 system
s. To each point p ∈ M, a neighborhood
Up of p is given along with a mapping φp : Up → 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 hp ∈ H, parametrized by p, such that
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 pt. Then pt supplies a notion of transport of the model space along the curve. Let τt : G/H → G/H be the (locally defined) composite map
Intuitively, τt is the transport map. A pseudogroup structure requires that τt be a symmetry of the model space for each t: τt ∈ G. A Cartan connection requires only that the derivative
of τt 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) = pt in M starting at p, we can associate the tangent vector
X, as well as a transport map τtγ. Taking the derivative determines a linear map
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
where ωH is the Maurer-Cartan form of H.
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
, and G is a Lie group G of dimension n+m, with Lie algebra 
, containing H as a subgroup.
of open sets U in M, along with a g-valued 1-form θU defined on each chart such that
By analogy with the case when the θU came from coordinate systems, condition 3 means that φU is related to φV by h.
The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by
ΩU satisfy the compatibility condition:
The definition can be made independent of the coordinate systems by forming the quotient space
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, h2) if and only if x ∈ U1 ∩ U2, θU1 is related to θU2 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 θU implies that they lift to a g-valued 1-form η defined on P (see below).
-valued 1-form η on P such that
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
, with values in 
, is horizontal. The vector space 
is a representation
of H using the adjoint representation of H on
, and the first condition implies that η mod 
is equivariant. Hence it defines a bundle homomorphism from TM to the associated bundle 
.
The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that η mod
is a solder form
.
The curvature of a Cartan connection is the
-valued 2-form Ω defined by
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 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)Rh*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 sU : U → P and letting θU = s*η be the pullbacks of the Cartan connection along the sections.
on a certain principal G-bundle. From this perspective, a Cartan connection consists of
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
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.
on the bundle E = Q ×G G/H of the preceding section. A Cartan connection then consists of
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 xt is a curve in M, then the Ehresmann connection on E supplies an associated parallel transport
map τt : Ext → Ex0 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(xt) transport back to the fibre over x0 and trace out a curve in Ex0. This curve is then called the development of the curve xt.
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.
.
In detail, η splits into h and m components:
Note that the 1-form ηh is a principal H-connection on the original Cartan bundle P. Moreover, the 1-form ηm satisfies:
In other words, η is a solder form
for the bundle P.
Hence, P equipped with the form ηm defines a (first order) H-structure
on M. The form ηh defines a connection on the H-structure.
p (i.e., p contains a maximal solvable subalgebra of g) and G and P are associated Lie groups, then a Cartan connection modelled on (G,P,g,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 space
s: this arises because the perpendicular subspace p⊥ of p in g with respect to the Killing form
of g is a subalgebra of p, and the Killing form induces a natural duality between p⊥ and g/p. Thus the bundle associated to 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:
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
where
denotes the space of k-forms on M with values in V so that
is the space of sections of V and 
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:
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 that V is a (
, H)-module: a representation of the group H with a compatible representation of the Lie algebra 
. 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 
to the tangent bundle of P. Define
.
In order to show that ∇v is well defined, it must:
For (1), the ambiguity in selecting a right-invariant lift of X is a transformation of the form
where 
is the right-invariant vertical vector field induced from 
. So, calculating the covariant derivative in terms of the new lift 
, one has
since
by taking the differential of the equivariance property 
at h equal to the identity element.
For (2), observe that since v is equivariant and
is right-invariant, 
is equivariant. On the other hand, since η is also equivariant, it follows that 
is equivariant as well.
be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism
given by
where 
and 
.
For each k, the exterior derivative is a first order operator differential operator
and so, for k=0, it defines a differential operator
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
. This is called the fundamental or universal derivative, or fundamental D-operator.
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
. It may also be regarded as a specialization of the general concept of a principal connection
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
, in which the geometry of the principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
is tied to the geometry of the base manifold using a solder form
Solder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
. Cartan connections describe the geometry of manifolds modelled on homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s.
The theory of Cartan connections was developed by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
, as part of (and a way of formulating) his method of moving frames (repère mobile). It operates with differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s and so is local in character. The main idea is to develop a suitable notion of the connection form
Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....
s and curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frame
Orthonormal frame
In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric...
s are used to obtain a description of the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
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
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
) Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, as well as the differential geometry of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s equipped with some non-metric structure, including Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s and homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s. The term Cartan connection most often refers to Cartan's formulation of a (pseudo-)Riemannian, affine
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
, projective
Projective connection
In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections,...
, or conformal connection
Conformal connection
In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space...
. 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 the method of moving frames, Cartan connection applications
Cartan connection applications
The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms...
and Einstein–Cartan theory
Einstein–Cartan theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory or the Cartan–Sciama–Kibble theory is a classical theory of gravitation similar to general relativity but relaxing the assumption that the metric be torsion-free. Introducing torsion allows...
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 groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
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
Klein geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.For background and motivation...
consists of a Lie group G together with a Lie subgroup H of G. Together G and H determine a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
.
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 KleinFelix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
'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
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
.
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
Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....
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
Conformal connection
In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space...
.
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 groupAffine groupIn mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
of the plane, and H = GL(2) the corresponding general linear group. - In the second case, G/H is the conformal (or celestialCelestial sphereIn astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
) sphere, with G = O+(3,1) the (orthochronous) Lorentz groupLorentz groupIn physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
, and H the stabilizer of a null line in R3,1.
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 connectionAffine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
on a manifold M is a connection (principal bundle)
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
on the frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
of M (or equivalently, a connection (vector bundle)
Connection (vector bundle)
In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
of M). A key aspect of the Cartan connection point of view is to elaborate this notion in the context of principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
s (which could be called the "general or abstract theory of frames").
Let H be a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
. Then a principal H-bundle is fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
P over M with a smooth action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
, then the orthonormal frame bundle is a principal O(n)-bundle.
Let Rh denote the (right) action of h ∈ H on P. The derivative of this action defines a vertical vector
Vertical bundle
The vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers...
field on P for each element ξ of
: 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 isA principal H-connection on P is a 1-form
on P,with values in the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of H, such that
-
- for any
, ω(Xξ) = 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
.Frame bundles have additional structure called the solder form
Solder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
, 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
Associated bundle
In mathematics, the theory of fiber bundles with a structure group G allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G...
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 ∈ TpP to the coordinates of dπp(X) ∈ Tπ(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
of the semidirect productSemidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
G of H with Rn, which provides an isomorphism of each tangent space TpP with
. 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 groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
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 spaceHomogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
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 transformations of classical Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) expressed as a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
of transformations. These generalized spaces turn out to be homogeneous smooth manifolds diffeomorphic to the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of a Lie group by a Lie subgroup
Lie subgroup
In mathematics, a Lie subgroup H of a Lie group G is a Lie group that is a subset of G and such that the inclusion map from H to G is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie...
. 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
and , respectively. Let P be the underlying principal homogeneous spacePrincipal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
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 Rhg = gh. Moreover, each fibre of π is a copy of H. P has the structure of a principal H-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
over P/H.
A vector field X on P is vertical if dπ(X) = 0. Any ξ ∈
gives rise to a canonical vertical vector field Xξ by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ. The Maurer-Cartan formMaurer-Cartan form
In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G...
η of P is the
-valued one-form on P which identifies each tangent space with the Lie algebra. It has the following properties:- Ad(h) Rh*η = η for all h in H
- η(Xξ) = ξ for all ξ in
- for all g∈P, η restricts a linear isomorphism of TgP with
(η is an absolute parallelism on P).
In addition to these properties, η satisfies the structure (or structural) equation
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
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
. Thus the Klein geometries are said to be the flat models for Cartan geometries.
Cartan connections and pseudogroups
Cartan connections are closely related to pseudogroupPseudogroup
In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra...
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
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
is modelled on Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. 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 system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
s. To each point p ∈ M, a neighborhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
Up of p is given along with a mapping φp : Up → 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 hp ∈ H, parametrized by p, such that
- φ′p = hp φp.
This freedom corresponds roughly to the physicists' notion of a gauge
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
.
Nearby points are related by joining them with a curve. Suppose that p and p′ are two points in M joined by a curve pt. Then pt supplies a notion of transport of the model space along the curve. Let τt : G/H → G/H be the (locally defined) composite map
- τt = φpt o φp0-1.
Intuitively, τt is the transport map. A pseudogroup structure requires that τt be a symmetry of the model space for each t: τt ∈ G. A Cartan connection requires only that the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of τt 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
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, 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
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
.
In this case, τ′0 defines a differential form at the point p as follows. For a curve γ(t) = pt in M starting at p, we can associate the tangent vector
Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
X, as well as a transport map τtγ. Taking the derivative determines a linear map
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
where ωH 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 manifoldRiemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
can be seen as a deformation of Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
; - a Lorentzian manifold can be seen as a deformation of Minkowski spaceMinkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
; - a conformal manifoldConformal geometryIn mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
can be seen as a deformation of the conformal sphereConformal geometryIn mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
; - a manifold equipped with an affine connectionAffine connectionIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
can be seen as a deformation of an affine spaceAffine spaceIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
.
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
, and G is a Lie group G of dimension n+m, with Lie algebra , containing H as a subgroup.Definition via gauge transitions
A Cartan connection consists of a coordinate atlasAtlas (topology)
In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
of open sets U in M, along with a g-valued 1-form θU defined on each chart such that
- θU : TU → g.
- θU mod h : TuU → 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
-
- where ωH is the Maurer-Cartan formMaurer-Cartan formIn mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G...
of H.
By analogy with the case when the θU came from coordinate systems, condition 3 means that φU is related to φV by h.
The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by
ΩU satisfy the compatibility condition:
- If the forms θU and θV are related by a function h : U ∩ V → H, as above, then ΩV = Ad(h-1) ΩU
The definition can be made independent of the coordinate systems by forming the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the disjoint union over all U in the atlas. The equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
~ is defined on pairs (x,h1) ∈ U1 × H and (x, h2) ∈ U2 × H, by ~ (x, h2) if and only if x ∈ U1 ∩ U2, θU1 is related to θU2 by h, and h2 = h(x)-1 h1.
Then P is a principal H-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
on M, and the compatibility condition on the connection forms θU implies that they lift to a 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-valued 1-form η on P such that- for all h in H, Ad(h)Rh*η = η
- for all ξ in
, η(Xξ) = ξ - for all p in P, the restriction of η defines a linear isomorphism from the tangent space TpP to
.
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
, with values in , is horizontal. The vector space is a representationGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of H using the adjoint representation of H on
, and the first condition implies that η mod is equivariant. Hence it defines a bundle homomorphism from TM to the associated bundle .The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that η mod
is a solder formSolder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
.
The curvature of a Cartan connection is the
-valued 2-form Ω defined byNote 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 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
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
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)Rh*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 sU : U → P and letting θU = s*η be the pullbacks of the Cartan connection along the sections.
Cartan connections as principal connections
Another way in which to define a Cartan connection is as a principal connectionConnection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
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 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
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
on any associated bundle
Associated bundle
In mathematics, the theory of fiber bundles with a structure group G allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G...
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
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 connectionEhresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
on the bundle E = Q ×G G/H of the preceding section. 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*θx : TxM → Vs(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 (differential geometry)
In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface at a point can be rolled around the surface to obtain the tangent-plane at other points.The tangential contact...
. If xt is a curve in M, then the Ehresmann connection on E supplies an associated parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
map τt : Ext → Ex0 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(xt) transport back to the fibre over x0 and trace out a curve in Ex0. This curve is then called the development of the curve xt.
To show that this definition is equivalent to the others above, one must introduce a suitable notion of a moving frame
Moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.-Introduction:...
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.
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⊕m, then the m-component of η generalizes the solder form for an affine connectionAffine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
.
In detail, η splits into h and m components:
- η = ηh + ηm.
Note that the 1-form ηh is a principal H-connection on the original Cartan bundle P. Moreover, the 1-form ηm satisfies:
- ηm(X) = 0 for every vertical vector X ∈ TP. (ηm is horizontal.)
- Rh*ηm = Ad(h-1)ηm for every h ∈ H. (ηm is equivariant under the right H-action.)
In other words, η is a solder form
Solder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
for the bundle P.
Hence, P equipped with the form ηm defines a (first order) H-structure
G-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
on M. The form ηh defines a connection on the H-structure.
Parabolic Cartan connections
If g is a semisimple Lie algebra with parabolic subalgebraParabolic Lie algebra
In algebra, a parabolic Lie algebra \mathfrak p is a subalgebra of a semisimple Lie algebra \mathfrak g satisfying one of the following two conditions:* \mathfrak p contains a maximal solvable subalgebra of \mathfrak g;...
p (i.e., p contains a maximal solvable subalgebra of g) and G and P are associated Lie groups, then a Cartan connection modelled on (G,P,g,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 space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
s: this arises because the perpendicular subspace p⊥ of p in g with respect to the Killing form
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...
of g is a subalgebra of p, and the Killing form induces a natural duality between p⊥ and g/p. Thus the bundle associated to p⊥ is isomorphic to the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
.
Parabolic geometries include many of those of interest in research and applications of Cartan connections, such as the following examples:
- Conformal connectionConformal connectionIn conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space...
s: Here G = SO(p+1,q+1), and P is the stabilizer of a null ray in Rn+2. - Projective connectionProjective connectionIn differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections,...
s: 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 hyperquadricQuadricIn mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
. - 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(Os) is the automorphism group of the algebra Os 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 Os).
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 representationGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
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
Connection (vector bundle)
In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
on V, which is a first order linear differential operator
where
denotes the space of k-forms on M with values in V so that is the space of sections of V and is the space of sections ofHom(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:
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 that V is a (
, H)-module: a representation of the group H with a compatible representation of the Lie algebra . 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 to the tangent bundle of P. Define.In order to show that ∇v is well defined, it must:
- be independent of the chosen lift
- 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
where is the right-invariant vertical vector field induced from . So, calculating the covariant derivative in terms of the new lift , one hassince
by taking the differential of the equivariance property at h equal to the identity element.For (2), observe that since v is equivariant and
is right-invariant, is equivariant. On the other hand, since η is also equivariant, it follows that 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 be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphismgiven by
where and .For each k, the exterior derivative is a first order operator differential operator
and so, for k=0, it defines a differential operator
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
. This is called the fundamental or universal derivative, or fundamental D-operator.Books
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- The section 3. Cartan Connections [pages 127-130] treats conformal and projective connections in a unified manner.