In the mathematical field of
differential geometry, a Cartan connection is a flexible generalization of the notion of an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
. It may also be regarded as a specialization of the general concept of a
principal connection, in which the geometry of the
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
is tied to the geometry of the base manifold using a
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
. Cartan connections describe the geometry of manifolds modelled on
homogeneous spaces.
The theory of Cartan connections was developed by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
, as part of (and a way of formulating) his
method of moving frames (''repère mobile''). The main idea is to develop a suitable notion of the
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.
Historically, connection forms were introduced by Élie Carta ...
s and
curvature using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry,
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. If ''M'' is a manifold equipped with a metric ''g'', then an orthonormal frame at a point ...
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
The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, ...
)
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 that varies smoothly from point to point ...
, 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
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
,
projective, or
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
:O+(n ...
. 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 formalism
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
and
Einstein–Cartan theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstei ...
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—those with zero curvature—are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
A
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 ...
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
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
.
Motivation
Consider a smooth surface ''S'' in 3-dimensional Euclidean space R
3. Near to any point, ''S'' can be approximated by its tangent plane at that point, which is an
affine subspace
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of Euclidean space. The affine subspaces are ''model'' surfaces—they are the simplest surfaces in R
3, and are homogeneous under the Euclidean group of the plane, hence they are ''Klein geometries'' in the sense of
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
's
Erlangen programme
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...
. Every smooth surface ''S'' has a unique affine plane tangent to it at each point. The family of all such planes in R
3, 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
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
.
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 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.
The ...
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 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
:O+(n ...
.
Differential geometers in the late 19th and early 20th centuries were very interested in using model families such as planes or spheres to describe the geometry of surfaces. A family of model spaces attached to each point of a surface ''S'' is called a congruence: in the previous examples there is a canonical choice of such a congruence. A Cartan connection provides an identification between the model spaces in the congruence along any curve in ''S''. An important feature of these identifications is that the point of contact of the model space with ''S'' ''always moves'' with the curve. This generic condition is characteristic of Cartan connections.
In the modern treatment of affine connections, the point of contact is viewed as the ''origin'' in the tangent plane (which is then a vector space), and the movement of the origin is corrected by a translation, and so Cartan connections are not needed. However, there is no canonical way to do this in general: in particular for the conformal connection of a sphere congruence, it is not possible to separate the motion of the point of contact from the rest of the motion in a natural way.
In both of these examples the model space is a homogeneous space ''G''/''H''.
* In the first case, ''G''/''H'' is the affine plane, with ''G'' = Aff(R
2) the
affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Rela ...
of the plane, and ''H'' = GL(2) the corresponding general linear group.
* In the second case, ''G''/''H'' is the conformal (or
celestial
Celestial may refer to:
Science
* Objects or events seen in the sky and the following astronomical terms:
** Astronomical object, a naturally occurring physical entity, association, or structure that exists in the observable universe
** Celes ...
) 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
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
on a manifold ''M'' is a
connection on the
frame bundle (principal bundle) of ''M'' (or equivalently, a
connection on the
tangent bundle (vector bundle) of ''M''). A key aspect of the Cartan connection point of view is to elaborate this notion in the context of
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s (which could be called the "general or abstract theory of frames").
Let ''H'' be a
Lie group,
its
Lie algebra. Then a principal ''H''-bundle is a
fiber bundle ''P'' over ''M'' with a smooth
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
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
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
''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
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
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
: 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
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
on ''P'',
with values in the
Lie algebra of ''H'', such that
#
# for any
, ''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
.
Frame bundles have additional structure called the
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
, which can be used to extend a principal connection on ''P'' to a trivialization of the tangent bundle of ''P'' called an absolute parallelism.
In general, suppose that ''M'' has dimension ''n'' and ''H'' acts on R
''n'' (this could be any ''n''-dimensional real vector space). A solder form on a principal ''H''-bundle ''P'' over ''M'' is an R
''n''-valued 1-form ''θ'': T''P'' → R
''n'' which is horizontal and equivariant so that it induces a
bundle homomorphism
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
from T''M'' to the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) 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 wit ...
''P'' ×
''H'' R
''n''. 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
''p''''P'' to the coordinates of d''π''
''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
of the
semidirect product ''G'' of ''H'' with R
''n'', which provides an isomorphism of each tangent space T
''p''''P'' 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 R
''n'' need not be effective. This allows, for example, the theory to include spin connections, in which ''H'' is the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
Spin(''n'') rather than the
orthogonal group O(''n'').
* The group ''G'' need not be a semidirect product of ''H'' with R
''n''.
Klein geometries as model spaces
Klein's
Erlangen programme
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...
suggested that geometry could be regarded as a study of
homogeneous spaces: in particular, it is the study of the many geometries of interest to geometers of 19th century (and earlier). A Klein geometry consisted of a space, along with a law for motion within the space (analogous to the
Euclidean transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations ...
s of classical
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
) expressed as a
Lie group of
transformations. These generalized spaces turn out to be homogeneous
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s 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
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
of ''G''. A Klein geometry is the homogeneous space given by the quotient ''P''/''H'' of ''P'' by the right action of ''H''. There is a right ''H''-action on the fibres of the canonical projection
:''π'': ''P'' → ''P''/''H''
given by ''R''
''h''''g'' = ''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 d''π''(''X'') = 0. Any ''ξ'' ∈
gives rise to a canonical vertical vector field ''X''
''ξ'' by taking the derivative of the right action of the 1-parameter subgroup of ''H'' associated to ξ. The
Maurer-Cartan form ''η'' of ''P'' is the
-valued one-form on ''P'' which identifies each tangent space with the Lie algebra. It has the following properties:
# Ad(''h'') ''R''
''h''*''η'' = ''η'' for all ''h'' in ''H''
# ''η''(''X''
''ξ'') = ''ξ'' for all ''ξ'' in
# for all ''g''∈''P'', ''η'' restricts a linear isomorphism of T
''g''''P'' 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 a 1-form ''η'' with these properties, then ''P'' is locally isomorphic as an ''H''-bundle to the principal homogeneous bundle ''G''→''G''/''H''. The structure equation is the
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
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.
Pseudogroups
Cartan connections are closely related to
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
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 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 that varies smoothly from point to point ...
is modelled on
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. On a manifold ''M'', one imagines attaching to each point of ''M'' a copy of the model space ''G''/''H''. The symmetry of the model space is then built into the Cartan geometry or pseudogroup structure by positing that the model spaces of nearby points are related by a transformation in ''G''. The fundamental difference between a Cartan geometry and pseudogroup geometry is that the symmetry for a Cartan geometry relates ''infinitesimally'' close points by an ''infinitesimal'' transformation in ''G'' (i.e., an element of the Lie algebra of ''G'') and the analogous notion of symmetry for a pseudogroup structure applies for points that are physically separated within the manifold.
The process of attaching spaces to points, and the attendant symmetries, can be concretely realized by using special
coordinate systems. To each point ''p'' ∈ ''M'', a
neighborhood ''U''
p of ''p'' is given along with a mapping φ
p : ''U''
p → ''G''/''H''. In this way, the model space is attached to each point of ''M'' by realizing ''M'' locally at each point as an open subset of ''G''/''H''. We think of this as a family of coordinate systems on ''M'', parametrized by the points of ''M''. Two such parametrized coordinate systems φ and φ′ are ''H''-related if there is an element ''h''
p ∈ ''H'', parametrized by ''p'', such that
: φ′
p = ''h''
p φ
p.
This freedom corresponds roughly to the physicists' notion of a
gauge
Gauge ( or ) may refer to:
Measurement
* Gauge (instrument), any of a variety of measuring instruments
* Gauge (firearms)
* Wire gauge, a measure of the size of a wire
** American wire gauge, a common measure of nonferrous wire diameter, ...
.
Nearby points are related by joining them with a curve. Suppose that ''p'' and ''p''′ are two points in ''M'' joined by a curve ''p''
t. Then ''p''
t supplies a notion of transport of the model space along the curve. Let τ
t : ''G''/''H'' → ''G''/''H'' be the (locally defined) composite map
:τ
t = φ
pt o φ
p0−1.
Intuitively, τ
t is the transport map. A pseudogroup structure requires that τ
t be a ''symmetry of the model space'' for each ''t'': τ
t ∈ ''G''. A Cartan connection requires only that the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of τ
t be a symmetry of the model space: τ′
0 ∈ g, the Lie algebra of ''G''.
Typical of Cartan, one motivation for introducing the notion of a Cartan connection was to study the properties of pseudogroups from an infinitesimal point of view. A Cartan connection defines a pseudogroup precisely when the derivative of the transport map τ′ can be
integrated, thus recovering a true (''G''-valued) transport map between the coordinate systems. There is thus an
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
at work, and Cartan's method for realizing integrability conditions was to introduce a
differential form.
In this case, τ′
0 defines a differential form at the point ''p'' as follows. For a curve γ(''t'') = ''p''
t in ''M'' starting at ''p'', we can associate the
tangent vector
In mathematics, 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 R''n''. More generally, tangent vectors are e ...
''X'', as well as a transport map τ
tγ. 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 ω
H is the Maurer-Cartan form of ''H''.
Formal definition
A Cartan geometry modelled on a homogeneous space ''G''/''H'' can be viewed as a ''deformation'' of this geometry which allows for the presence of ''curvature''. For example:
* a
Riemannian manifold can be seen as a deformation of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
;
* a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
can be seen as a deformation of
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
;
* a
conformal manifold
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
can be seen as a deformation of the
conformal sphere;
* a manifold equipped with an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
can be seen as a deformation of an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
.
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 of dimension ''n''+''m'', with Lie algebra
, containing ''H'' as a subgroup.
Definition via gauge transitions
A Cartan connection consists of a
coordinate atlas
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
of open sets ''U'' in ''M'', along with a
-valued 1-form θ
U defined on each chart such that
# θ
U : T''U'' →
.
# θ
U mod
: T
u''U'' →
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 ω
H is the
Maurer-Cartan form of ''H''.
By analogy with the case when the θ
U came from coordinate systems, condition 3 means that φ
U is related to φ
V by ''h''.
The curvature of a Cartan connection consists of a system of 2-forms defined on the charts, given by
:
Ω
U satisfy the compatibility condition:
:If the forms θ
U and θ
V are related by a function ''h'' : ''U'' ∩ ''V'' → ''H'', as above, then Ω
V = Ad(''h''
−1) Ω
U
The definition can be made independent of the coordinate systems by forming the
quotient space
:
of the disjoint union over all ''U'' in the atlas. The
equivalence relation ~ is defined on pairs (''x'',''h''
1) ∈ ''U''
1 × ''H'' and (''x'', ''h''
2) ∈ ''U''
2 × ''H'', by
:(''x'',''h''
1) ~ (''x'', ''h''
2) if and only if ''x'' ∈ ''U''
1 ∩ ''U''
2, θ
''U''1 is related to θ
''U''2 by ''h'', and ''h''
2 = ''h''(''x'')
−1 ''h''
1.
Then ''P'' is a
principal ''H''-bundle on ''M'', and the compatibility condition on the connection forms θ
U implies that they lift to a
-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'')''R''
''h''*''η'' = ''η''
# for all ''ξ'' in
, ''η''(''X''
''ξ'') = ''ξ''
# for all ''p'' in ''P'', the restriction of ''η'' defines a linear isomorphism from the tangent space T
''p''''P'' 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
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 T''M'' to the associated bundle
.
The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that ''η'' mod
is a
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
.
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
, but is only equivariant under the action of ''H''. Indeed, it cannot be equivariant under the full group ''G'' because there is no ''G'' bundle and no ''G'' action. Secondly, the 1-form is an absolute parallelism, which intuitively means that η yields information about the behavior of additional directions in the principal bundle (rather than simply being a projection operator onto the vertical space). Concretely, the existence of a solder form binds (or solders) the Cartan connection to the underlying
differential topology of the manifold.
An intuitive interpretation of the Cartan connection in this form is that it determines a ''fracturing'' of the tautological principal bundle associated to a Klein geometry. Thus Cartan geometries are deformed analogues of Klein geometries. This deformation is roughly a prescription for attaching a copy of the model space ''G''/''H'' to each point of ''M'' and thinking of that model space as being ''tangent'' to (and ''infinitesimally identical'' with) the manifold at a point of contact. The fibre of the tautological bundle ''G'' → ''G''/''H'' of the Klein geometry at the point of contact is then identified with the fibre of the bundle ''P''. Each such fibre (in ''G'') carries a Maurer-Cartan form for ''G'', and the Cartan connection is a way of assembling these Maurer-Cartan forms gathered from the points of contact into a coherent 1-form η defined on the whole bundle. The fact that only elements of ''H'' contribute to the Maurer-Cartan equation Ad(''h'')''R''
''h''*''η'' = ''η'' has the intuitive interpretation that any other elements of ''G'' would move the model space away from the point of contact, and so no longer be tangent to the manifold.
From the Cartan connection, defined in these terms, one can recover a Cartan connection as a system of 1-forms on the manifold (as in the gauge definition) by taking a collection of
local trivializations of ''P'' given as sections ''s''
U : ''U'' → ''P'' and letting θ
U = ''s''
*η be the
pullbacks of the Cartan connection along the sections.
As principal connections
Another way in which to define a Cartan connection is as a
principal connection on a certain principal ''G''-bundle. From this perspective, a Cartan connection consists of
* a principal ''G''-bundle ''Q'' over ''M''
* a principal ''G''-connection ''α'' on ''Q'' (the Cartan connection)
* a principal ''H''-subbundle ''P'' of ''Q'' (i.e., a reduction of structure group)
such that the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
''η'' of ''α'' to ''P'' satisfies the Cartan condition.
The principal connection ''α'' on ''Q'' can be recovered from the form ''η'' by taking ''Q'' to be the associated bundle ''P'' ×
''H'' ''G''. Conversely, the form η can be recovered from α by pulling back along the inclusion ''P'' ⊂ ''Q''.
Since ''α'' is a principal connection, it induces a
connection on any
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) 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 wit ...
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 connection on the bundle ''E'' = ''Q'' ×
''G'' ''G''/''H'' of the preceding section. A Cartan connection then consists of
*A
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
π : ''E'' → ''M'' with fibre ''G''/''H'' and vertical space V''E'' ⊂ T''E''.
*A section ''s'' : ''M'' → ''E''.
*A
G-connection θ : T''E'' → V''E'' such that
::''s''
*θ
x : T
x''M'' → V
''s''(''x'')''E'' is a linear isomorphism of vector spaces for all ''x'' ∈ ''M''.
This definition makes rigorous the intuitive ideas presented in the introduction. First, the preferred section ''s'' can be thought of as identifying a point of contact between the manifold and the tangent space. The last condition, in particular, means that the tangent space of ''M'' at ''x'' is isomorphic to the tangent space of the model space at the point of contact. So the model spaces are, in this way, tangent to the manifold.
This definition also brings prominently into focus the idea of
development
Development or developing may refer to:
Arts
*Development hell, when a project is stuck in development
*Filmmaking, development phase, including finance and budgeting
*Development (music), the process thematic material is reshaped
* Photograph ...
. If ''x''
t is a curve in ''M'', then the Ehresmann connection on ''E'' supplies an associated
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
map τ
t : ''E''
xt → ''E''
x0 from the fibre over the endpoint of the curve to the fibre over the initial point. In particular, since ''E'' is equipped with a preferred section ''s'', the points ''s''(''x''
t) transport back to the fibre over ''x''
0 and trace out a curve in ''E''
x0. This curve is then called the ''development'' of the curve ''x''
t.
To show that this definition is equivalent to the others above, one must introduce a suitable notion of a
moving frame
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
In lay te ...
for the bundle ''E''. In general, this is possible for any ''G''-connection on a fibre bundle with structure group ''G''. See
Ehresmann connection#Associated bundles for more details.
Special Cartan connections
Reductive Cartan connections
Let ''P'' be a principal ''H''-bundle on ''M'', equipped with a Cartan connection η : T''P'' →
. If
is a
reductive module for ''H'', meaning that
admits an Ad(''H'')-invariant splitting of vector spaces
, then the
-component of η generalizes the solder form for an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
.
In detail, η splits into
and
components:
:η = η
+ η
.
Note that the 1-form η
is a principal ''H''-connection on the original Cartan bundle ''P''. Moreover, the 1-form η
satisfies:
:η
(''X'') = 0 for every vertical vector ''X'' ∈ T''P''. (η
is ''horizontal''.)
:R
h*η
= Ad(''h''
−1)η
for every ''h'' ∈ ''H''. (η
is ''equivariant'' under the right ''H''-action.)
In other words, η is a
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
for the bundle ''P''.
Hence, ''P'' equipped with the form η
defines a (first order)
''H''-structure on ''M''. The form η
defines a connection on the ''H''-structure.
Parabolic Cartan connections
If
is a
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
with
parabolic subalgebra (i.e.,
contains a
maximal solvable subalgebra of
) and ''G'' and ''P'' are associated Lie groups, then a Cartan connection modelled on (''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
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
s: this arises because the perpendicular subspace
⊥ of
in
with respect to the
Killing form of
is a subalgebra of
, and the Killing form induces a natural duality between
⊥ and
. Thus the bundle associated to
⊥ is isomorphic to the
cotangent bundle.
Parabolic geometries include many of those of interest in research and applications of Cartan connections, such as the following examples:
*
Conformal 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
:O+(n ...
s: Here ''G'' = ''SO''(''p''+1,''q''+1), and ''P'' is the stabilizer of a null ray in R
n+2.
*
Projective connections: Here ''G'' = ''PGL''(n+1) and ''P'' is the stabilizer of a point in RP
n.
*
CR structure In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
Forma ...
s and Cartan-Chern-Tanaka connections: ''G'' = ''PSU''(''p''+1,''q''+1), ''P'' = stabilizer of a point on the projective null
hyperquadric.
* Contact projective connections: Here ''G'' = ''SP''(2n+2) and ''P'' is the stabilizer of the ray generated by the first standard basis vector in R
n+2.
* Generic rank 2 distributions on 5-manifolds: Here ''G'' = ''Aut''(O
s) is the automorphism group of the algebra O
s of
split octonions, a
closed subgroup
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...
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 R
7 viewed as the purely imaginary split octonions (orthogonal complement of the unit element in O
s).
Associated differential operators
Covariant differentiation
Suppose that ''M'' is a Cartan geometry modelled on ''G''/''H'', and let (''Q'',''α'') be the principal ''G''-bundle with connection, and (''P'',''η'') the corresponding reduction to ''H'' with ''η'' equal to the pullback of ''α''. Let ''V'' a
representation of ''G'', and form the vector bundle V = ''Q'' ×
''G'' ''V'' over ''M''. Then the principal ''G''-connection ''α'' on ''Q'' induces a
covariant derivative on V, which is a first order
linear differential operator
:
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(T''M'',V). For any section ''v'' of V, the contraction of the covariant derivative ∇''v'' with a vector field ''X'' on ''M'' is denoted ∇
''X''''v'' 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''.
[See, for instance, .] Suppose instead 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 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.
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 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''(d''v''). It follows that this composite descends to a first order differential operator ''D'' from sections of V=''P''×
''H''''V'' to sections of the bundle
. This is called the fundamental or universal derivative, or fundamental D-operator.
Notes
References
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Books
* .
:: The section 3. Cartan Connections
ages 127–130treats conformal and projective connections in a unified manner.
External links
*
{{Manifolds
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