Affine connection
In differential geometry, an affine connection[a] 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 in a fixed vector space.
[3] The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity).
The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule).
The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection.
Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.
The origins of this idea can be traced back to two main sources: surface theory and tensor calculus.
Differential geometers in the 19th century were interested in the notion of development in which one surface was rolled along another, without slipping or twisting.
In particular, the tangent plane to a point of S can be rolled on S: this should be easy to imagine when S is a surface like the 2-sphere, which is the smooth boundary of a convex region.
The second motivation for affine connections comes from the notion of a covariant derivative of vector fields.
The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept.
As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of parallel transport of vector fields along a curve.
The pair (θ, ω) defines a bundle isomorphism of T(FM) with the trivial bundle FM × aff(n), where aff(n) is the Cartesian product of Rn and gl(n) (viewed as the Lie algebra of the affine group, which is actually a semidirect product – see below).
[7] In the modern approach, this is closely related to the definition of affine connections on the frame bundle.
Indeed, in one formulation, a Cartan connection is an absolute parallelism of a principal bundle satisfying suitable properties.
However, Cartan's original approach was different from this in a number of ways: The points just raised are easiest to explain in reverse, starting from the motivation provided by surface theory.
To motivate this, it suffices to consider how affine frames of reference transform infinitesimally with respect to parallel transport.
The affine connection is then given symbolically by a first order differential system defined by a collection of one-forms (θ j, ω ji).
It also draws on the inspiration of Felix Klein's Erlangen programme,[10] in which a geometry is defined to be a homogeneous space.
The operation thus described p → p + v is the translation of p along v. In technical terms, affine n-space is a set An equipped with a free transitive action of the vector group Rn on it through this operation of translation of points: An is thus a principal homogeneous space for the vector group Rn.
An affine frame for A consists of a point p ∈ A and a basis (e1,… en) of the vector space TpA = Rn.
The general linear group GL(n) acts freely on the set FA of all affine frames by fixing p and transforming the basis (e1,… en) in the usual way, and the map π sending an affine frame (p; e1,… en) to p is the quotient map.
Since the functions εi yield a basis for Rn at each point of FA, these 1-forms must be expressible as sums of the form for some collection (θ i, ω kj)1 ≤ i, j, k ≤ n of real-valued one-forms on Aff(n).
Taking the exterior derivative a second time, and using the fact that d2 = 0 as well as the linear independence of the εi, the following relations are obtained: These are the Maurer–Cartan equations for the Lie group Aff(n) (identified with FA by the choice of a reference frame).
An affine space, as with essentially any smooth Klein geometry, is a manifold equipped with a flat Cartan connection.
Let M be a manifold, and P a principal GL(n)-bundle over M. Then an affine connection is a 1-form η on P with values in aff(n) satisfying the following properties The last condition means that η is an absolute parallelism on P, i.e., it identifies the tangent bundle of P with a trivial bundle (in this case P × aff(n)).
The affine Lie algebra aff(n) splits as a semidirect product of Rn and gl(n) and so η may be written as a pair (θ, ω) where θ takes values in Rn and ω takes values in gl(n).
The curvature is given by the formula Note that [X, Y] is the Lie bracket of vector fields in Einstein notation.
When both curvature and torsion vanish, the connection defines a pre-Lie algebra structure on the space of global sections of the tangent bundle.
In formal terms, let τ0t : TxtM → Tx0M be the linear parallel transport map associated to the affine connection.
From the linear connection point of view, the covariant derivative of a vector field is defined by differentiating the vector field, viewed as a map from M to R3, and then projecting the result orthogonally back onto the tangent spaces of M. It is easy to see that this affine connection is torsion-free.
