User:Silly rabbit/Sandbox/Affine connection

From Affine connection

The following is an abandoned attempt to include an explicit description of the affine group in terms of matrices into the Affine connection article. It's too long, and doesn't fit with the flow of things, but I still feel as though some kind of concrete realization of the affine group is justified.

Affine frames and the affine group

An affine frame for Aⁿ consists of a point O ∈ Aⁿ corresponding to a choice of origin of the affine space, and a basis (e₁,...,e_n) of the vector space Rⁿ consisting of vectors whose initial point is O (which can be naturally identified with the tangent space T_oAⁿ.)

In terms of a fixed background affine reference frame, the affine group can be realized concretely as a matrix group (by analogy with the representation of the general linear group as a group of invertible matrices relative to a fixed basis.) The action of φ ∈ Aff(n) on a point of the form O + v is

$\phi (O+v)=(O+\xi )+Av\,\,(=\xi +Av)$

where ξ is a pure translation and A is an invertible matrix. Thus Aff(n) is realized as a group of block matrices, whose action is given by

${\begin{bmatrix}1&0\\\xi &A\end{bmatrix}}{\begin{bmatrix}1\\v\end{bmatrix}}={\begin{bmatrix}1\\\xi +Av\end{bmatrix}}.$

where ξ has been represented as a column vector. Geometrically, the matrix realization of the affine group corresponds to embedding Aⁿ as the hyperplane x₀=1 in the ambient space Rⁿ⁺¹. The affine group is then the group of linear transformations which preserve this hyperplane. The stabilizer of the origin O in Aⁿ is the subgroup consisting of transformations whose translational part ξ vanishes:

$H=\left\{{\begin{bmatrix}A&0\\0&1\end{bmatrix}};A\in GL(n)\right\}$

The normal subgroup of translations consists of matrices of the form

$T=\left\{{\begin{bmatrix}I&0\\\xi &1\end{bmatrix}}\right\}.$

The general linear group GL(n) acts freely on the set FA of all affine frames by fixing p and transforming the basis (e₁,...,e_n) in the usual way, and the map π sending an affine frame (p;e₁,...,e_n) to p is the quotient map. Thus FA is a principal GL(n)-bundle over A. The action of GL(n) extends naturally to a free transitive action of the affine group Aff(n) on FA, so that FA is an Aff(n)-torsor, and the choice of a reference frame identifies FA → A with the principal bundle Aff(n) → Aff(n)/GL(n).

Cartan connections as deformed model spaces

he prototypical example of a Cartan connection is an affine connection. In this case, the model geometry is that of the homogeneous action of the affine group on affine space in n dimensions. The Klein model has G=Aff(n) and H=GL(n), and the model space is the affine space Aⁿ = P/H, where P is the underlying principal homogeneous space of Aff(n).

An affine connection on a manifold M gives a way of regarding M as infinitesimally identical to the affine model space. In order to understand this a little better, we first consider the model space Aⁿ, and its infinitesimal properties. From the infinitesimal properties, it is then possible to extrapolate the appropriate notion of a Cartan connection (affine connection in this case) on M.

The model Klein geometry Aⁿ is equipped with a canonical principal H-bundle given by the quotient map P → G/H = Aⁿ. The fibre of this bundle over a point x ∈ Aⁿ consists of all linear frames for the affine space based at x. As above, let ω be the Maurer-Cartan form of the Lie group Aff(n). This defines a canonical absolute parallelism on P: a linear isomorphism of T_uP with the Lie algebra aff(n). Under the right action of G on P it transforms under pullback by

$R_{g}^{*}\omega =Ad(g^{-1})\omega .$ (1)

Consider now a manifold M. Suppose that a point x ∈ M is given. Imagine that a copy of the model space is tangent to M at x, so that x is the point of contact between M and a copy of Aⁿ. A linear frame at x, lying in the affine space, can also be attached to M at x. Applying this to all points of the manifold, the totality of all linear frames attached to each point of M forms a principal GL(n) bundle P_M → M. An affine connection prescribes a manner of assembling the Maurer-Cartan forms from the fibres of this principal bundle into a new g-valued 1-form ω_M.

A basic requirement is that ω_M must respect the infinitesimal (i.e., first order) properties of the affine model space at the point of contact. Firstly, this means that ω_M must be an absolute parallelism on P_M. In other words, it must define a linear isomorphism T_uP_M ≈ g. Secondly, the equivariance condition (1) must hold, but only for those elements of G which act tangentially to the fibre of P, since the transverse actions move away from the point of contact. Hence,

$R_{h}^{*}\omega _{M}=Ad(h^{-1})\omega _{M}$ for all h ∈ H.

Affine connections

The prototypical example of a Cartan connection is an affine connection. In this case, the model geometry is that of the homogeneous action of the affine group on affine space in n dimensions. The Klein model has G=Aff(n) and H=GL(n), and the model space is the affine space Aⁿ = P/H, where P is the underlying principal homogeneous space of Aff(n).

An affine connection on a manifold M gives a way of regarding M as infinitesimally identical to the affine model space. In order to understand this a little better, we first consider the model space Aⁿ, and its infinitesimal properties. From the infinitesimal properties, it is then possible to extrapolate the appropriate notion of a Cartan connection (affine connection in this case) on M.

The model Klein geometry Aⁿ is equipped with a canonical principal H-bundle given by the quotient map P → G/H = Aⁿ. The fibre of this bundle over a point x ∈ Aⁿ consists of all linear frames for the affine space based at x. As above, let ω be the Maurer-Cartan form of the Lie group Aff(n). This defines a canonical absolute parallelism on P: a linear isomorphism of T_uP with the Lie algebra aff(n). Under the right action of G on P it transforms under pullback by

$R_{g}^{*}\omega =Ad(g^{-1})\omega .$ (1)

Consider now a manifold M. Suppose that a point x ∈ M is given. Imagine that a copy of the model space is tangent to M at x, so that x is the point of contact between M and a copy of Aⁿ. A linear frame at x, lying in the affine space, can also be attached to M at x. Applying this to all points of the manifold, the totality of all linear frames attached to each point of M forms a principal GL(n) bundle P_M → M. An affine connection prescribes a manner of assembling the Maurer-Cartan forms from the fibres of this principal bundle into a new g-valued 1-form ω_M.

A basic requirement is that ω_M must respect the infinitesimal (i.e., first order) properties of the affine model space at the point of contact. Firstly, this means that ω_M must be an absolute parallelism on P_M. In other words, it must define a linear isomorphism T_uP_M ≈ g. Secondly, the equivariance condition (1) must hold, but only for those elements of G which act tangentially to the fibre of P, since the transverse actions move away from the point of contact. Hence,

$R_{h}^{*}\omega _{M}=Ad(h^{-1})\omega _{M}$ for all h ∈ H.

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

Let H be a Lie group. Then a principal H-bundle is fiber bundle P over M with a smooth action of H on P which is free and transitive on the fibers. Thus P is a smooth manifold with a smooth map π: P → M which looks locally like the trivial bundle M × H → M. The frame bundle of M is a principal GL(n)-bundle, while if M is a Riemannian manifold, then the orthonormal frame bundle is a principal O(n)-bundle.

Let R_h denote the (right) action of h ∈ H on P. The derivative of this action defines a vertical vector field on P for each element ξ of ${\mathfrak {h}}$: if h(t) is a 1-parameter subgroup with h(0)=e (the identity element) and h '(0)=ξ, then the corresponding vertical vector field is

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

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

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

The intuitive idea is that ω(X) provides a vertical component of X, using the isomorphism of the fibers of π with H to identify vertical vectors with elements of ${\mathfrak {h}}$.

Frame bundles have additional structure called the solder form, which can be used to extend a principal connection on P to a trivialization of the tangent bundle of P called an absolute parallelism.

In general, suppose that M has dimension n and H acts on Rⁿ (this could be any n-dimensional real vector space). A solder form on a principal H-bundle P over M is an Rⁿ-valued 1-form θ: TP → Rⁿ which is horizontal and equivariant so that it induces a bundle homomorphism from TM to the associated bundle P ×_H Rⁿ. This is furthermore required to be a bundle isomorphism. Frame bundles have a (canonical or tautological) solder form which sends a tangent vector X ∈ T_pP to the coordinates of dπ_p(X) ∈ T_π(p)M with respect to the frame p.

The pair (ω, θ) (a principal connection and a solder form) defines a 1-form η on P, with values in the Lie algebra ${\mathfrak {g}}$ of the semidirect product G of H with Rⁿ, which provides an isomorphism of each tangent space T_pP with ${\mathfrak {g}}$. It induces a principal connection α on the associated principal G-bundle P ×_H G. This is a Cartan connection.

Cartan connections generalize affine connections in two ways.

• The action of H on Rⁿ need not be effective. This allows, for example, the theory to include spin connections, in which H is the spin group Spin(n) rather than the orthogonal group O(n).
• The group G need not be a semidirect product of H with Rⁿ.