In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. See Method of moving frames, Cartan connection applications and Einstein-Cartan theory for some applications of the method. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... Einstein-Cartan theory extends general relativity to correctly handle spin angular momentum. ...

Conceptual aspects of the theory

It was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames. It operates with differential forms and so is computational in character, but has two other major aspects, both more geometric. Cartan reformulated the differential geometry of (pseudo) Riemannian geometry; and not just those (metric) manifolds, but theories for an arbitrary manifold, including Lie group manifolds. This was in terms of moving frames (repère mobile), as an alternative reformulation in particular of general relativity. Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... Two-dimensional visualization of space-time distortion. ...

The main idea is to develop expressions for connection forms and curvature using orthogonal frames. In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... Curvature is the amount by which a geometric object deviates from being flat. ...

Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames. Although it is frame-dependent in its most basic form, it is very well suited for computations. It can also be understood in terms of frame bundles, and it allows generalizations like the spinor bundle. This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... Curvature is the amount by which a geometric object deviates from being flat. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... Given a differentiable manifold M with a tetrad of signature (p,q) over it, a spinor bundle over M is a vector SO(p,q)-bundle over M such that its fiber is a spinor representation of Spin(p,q), the double cover of the special orthogonal group SO(p...

A general theory of frames

The first aspect of the theory looks first to the theory of principal bundles (which one can call the general theory of frames). The idea of a connection on a principal bundle for a Lie group G is relatively easy to formulate, because in the 'vertical direction' one can see that the required datum is given by translating all tangent vectors back to the identity element (into the Lie algebra), and the connection definition should simply add a 'horizontal' component, compatible with that. If G is a type of affine group with respect to another Lie group H - meaning that G is a semidirect product of H with a vector translation group T on which H acts, an H-bundle can be made into a G-bundle by the associated bundle construction. There is a T-bundle associated, too: a vector bundle, on which H acts by automorphisms that become inner automorphisms in G. In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ...

The first type of definition in this set-up is that a Cartan connection for H is a specific type of principal G-connection.

Identifying the tangent bundle

The second type of definition looks directly at the tangent bundle TM of the smooth manifold M assumed as the base. Here the datum is a certain type of identification of TM, as a bundle, as the 'vertical' tangent vectors in the T-bundle mentioned before (where M is natural identified as the zero section). This is called a soldering (sometimes welding): we now have TM within a richer setting, expressed by the H-valued transition data. A major point here, as with the previous discussion, is that it is not assumed that H acts faithfully on T. That immediately allows spinor bundles to take their place in the theory, with H a spin group rather than simply an orthogonal group. In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

General theory in formal terms

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. Of course, a flat Cartan geometry should be a geometry without curvature. Beginning then with the flat case, we describe what is meant by a Cartan Geometry in general formal mathematical terms. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... Curvature is the amount by which a geometric object deviates from being flat. ...

The flat case

Motivation

The Erlangen program focuses upon the study of homogeneous spaces of topological groups, and in particular, most geometries of interest (at least during the 19^th century and early 20^th century) turn out to be homogeneous differential manifolds isomorphic to the quotient space of a Lie group by a Lie subgroup. It is precisely the differential structure which is inherited from the differential structure of the Lie group which endows these homogeneous spaces with more structure (of a differential kind) than homogeneous spaces in general. An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartans theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is...

Mathematical details

The general approach to Cartan is to begin with a Lie group G and a Lie subgroup H with associated Lie algebras and , respectively. There is a right H-action on the fibres of the canonical homomorphism In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartans theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is...

given by R_hg = gh. A vector field is vertical if π _* X = 0. Any gives rise to a canonical vertical vector field X ⁺ by taking the differential of the right action. So for instance, if h(t) is a 1-parameter subgroup with tangent vector at the identity h'(e)=X, then the vertical vector field is

The Maurer-Cartan form w for G can be reinterpreted in terms of such principal bundles over homogeneous spaces axiomatically as follows: In mathematics, the Maurer-Cartan form for a Lie group G is a distinguished differential form on G that carries within itself the basic infinitesimal information about the structure of G. It was much used by Elie Cartan, as a basic ingredient of his method of moving frames. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

1. w is a g-valued one-form on G, which is a linear isomorphism of the tangent space of G.
2. (R_h) _* w = Ad(h ^{− 1})w for all h in H.
3. w(X ⁺ ) = X for all X in .
4. (the structural equation)

Conversely, one can show that given a manifold M and a principal H-bundle over M, if a form w obeying these properties is given on the bundle, then that principal bundle is locally isomorphic as an H-bundle to the principal homogeneous bundle . Property 4 of the Maurer-Cartan form is tantamount to an integrability condition for the problem of establishing such an isomorphism. A Cartan geometry is a fracturing of the integrability condition in this picture, allowing for the presence of curvature. 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. ... Curvature is the amount by which a geometric object deviates from being flat. ...

The curved case

Starting with the basic data for a homogeneous space as above, we are now prepared to define a Cartan geometry as a certain deformation of this structure, allowing for the presence of curvature.

Motivation

Riemannian geometry can be seen as a "deformation" of Euclidean geometry, a pseudo-Riemannian manifold as a deformation of Minkowski space, a differential manifold equipped with a conformal structure (a Weyl manifold) can be seen as a deformation of a conformal geometry, a differential manifold equipped with an affine connection (but no Riemannian metric) can be seen as a deformation of an affine geometry, etc. In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ...

Mathematical details

A Cartan geometry consists of the following. A smooth manifold M of dimension n, a Lie group H of dimension r having Lie algebra , a principal H-bundle P on M, and Lie group G of dimension n+r with Lie algebra containing H as a subgroup. A Cartan connection is a -valued 1-form on P satisfying

1. w is a linear isomorphism of the tangent space of P.
2. (R_h) _* w = Ad(h ^{− 1})w for all h in H.
3. w(X ⁺ ) = X for all X in .

The curvature of a Cartan connection is the -valued 2-form

.

If M is equipped with a Cartan geometry, the tangent space of M carries a canonical H-representation. Indeed, the projection has differential . The kernel of π _* consists of the subbundle of vertical vectors, which the Cartan connection trivializes to . Thus the tangent bundle of M is isomorphic to the fibre product In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...

where is acted upon by the adjoint representation of H.

Gauges for a Cartan connection

In performing actual calculations with a Cartan connection, it is traditional to work in a particular gauge. A gauge on M is just a -valued 1-form θ on (an open subset of) M such that the quotient mapping is an isomorphism of vector spaces.

In terms of the connection w, a gauge can be determined by choosing a section , and setting θ = s ^* w. Such a section of the bundle is called a moving frame. If a pair of sections s and t are given, then they are related by the H-action, so s = kt where k is an H-valued function on M. The induced gauges s ^* w and t ^* w are related by the equation In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ...

s ^* w = Ad(k ^{− 1})t ^* w + k ^* ω_H

where ω_H is the Maurer-Cartan form of H.

The fundamental D operator

Let V be a real or complex representation of H, with the action of H denoted by ρ. Let A⁰(P,V) be the space of equivariant V-valued functions on P, so that

for all .

Or equivalently

.

Let A^q(P,V) be the space of equivariant V-valued q-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism

given by

The de Rham operator preserves equivariance and so descends to give a first order differential operator

.

The fundamental D operator is then the composite operator

.

Acting on functions in A⁰(P,V), one has

D_Xf = w ^{− 1}(X)f.

Covariant differentiation

The covariant derivative is a first order differential operator which can be defined in a wide class of Cartan geometries. As in the previous section, let the data specify a Cartan geometry, and let (V,ρ) be a representation of H, and form the vector bundle over M. The covariant derivative is a first-order differential operator In mathematics, a group representation is a way of viewing a group in some more concrete way. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

for each satisfying the usual axioms: If v and w are sections of , k is a function on M, and X and Y are sections of TM, then

To construct the covariant derivative, let v be any section of . Recall that v may be thought of as an H-equivariant map . This is the point of view we shall adopt. Let X be a section of the tangent bundle of M. Choose any right-invariant lift to the tangent bundle of P. Define

.

In order to show that has the required properties, it must: (1) be independent of the chosen lift , (2) be equivariant, so that it descends to a section of the bundle .

For (1), the ambiguity in selecting a right-invariant lift of X is a transformation of the form where ζ ⁺ is the right-invariant vertical vector field induced from . So, calculating the covariant derivative in terms of the new lift , one has

since ρ(ζ)(v) = − ζ ⁺ (v) by taking the differential of the equivariance property .

For (2), since is right-invariant,

and furthermore

so as required.

See also: Riemannian geometry, General relativity In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... Two-dimensional visualization of space-time distortion. ...

Further reading

• M. Nakahara, "Geometry, Topology and Physics", ISBN 0750306068 (2nd ed, paperback)