Affine connection

File:Parallel transport sphere.svg

An affine connection on the sphere rolls the affine tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the development.

In differential geometry, an affine connectionTemplate:Efn 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. Connections are among the simplest methods of defining differentiation of the sections of vector bundles.Template:Sfn

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 terminology is due to CartanTemplate:Efn and has its origins in the identification of tangent spaces in Euclidean space Template:Math by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a metric tensor then there is a natural choice of affine connection, called the Levi-Civita connection. 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). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.

The main invariants of an affine connection are its torsion and its curvature. 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.

Motivation and history

A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space Template:Math: for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane. Smooth functions and vector fields can be defined on manifolds, just as they can on Euclidean space, and scalar functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point Template:Mvar can be identified naturally (by translation) with the tangent space at a nearby point Template:Mvar. On a general manifold, there is no such natural identification between nearby tangent spaces, and so tangent vectors at nearby points cannot be compared in a well-defined way. The notion of an affine connection was introduced to remedy this problem by connecting nearby tangent spaces. The origins of this idea can be traced back to two main sources: surface theory and tensor calculus.

Motivation from surface theory

Consider a smooth surface Template:Mvar in a 3-dimensional Euclidean space. Near any point, Template:Mvar can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. 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 Template:Mvar can be rolled on Template:Mvar: this should be easy to imagine when Template:Mvar is a surface like the 2-sphere, which is the smooth boundary of a convex region. As the tangent plane is rolled on Template:Mvar, the point of contact traces out a curve on Template:Mvar. Conversely, given a curve on Template:Mvar, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve. These identifications are always given by affine transformations from one tangent plane to another.

This notion of parallel transport of tangent vectors, by affine transformations, along a curve has a characteristic feature: the point of contact of the tangent plane with the surface always moves with the curve under parallel translation (i.e., as the tangent plane is rolled along the surface, the point of contact moves). This generic condition is characteristic of Cartan connections. In more modern approaches, 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, so that parallel transport is linear, rather than affine.

In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are model surfaces — they are the simplest surfaces in Euclidean 3-space, and are homogeneous under the affine group of the plane — and every smooth surface has a unique model surface tangent to it at each point. These model surfaces are Klein geometries in the sense of Felix Klein's Erlangen programme. More generally, an Template:Mvar-dimensional affine space is a Klein geometry for the affine group Template:Math, the stabilizer of a point being the general linear group Template:Math. An affine Template:Mvar-manifold is then a manifold which looks infinitesimally like Template:Mvar-dimensional affine space.

Motivation from tensor calculus

File:Affine connection example.svg

Historically, people used the covariant derivative (or Levi-Civita connection given by the metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2-dimensional Euclidean space, the blue vector field Template:Mvar sends the one-form Template:Math to 0.07 everywhere. The red vector field Template:Mvar sends the one-form Template:Math to Template:Math everywhere. Endorsed by the metric Template:Math, the Levi-Civita connection Template:Math is 0 everywhere, indicating Template:Mvar has no change along Template:Mvar. In other words, Template:Mvar parallel transports along each concentric circle. Template:Math everywhere, which sends Template:Math to 0.5 everywhere, implying Template:Mvar has a "constant" changing rate on the radial direction.

The second motivation for affine connections comes from the notion of a covariant derivative of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields by embedding their respective Euclidean vectors into an atlas. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates.Template:Citation needed Correction terms were introduced by Elwin Bruno Christoffel (following ideas of Bernhard Riemann) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols.

This idea was developed into the theory of absolute differential calculus (now known as tensor calculus) by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita between 1880 and the turn of the 20th century.

Tensor calculus really came to life, however, with the advent of Albert Einstein's theory of general relativity in 1915. A few years after this, Levi-Civita formalized the unique connection associated to a Riemannian metric, now known as the Levi-Civita connection. More general affine connections were then studied around 1920, by Hermann Weyl,<ref>Template:Harvnb, 5 editions to 1922.</ref> who developed a detailed mathematical foundation for general relativity, and Élie Cartan,<ref name="Cartan-affine">Template:Harvnb.</ref> who made the link with the geometrical ideas coming from surface theory.

Approaches

The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept.

The most popular approach is probably the definition motivated by covariant derivatives. On the one hand, the ideas of Weyl were taken up by physicists in the form of gauge theory and gauge covariant derivatives. On the other hand, the notion of covariant differentiation was abstracted by Jean-Louis Koszul, who defined (linear or Koszul) connections on vector bundles. In this language, an affine connection is simply a covariant derivative or (linear) connection on the tangent bundle.

However, this approach does not explain the geometry behind affine connections nor how they acquired their name.Template:Efn The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean Template:Mvar-space is an affine space. (Alternatively, Euclidean space is a principal homogeneous space or torsor under the group of translations, which is a subgroup of the affine group.) 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. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group Template:Math or as a principal Template:Math connection on the frame bundle.

Formal definition as a differential operator

Let Template:Mvar be a smooth manifold and let Template:Math be the space of vector fields on Template:Mvar, that is, the space of smooth sections of the tangent bundle Template:Math. Then an affine connection on Template:Mvar is a bilinear map

<math>\begin{align}

\Gamma(\mathrm{T}M)\times \Gamma(\mathrm{T}M) & \rightarrow \Gamma(\mathrm{T}M)\\ (X,Y) & \mapsto \nabla_X Y\,,\end{align}</math> such that for all Template:Mvar in the set of smooth functions on Template:Math, written Template:Math, and all vector fields Template:Math on Template:Mvar:

1. Template:Math, that is, Template:Math is Template:Math-linear in the first variable;
2. Template:Math, where Template:Math denotes the directional derivative; that is, Template:Math satisfies Leibniz rule in the second variable.

Elementary properties

• It follows from property 1 above that the value of Template:Math at a point Template:Math depends only on the value of Template:Mvar at Template:Mvar and not on the value of Template:Mvar on Template:Math. It also follows from property 2 above that the value of Template:Math at a point Template:Math depends only on the value of Template:Mvar on a neighbourhood of Template:Mvar.
• If Template:Math are affine connections then the value at Template:Mvar of Template:Math may be written Template:Math where <math display="block">\Gamma_x : \mathrm{T}_xM \times \mathrm{T}_xM \to \mathrm{T}_xM</math> is bilinear and depends smoothly on Template:Mvar (i.e., it defines a smooth bundle homomorphism). Conversely if Template:Math is an affine connection and Template:Math is such a smooth bilinear bundle homomorphism (called a connection form on Template:Mvar) then Template:Math is an affine connection.
• If Template:Mvar is an open subset of Template:Math, then the tangent bundle of Template:Mvar is the trivial bundle Template:Math. In this situation there is a canonical affine connection Template:Math on Template:Mvar: any vector field Template:Mvar is given by a smooth function Template:Mvar from Template:Mvar to Template:Math; then Template:Math is the vector field corresponding to the smooth function Template:Math from Template:Mvar to Template:Math. Any other affine connection Template:Math on Template:Mvar may therefore be written Template:Math, where Template:Math is a connection form on Template:Mvar.
• More generally, a local trivialization of the tangent bundle is a bundle isomorphism between the restriction of Template:Math to an open subset Template:Mvar of Template:Mvar, and Template:Math. The restriction of an affine connection Template:Math to Template:Mvar may then be written in the form Template:Math where Template:Math is a connection form on Template:Mvar.

Parallel transport for affine connections

File:Parallel transport sphere2.svg

Parallel transport of a tangent vector along a curve in the sphere.

Comparison of tangent vectors at different points on a manifold is generally not a well-defined process. An affine connection provides one way to remedy this using the notion of parallel transport, and indeed this can be used to give a definition of an affine connection.

Let Template:Mvar be a manifold with an affine connection Template:Math. Then a vector field Template:Mvar is said to be parallel if Template:Math in the sense that for any vector field Template:Mvar, Template:Math. Intuitively speaking, parallel vectors have all their derivatives equal to zero and are therefore in some sense constant. By evaluating a parallel vector field at two points Template:Mvar and Template:Mvar, an identification between a tangent vector at Template:Mvar and one at Template:Mvar is obtained. Such tangent vectors are said to be parallel transports of each other.

Nonzero parallel vector fields do not, in general, exist, because the equation Template:Math is a partial differential equation which is overdetermined: the integrability condition for this equation is the vanishing of the curvature of Template:Math (see below). However, if this equation is restricted to a curve from Template:Mvar to Template:Mvar it becomes an ordinary differential equation. There is then a unique solution for any initial value of Template:Mvar at Template:Mvar.

More precisely, if Template:Math a smooth curve parametrized by an interval Template:Math and Template:Math, where Template:Math, then a vector field Template:Mvar along Template:Mvar (and in particular, the value of this vector field at Template:Math) is called the parallel transport of Template:Mvar along Template:Mvar if

1. Template:Math, for all Template:Math
2. Template:Math.

Formally, the first condition means that Template:Mvar is parallel with respect to the pullback connection on the pullback bundle Template:Math. However, in a local trivialization it is a first-order system of linear ordinary differential equations, which has a unique solution for any initial condition given by the second condition (for instance, by the Picard–Lindelöf theorem).

Thus parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve. The isomorphism obtained in this way will in general depend on the choice of the curve: if it does not, then parallel transport along every curve can be used to define parallel vector fields on Template:Mvar, which can only happen if the curvature of Template:Math is zero.

A linear isomorphism is determined by its action on an ordered basis or frame. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent) frame bundle Template:Math along a curve. In other words, the affine connection provides a lift of any curve Template:Mvar in Template:Mvar to a curve Template:Mvar in Template:Math.

Formal definition on the frame bundle

An affine connection may also be defined as a [[connection (principal bundle)|principal Template:Math connection]] Template:Mvar on the frame bundle Template:Math or Template:Math of a manifold Template:Mvar. In more detail, Template:Mvar is a smooth map from the tangent bundle Template:Math of the frame bundle to the space of Template:Math matrices (which is the Lie algebra Template:Math of the Lie group Template:Math of invertible Template:Math matrices) satisfying two properties:

1. Template:Mvar is equivariant with respect to the action of Template:Math on Template:Math and Template:Math;
2. Template:Math for any Template:Mvar in Template:Math, where Template:Mvar is the vector field on Template:Math corresponding to Template:Mvar.

Such a connection Template:Mvar immediately defines a covariant derivative not only on the tangent bundle, but on vector bundles associated to any group representation of Template:Math, including bundles of tensors and tensor densities. Conversely, an affine connection on the tangent bundle determines an affine connection on the frame bundle, for instance, by requiring that Template:Mvar vanishes on tangent vectors to the lifts of curves to the frame bundle defined by parallel transport.

The frame bundle also comes equipped with a solder form Template:Math which is horizontal in the sense that it vanishes on vertical vectors such as the point values of the vector fields Template:Mvar: Indeed Template:Mvar is defined first by projecting a tangent vector (to Template:Math at a frame Template:Mvar) to Template:Mvar, then by taking the components of this tangent vector on Template:Mvar with respect to the frame Template:Mvar. Note that Template:Mvar is also Template:Math-equivariant (where Template:Math acts on Template:Math by matrix multiplication).

The pair Template:Math defines a bundle isomorphism of Template:Math with the trivial bundle Template:Math, where Template:Math is the Cartesian product of Template:Math and Template:Math (viewed as the Lie algebra of the affine group, which is actually a semidirect product – see below).

Affine connections as Cartan connections

Affine connections can be defined within Cartan's general framework.<ref>Template:Harvnb.</ref> 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. From this point of view the Template:Math-valued one-form Template:Math on the frame bundle (of an affine manifold) is a Cartan connection. However, Cartan's original approach was different from this in a number of ways:

• the concept of frame bundles or principal bundles did not exist;
• a connection was viewed in terms of parallel transport between infinitesimally nearby points;Template:Efn
• this parallel transport was affine, rather than linear;
• the objects being transported were not tangent vectors in the modern sense, but elements of an affine space with a marked point, which the Cartan connection ultimately identifies with the tangent space.

Explanations and historical intuition

The points just raised are easiest to explain in reverse, starting from the motivation provided by surface theory. In this situation, although the planes being rolled over the surface are tangent planes in a naive sense, the notion of a tangent space is really an infinitesimal notion,Template:Efn whereas the planes, as affine subspaces of Template:Math, are infinite in extent. However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point. The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point. However, the parallel transport defined by rolling does not fix this origin: it is affine rather than linear; the linear parallel transport can be recovered by applying a translation.

Abstracting this idea, an affine manifold should therefore be an Template:Mvar-manifold Template:Mvar with an affine space Template:Math, of dimension Template:Mvar, attached to each Template:Math at a marked point Template:Math, together with a method for transporting elements of these affine spaces along any curve Template:Mvar in Template:Mvar. This method is required to satisfy several properties:

1. for any two points Template:Math on Template:Mvar, parallel transport is an affine transformation from Template:Math to Template:Math;
2. parallel transport is defined infinitesimally in the sense that it is differentiable at any point on Template:Mvar and depends only on the tangent vector to Template:Mvar at that point;
3. the derivative of the parallel transport at Template:Mvar determines a linear isomorphism from Template:Math to Template:Math.

These last two points are quite hard to make precise,<ref>For details, see Template:Harvtxt. The following intuitive treatment is that of Template:Harvtxt and Template:Harvtxt.</ref> so affine connections are more often defined infinitesimally. To motivate this, it suffices to consider how affine frames of reference transform infinitesimally with respect to parallel transport. (This is the origin of Cartan's method of moving frames.) An affine frame at a point consists of a list Template:Math, where Template:MathTemplate:Efn and the Template:Math form a basis of Template:Math. The affine connection is then given symbolically by a first order differential system

<math>(*) \begin{cases}

\mathrm{d}{p} &= \theta^1\mathbf{e}_1 + \cdots + \theta^n\mathbf{e}_n \\ \mathrm{d}\mathbf{e}_i &= \omega^1_i\mathbf{e}_1 + \cdots + \omega^n_i\mathbf{e}_n \end{cases} \quad i=1,2,\ldots,n</math>

defined by a collection of one-forms Template:Math. Geometrically, an affine frame undergoes a displacement travelling along a curve Template:Mvar from Template:Math to Template:Math given (approximately, or infinitesimally) by

<math>\begin{align}

p(\gamma(t+\delta t)) - p(\gamma(t)) &= \left(\theta^1\left(\gamma'(t)\right)\mathbf{e}_1 + \cdots + \theta^n\left(\gamma'(t)\right)\mathbf{e}_n\right)\mathrm \delta t \\ \mathbf{e}_i(\gamma(t+\delta t)) - \mathbf{e}_i(\gamma(t)) &= \left(\omega^1_i\left(\gamma'(t)\right)\mathbf{e}_1 + \cdots + \omega^n_i\left(\gamma'(t)\right) \mathbf{e}_n\right)\delta t\,. \end{align}</math>

Furthermore, the affine spaces Template:Math are required to be tangent to Template:Mvar in the informal sense that the displacement of Template:Math along Template:Mvar can be identified (approximately or infinitesimally) with the tangent vector Template:Math to Template:Mvar at Template:Math (which is the infinitesimal displacement of Template:Mvar). Since

<math>a_x (\gamma(t + \delta t)) - a_x (\gamma(t)) = \theta\left(\gamma'(t)\right) \delta t \,,</math>

where Template:Mvar is defined by Template:Math, this identification is given by Template:Mvar, so the requirement is that Template:Mvar should be a linear isomorphism at each point.

The tangential affine space Template:Math is thus identified intuitively with an infinitesimal affine neighborhood of Template:Mvar.

The modern point of view makes all this intuition more precise using principal bundles (the essential idea is to replace a frame or a variable frame by the space of all frames and functions on this space). It also draws on the inspiration of Felix Klein's Erlangen programme,<ref>Cf. R. Hermann (1983), Appendix 1–3 to Template:Harvtxt, and also Template:Harvtxt.</ref> in which a geometry is defined to be a homogeneous space. Affine space is a geometry in this sense, and is equipped with a flat Cartan connection. Thus a general affine manifold is viewed as curved deformation of the flat model geometry of affine space.

Affine space as the flat model geometry

Definition of an affine space

Informally, an affine space is a vector space without a fixed choice of origin. It describes the geometry of points and free vectors in space. As a consequence of the lack of origin, points in affine space cannot be added together as this requires a choice of origin with which to form the parallelogram law for vector addition. However, a vector Template:Mvar may be added to a point Template:Mvar by placing the initial point of the vector at Template:Mvar and then transporting Template:Mvar to the terminal point. The operation thus described Template:Math is the translation of Template:Mvar along Template:Mvar. In technical terms, affine Template:Mvar-space is a set Template:Math equipped with a free transitive action of the vector group Template:Math on it through this operation of translation of points: Template:Math is thus a principal homogeneous space for the vector group Template:Math.

The general linear group Template:Math is the group of transformations of Template:Math which preserve the linear structure of Template:Math in the sense that Template:Math. By analogy, the affine group Template:Math is the group of transformations of Template:Math preserving the affine structure. Thus Template:Math must preserve translations in the sense that

<math>\varphi(p+v)=\varphi(p)+T(v)</math>

where Template:Mvar is a general linear transformation. The map sending Template:Math to Template:Math is a group homomorphism. Its kernel is the group of translations Template:Math. The stabilizer of any point Template:Mvar in Template:Mvar can thus be identified with Template:Math using this projection: this realises the affine group as a semidirect product of Template:Math and Template:Math, and affine space as the homogeneous space Template:Math.

Affine frames and the flat affine connection

An affine frame for Template:Mvar consists of a point Template:Math and a basis Template:Math of the vector space Template:Math. The general linear group Template:Math acts freely on the set Template:Math of all affine frames by fixing Template:Mvar and transforming the basis Template:Math in the usual way, and the map Template:Mvar sending an affine frame Template:Math to Template:Mvar is the quotient map. Thus Template:Math is a [[principal bundle|principal Template:Math-bundle]] over Template:Mvar. The action of Template:Math extends naturally to a free transitive action of the affine group Template:Math on Template:Math, so that Template:Math is an Template:Math-torsor, and the choice of a reference frame identifies Template:Math with the principal bundle Template:Math.

On Template:Math there is a collection of Template:Math functions defined by

<math>\pi(p;\mathbf{e}_1, \dots ,\mathbf{e}_n) = p</math>

(as before) and

<math>\varepsilon_i(p;\mathbf{e}_1,\dots , \mathbf{e}_n) = \mathbf{e}_i\,.</math>

After choosing a basepoint for Template:Mvar, these are all functions with values in Template:Math, so it is possible to take their exterior derivatives to obtain differential 1-forms with values in Template:Math. Since the functions Template:Mvar yield a basis for Template:Math at each point of Template:Math, these 1-forms must be expressible as sums of the form

<math>\begin{align}

\mathrm{d}\pi &= \theta^1\varepsilon_1+\cdots+\theta^n\varepsilon_n\\ \mathrm{d}\varepsilon_i &= \omega^1_i\varepsilon_1+\cdots+\omega^n_i\varepsilon_n \end{align}</math>

for some collection Template:Math of real-valued one-forms on Template:Math. This system of one-forms on the principal bundle Template:Math defines the affine connection on Template:Mvar.

Taking the exterior derivative a second time, and using the fact that Template:Math as well as the linear independence of the Template:Mvar, the following relations are obtained:

<math>\begin{align}

\mathrm{d}\theta^j - \sum_i\omega^j_i\wedge\theta^i &=0\\ \mathrm{d}\omega^j_i - \sum_k \omega^j_k\wedge\omega^k_i &=0\,. \end{align}</math>

These are the Maurer–Cartan equations for the Lie group Template:Math (identified with Template:Math by the choice of a reference frame). Furthermore:

• the Pfaffian system Template:Math (for all Template:Mvar) is integrable, and its integral manifolds are the fibres of the principal bundle Template:Math.
• the Pfaffian system Template:Math (for all Template:Math) is also integrable, and its integral manifolds define parallel transport in Template:Math.

Thus the forms Template:Math define a flat principal connection on Template:Math.

For a strict comparison with the motivation, one should actually define parallel transport in a principal Template:Math-bundle over Template:Mvar. This can be done by pulling back Template:Math by the smooth map Template:Math defined by translation. Then the composite Template:Math is a principal Template:Math-bundle over Template:Mvar, and the forms Template:Math pull back to give a flat principal Template:Math-connection on this bundle.

General affine geometries: formal definitions

An affine space, as with essentially any smooth Klein geometry, is a manifold equipped with a flat Cartan connection. More general affine manifolds or affine geometries are obtained easily by dropping the flatness condition expressed by the Maurer-Cartan equations. There are several ways to approach the definition and two will be given. Both definitions are facilitated by the realisation that 1-forms Template:Math in the flat model fit together to give a 1-form with values in the Lie algebra Template:Math of the affine group Template:Math.

In these definitions, Template:Mvar is a smooth Template:Mvar-manifold and Template:Math is an affine space of the same dimension.

Definition via absolute parallelism

Let Template:Mvar be a manifold, and Template:Mvar a principal Template:Math-bundle over Template:Mvar. Then an affine connection is a 1-form Template:Mvar on Template:Mvar with values in Template:Math satisfying the following properties

1. Template:Mvar is equivariant with respect to the action of Template:Math on Template:Mvar and Template:Math;
2. Template:Math for all Template:Mvar in the Lie algebra Template:Math of all Template:Math matrices;
3. Template:Mvar is a linear isomorphism of each tangent space of Template:Mvar with Template:Math.

The last condition means that Template:Mvar is an absolute parallelism on Template:Mvar, i.e., it identifies the tangent bundle of Template:Mvar with a trivial bundle (in this case Template:Math). The pair Template:Math defines the structure of an affine geometry on Template:Mvar, making it into an affine manifold.

The affine Lie algebra Template:Math splits as a semidirect product of Template:Math and Template:Math and so Template:Mvar may be written as a pair Template:Math where Template:Mvar takes values in Template:Math and Template:Mvar takes values in Template:Math. Conditions 1 and 2 are equivalent to Template:Mvar being a principal Template:Math-connection and Template:Mvar being a horizontal equivariant 1-form, which induces a bundle homomorphism from Template:Math to the associated bundle Template:Math. Condition 3 is equivalent to the fact that this bundle homomorphism is an isomorphism. (However, this decomposition is a consequence of the rather special structure of the affine group.) Since Template:Mvar is the frame bundle of Template:Math, it follows that Template:Mvar provides a bundle isomorphism between Template:Mvar and the frame bundle Template:Math of Template:Mvar; this recovers the definition of an affine connection as a principal Template:Math-connection on Template:Math.

The 1-forms arising in the flat model are just the components of Template:Mvar and Template:Mvar.

Definition as a principal affine connection

An affine connection on Template:Mvar is a principal Template:Math-bundle Template:Mvar over Template:Mvar, together with a principal Template:Math-subbundle Template:Mvar of Template:Mvar and a principal Template:Math-connection Template:Mvar (a 1-form on Template:Mvar with values in Template:Math) which satisfies the following (generic) Cartan condition. The Template:Math component of pullback of Template:Mvar to Template:Mvar is a horizontal equivariant 1-form and so defines a bundle homomorphism from Template:Math to Template:Math: this is required to be an isomorphism.

Relation to the motivation

Since Template:Math acts on Template:Mvar, there is, associated to the principal bundle Template:Mvar, a bundle Template:Math, which is a fiber bundle over Template:Mvar whose fiber at Template:Mvar in Template:Mvar is an affine space Template:Math. A section Template:Mvar of Template:Mvar (defining a marked point Template:Mvar in Template:Mvar for each Template:Mvar) determines a principal Template:Math-subbundle Template:Mvar of Template:Mvar (as the bundle of stabilizers of these marked points) and vice versa. The principal connection Template:Mvar defines an Ehresmann connection on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section Template:Mvar always moves under parallel transport.

Further properties

Curvature and torsion

Curvature and torsion are the main invariants of an affine connection. As there are many equivalent ways to define the notion of an affine connection, so there are many different ways to define curvature and torsion.

From the Cartan connection point of view, the curvature is the failure of the affine connection Template:Mvar to satisfy the Maurer–Cartan equation

<math>\mathrm{d}\eta + \tfrac12[\eta\wedge\eta] = 0,</math>

where the second term on the left hand side is the wedge product using the Lie bracket in Template:Math to contract the values. By expanding Template:Mvar into the pair Template:Math and using the structure of the Lie algebra Template:Math, this left hand side can be expanded into the two formulae

<math> \mathrm{d}\theta + \omega\wedge\theta \quad \text{and} \quad \mathrm{d}\omega + \omega\wedge\omega\,,</math>

where the wedge products are evaluated using matrix multiplication. The first expression is called the torsion of the connection, and the second is also called the curvature.

These expressions are differential 2-forms on the total space of a frame bundle. However, they are horizontal and equivariant, and hence define tensorial objects. These can be defined directly from the induced covariant derivative Template:Math on Template:Math as follows.

The torsion is given by the formula

<math>T^\nabla(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y].</math>

If the torsion vanishes, the connection is said to be torsion-free or symmetric.

The curvature is given by the formula

<math>R^\nabla_{X,Y}Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z.</math>

Note that Template:Math is the Lie bracket of vector fields

<math>[X,Y]=\left(X^j \partial_j Y^i - Y^j \partial_j X^i\right)\partial_i</math>

in Einstein notation. This is independent of coordinate system choice and

<math>\partial_i = \left(\frac{\partial}{\partial\xi^i}\right)_p\,,</math>

the tangent vector at point Template:Mvar of the Template:Mvarth coordinate curve. The Template:Math are a natural basis for the tangent space at point Template:Mvar, and the Template:Mvar the corresponding coordinates for the vector field Template:Math.

When both curvature and torsion vanish, the connection defines a pre-Lie algebra structure on the space of global sections of the tangent bundle.

The Levi-Civita connection

If Template:Math is a Riemannian manifold then there is a unique affine connection Template:Math on Template:Mvar with the following two properties:

• the connection is torsion-free, i.e., Template:Math is zero, so that Template:Math;
• parallel transport is an isometry, i.e., the inner products (defined using Template:Mvar) between tangent vectors are preserved.

This connection is called the Levi-Civita connection.

The term "symmetric" is often used instead of torsion-free for the first property. The second condition means that the connection is a metric connection in the sense that the Riemannian metric Template:Mvar is parallel: Template:Math. For a torsion-free connection, the condition is equivalent to the identity Template:Math = Template:Math + Template:Math, "compatibility with the metric".<ref>Template:Harvnb, Vol. I</ref> In local coordinates the components of the form are called Christoffel symbols: because of the uniqueness of the Levi-Civita connection, there is a formula for these components in terms of the components of Template:Mvar.

Geodesics

Since straight lines are a concept in affine geometry, affine connections define a generalized notion of (parametrized) straight lines on any affine manifold, called affine geodesics. Abstractly, a parametric curve Template:Math is a straight line if its tangent vector remains parallel and equipollent with itself when it is transported along Template:Mvar. From the linear point of view, an affine connection Template:Mvar distinguishes the affine geodesics in the following way: a smooth curve Template:Math is an affine geodesic if <math>\dot\gamma</math> is parallel transported along Template:Mvar, that is

<math>\tau_t^s\dot\gamma(s) = \dot\gamma(t)</math>

where Template:Math is the parallel transport map defining the connection.

In terms of the infinitesimal connection Template:Math, the derivative of this equation implies

<math>\nabla_{\dot\gamma(t)}\dot\gamma(t) = 0</math>

for all Template:Math.

Conversely, any solution of this differential equation yields a curve whose tangent vector is parallel transported along the curve. For every Template:Math and every Template:Math, there exists a unique affine geodesic Template:Math with Template:Math and Template:Math and where Template:Mvar is the maximal open interval in Template:Math, containing 0, on which the geodesic is defined. This follows from the Picard–Lindelöf theorem, and allows for the definition of an exponential map associated to the affine connection.

In particular, when Template:Mvar is a (pseudo-)Riemannian manifold and Template:Math is the Levi-Civita connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

The geodesics defined here are sometimes called affinely parametrized, since a given straight line in Template:Mvar determines a parametric curve Template:Mvar through the line up to a choice of affine reparametrization Template:Math, where Template:Mvar and Template:Mvar are constants. The tangent vector to an affine geodesic is parallel and equipollent along itself. An unparametrized geodesic, or one which is merely parallel along itself without necessarily being equipollent, need only satisfy

<math>\nabla_{\dot{\gamma}}\dot{\gamma} = k\dot{\gamma}</math>

for some function Template:Mvar defined along Template:Mvar. Unparametrized geodesics are often studied from the point of view of projective connections.

Development

An affine connection defines a notion of development of curves. Intuitively, development captures the notion that if Template:Mvar is a curve in Template:Mvar, then the affine tangent space at Template:Math may be rolled along the curve. As it does so, the marked point of contact between the tangent space and the manifold traces out a curve Template:Mvar in this affine space: the development of Template:Mvar.

In formal terms, let Template:Math be the linear parallel transport map associated to the affine connection. Then the development Template:Mvar is the curve in Template:Math starts off at 0 and is parallel to the tangent of Template:Mvar for all time Template:Mvar:

<math>\dot{C}_t = \tau_t^0\dot{x}_t\,,\quad C_0 = 0.</math>

In particular, Template:Mvar is a geodesic if and only if its development is an affinely parametrized straight line in Template:Math.<ref>This treatment of development is from Template:Harvtxt; see section III.3 for a more geometrical treatment. See also Template:Harvtxt for a thorough discussion of development in other geometrical situations.</ref>

Surface theory revisited

If Template:Mvar is a surface in Template:Math, it is easy to see that Template:Mvar has a natural 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 Template:Mvar to Template:Math, and then projecting the result orthogonally back onto the tangent spaces of Template:Mvar. It is easy to see that this affine connection is torsion-free. Furthermore, it is a metric connection with respect to the Riemannian metric on Template:Mvar induced by the inner product on Template:Math, hence it is the Levi-Civita connection of this metric.

Example: the unit sphere in Euclidean space

Let Template:Math be the usual scalar product on Template:Math, and let Template:Math be the unit sphere. The tangent space to Template:Math at a point Template:Mvar is naturally identified with the vector subspace of Template:Math consisting of all vectors orthogonal to Template:Mvar. It follows that a vector field Template:Mvar on Template:Math can be seen as a map Template:Math which satisfies

<math>\langle Y_x, x\rangle = 0\,, \quad \forall x\in \mathbf{S}^2.</math>

Denote as Template:Math the differential (Jacobian matrix) of such a map. Then we have:

Lemma. The formula

<math>(\nabla_Z Y)_x = \mathrm{d}Y_x(Z_x) + \langle Z_x,Y_x\rangle x</math>

defines an affine connection on Template:Math with vanishing torsion.

Proof. It is straightforward to prove that Template:Math satisfies the Leibniz identity and is Template:Math linear in the first variable. So all that needs to be proved here is that the map above does indeed define a tangent vector field. That is, we need to prove that for all Template:Mvar in Template:Math

<math>\bigl\langle(\nabla_Z Y)_x,x\bigr\rangle = 0\,.\qquad \text{(Eq.1)}</math>

Consider the map

<math>\begin{align} f: \mathbf{S}^2&\to \mathbf{R}\\ x &\mapsto \langle Y_x, x\rangle\,.\end{align}</math>

The map f is constant, hence its differential vanishes. In particular

<math>\mathrm{d}f_x(Z_x) = \bigl\langle (\mathrm{d} Y)_x(Z_x),x(\gamma'(t))\bigr\rangle + \langle Y_x, Z_x\rangle = 0\,.</math>

Equation 1 above follows. Q.E.D.

See also

• Atlas (topology)
• Connection (mathematics)
• Connection (fibred manifold)
• Connection (affine bundle)
• Differentiable manifold
• Differential geometry
• Introduction to the mathematics of general relativity
• Levi-Civita connection
• List of formulas in Riemannian geometry
• Riemannian geometry

References

• Template:Cite book
• Template:Lee Riemannian Manifolds An Introduction to Curvature
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Bibliography

Primary historical references

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Cartan's treatment of affine connections as motivated by the study of relativity theory. Includes a detailed discussion of the physics of reference frames, and how the connection reflects the physical notion of transport along a worldline.

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A more mathematically motivated account of affine connections.

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Affine connections from the point of view of Riemannian geometry. Robert Hermann's appendices discuss the motivation from surface theory, as well as the notion of affine connections in the modern sense of Koszul. He develops the basic properties of the differential operator ∇, and relates them to the classical affine connections in the sense of Cartan.

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Secondary references

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This is the main reference for the technical details of the article. Volume 1, chapter III gives a detailed account of affine connections from the perspective of principal bundles on a manifold, parallel transport, development, geodesics, and associated differential operators. Volume 1 chapter VI gives an account of affine transformations, torsion, and the general theory of affine geodesy. Volume 2 gives a number of applications of affine connections to homogeneous spaces and complex manifolds, as well as to other assorted topics.

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Two articles by Lumiste, giving precise conditions on parallel transport maps in order that they define affine connections. They also treat curvature, torsion, and other standard topics from a classical (non-principal bundle) perspective.

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This fills in some of the historical details, and provides a more reader-friendly elementary account of Cartan connections in general. Appendix A elucidates the relationship between the principal connection and absolute parallelism viewpoints. Appendix B bridges the gap between the classical "rolling" model of affine connections, and the modern one based on principal bundles and differential operators.