Английская Википедия:Affine connection

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Файл: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 connectionШаблон: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.Шаблон: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 CartanШаблон:Efn and has its origins in the identification of tangent spaces in Euclidean space Шаблон: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 Шаблон: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 Шаблон:Mvar can be identified naturally (by translation) with the tangent space at a nearby point Шаблон: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

Шаблон:See also

Consider a smooth surface Шаблон:Mvar in a 3-dimensional Euclidean space. Near any point, Шаблон: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 Шаблон:Mvar can be rolled on Шаблон:Mvar: this should be easy to imagine when Шаблон:Mvar is a surface like the 2-sphere, which is the smooth boundary of a convex region. As the tangent plane is rolled on Шаблон:Mvar, the point of contact traces out a curve on Шаблон:Mvar. Conversely, given a curve on Шаблон: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 Шаблон:Mvar-dimensional affine space is a Klein geometry for the affine group Шаблон:Math, the stabilizer of a point being the general linear group Шаблон:Math. An affine Шаблон:Mvar-manifold is then a manifold which looks infinitesimally like Шаблон:Mvar-dimensional affine space.

Motivation from tensor calculus

Шаблон:See also

Файл: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 Шаблон:Mvar sends the one-form Шаблон:Math to 0.07 everywhere. The red vector field Шаблон:Mvar sends the one-form Шаблон:Math to Шаблон:Math everywhere. Endorsed by the metric Шаблон:Math, the Levi-Civita connection Шаблон:Math is 0 everywhere, indicating Шаблон:Mvar has no change along Шаблон:Mvar. In other words, Шаблон:Mvar parallel transports along each concentric circle. Шаблон:Math everywhere, which sends Шаблон:Math to 0.5 everywhere, implying Шаблон: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.Шаблон: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,[1] who developed a detailed mathematical foundation for general relativity, and Élie Cartan,[2] 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.Шаблон:Efn The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean Шаблон: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 Шаблон:Math or as a principal Шаблон:Math connection on the frame bundle.

Formal definition as a differential operator

Шаблон:See also

Let Шаблон:Mvar be a smooth manifold and let Шаблон:Math be the space of vector fields on Шаблон:Mvar, that is, the space of smooth sections of the tangent bundle Шаблон:Math. Then an affine connection on Шаблон: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 Шаблон:Mvar in the set of smooth functions on Шаблон:Math, written Шаблон:Math, and all vector fields Шаблон:Math on Шаблон:Mvar:

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

Elementary properties

Parallel transport for affine connections

Шаблон:See also

Файл: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 Шаблон:Mvar be a manifold with an affine connection Шаблон:Math. Then a vector field Шаблон:Mvar is said to be parallel if Шаблон:Math in the sense that for any vector field Шаблон:Mvar, Шаблон: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 Шаблон:Mvar and Шаблон:Mvar, an identification between a tangent vector at Шаблон:Mvar and one at Шаблон: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 Шаблон:Math is a partial differential equation which is overdetermined: the integrability condition for this equation is the vanishing of the curvature of Шаблон:Math (see below). However, if this equation is restricted to a curve from Шаблон:Mvar to Шаблон:Mvar it becomes an ordinary differential equation. There is then a unique solution for any initial value of Шаблон:Mvar at Шаблон:Mvar.

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

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

Formally, the first condition means that Шаблон:Mvar is parallel with respect to the pullback connection on the pullback bundle Шаблон: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 Шаблон:Mvar, which can only happen if the curvature of Шаблон: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 Шаблон:Math along a curve. In other words, the affine connection provides a lift of any curve Шаблон:Mvar in Шаблон:Mvar to a curve Шаблон:Mvar in Шаблон:Math.

Formal definition on the frame bundle

Шаблон:See also

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

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

Such a connection Шаблон:Mvar immediately defines a covariant derivative not only on the tangent bundle, but on vector bundles associated to any group representation of Шаблон: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 Шаблон: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 Шаблон:Math which is horizontal in the sense that it vanishes on vertical vectors such as the point values of the vector fields Шаблон:Mvar: Indeed Шаблон:Mvar is defined first by projecting a tangent vector (to Шаблон:Math at a frame Шаблон:Mvar) to Шаблон:Mvar, then by taking the components of this tangent vector on Шаблон:Mvar with respect to the frame Шаблон:Mvar. Note that Шаблон:Mvar is also Шаблон:Math-equivariant (where Шаблон:Math acts on Шаблон:Math by matrix multiplication).

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

Affine connections as Cartan connections

Шаблон:See also

Affine connections can be defined within Cartan's general framework.[3] 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 Шаблон:Math-valued one-form Шаблон: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;Шаблон: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,Шаблон:Efn whereas the planes, as affine subspaces of Шаблон: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 Шаблон:Mvar-manifold Шаблон:Mvar with an affine space Шаблон:Math, of dimension Шаблон:Mvar, attached to each Шаблон:Math at a marked point Шаблон:Math, together with a method for transporting elements of these affine spaces along any curve Шаблон:Mvar in Шаблон:Mvar. This method is required to satisfy several properties:

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

These last two points are quite hard to make precise,[4] 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 Шаблон:Math, where Шаблон:MathШаблон:Efn and the Шаблон:Math form a basis of Шаблон: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 Шаблон:Math. Geometrically, an affine frame undergoes a displacement travelling along a curve Шаблон:Mvar from Шаблон:Math to Шаблон: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 Шаблон:Math are required to be tangent to Шаблон:Mvar in the informal sense that the displacement of Шаблон:Math along Шаблон:Mvar can be identified (approximately or infinitesimally) with the tangent vector Шаблон:Math to Шаблон:Mvar at Шаблон:Math (which is the infinitesimal displacement of Шаблон:Mvar). Since

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

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

The tangential affine space Шаблон:Math is thus identified intuitively with an infinitesimal affine neighborhood of Шаблон: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,[5] 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 Шаблон:Mvar may be added to a point Шаблон:Mvar by placing the initial point of the vector at Шаблон:Mvar and then transporting Шаблон:Mvar to the terminal point. The operation thus described Шаблон:Math is the translation of Шаблон:Mvar along Шаблон:Mvar. In technical terms, affine Шаблон:Mvar-space is a set Шаблон:Math equipped with a free transitive action of the vector group Шаблон:Math on it through this operation of translation of points: Шаблон:Math is thus a principal homogeneous space for the vector group Шаблон:Math.

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

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

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

Affine frames and the flat affine connection

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

On Шаблон:Math there is a collection of Шаблон: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 Шаблон:Mvar, these are all functions with values in Шаблон:Math, so it is possible to take their exterior derivatives to obtain differential 1-forms with values in Шаблон:Math. Since the functions Шаблон:Mvar yield a basis for Шаблон:Math at each point of Шаблон: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 Шаблон:Math of real-valued one-forms on Шаблон:Math. This system of one-forms on the principal bundle Шаблон:Math defines the affine connection on Шаблон:Mvar.

Taking the exterior derivative a second time, and using the fact that Шаблон:Math as well as the linear independence of the Шаблон: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 Шаблон:Math (identified with Шаблон:Math by the choice of a reference frame). Furthermore:

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

For a strict comparison with the motivation, one should actually define parallel transport in a principal Шаблон:Math-bundle over Шаблон:Mvar. This can be done by pulling back Шаблон:Math by the smooth map Шаблон:Math defined by translation. Then the composite Шаблон:Math is a principal Шаблон:Math-bundle over Шаблон:Mvar, and the forms Шаблон:Math pull back to give a flat principal Шаблон: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 Шаблон:Math in the flat model fit together to give a 1-form with values in the Lie algebra Шаблон:Math of the affine group Шаблон:Math.

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

Definition via absolute parallelism

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

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

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

The affine Lie algebra Шаблон:Math splits as a semidirect product of Шаблон:Math and Шаблон:Math and so Шаблон:Mvar may be written as a pair Шаблон:Math where Шаблон:Mvar takes values in Шаблон:Math and Шаблон:Mvar takes values in Шаблон:Math. Conditions 1 and 2 are equivalent to Шаблон:Mvar being a principal Шаблон:Math-connection and Шаблон:Mvar being a horizontal equivariant 1-form, which induces a bundle homomorphism from Шаблон:Math to the associated bundle Шаблон: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 Шаблон:Mvar is the frame bundle of Шаблон:Math, it follows that Шаблон:Mvar provides a bundle isomorphism between Шаблон:Mvar and the frame bundle Шаблон:Math of Шаблон:Mvar; this recovers the definition of an affine connection as a principal Шаблон:Math-connection on Шаблон:Math.

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

Definition as a principal affine connection

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

Relation to the motivation

Since Шаблон:Math acts on Шаблон:Mvar, there is, associated to the principal bundle Шаблон:Mvar, a bundle Шаблон:Math, which is a fiber bundle over Шаблон:Mvar whose fiber at Шаблон:Mvar in Шаблон:Mvar is an affine space Шаблон:Math. A section Шаблон:Mvar of Шаблон:Mvar (defining a marked point Шаблон:Mvar in Шаблон:Mvar for each Шаблон:Mvar) determines a principal Шаблон:Math-subbundle Шаблон:Mvar of Шаблон:Mvar (as the bundle of stabilizers of these marked points) and vice versa. The principal connection Шаблон:Mvar defines an Ehresmann connection on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section Шаблон: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 Шаблон: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 Шаблон:Math to contract the values. By expanding Шаблон:Mvar into the pair Шаблон:Math and using the structure of the Lie algebra Шаблон: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 Шаблон:Math on Шаблон: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 Шаблон: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 Шаблон:Mvar of the Шаблон:Mvarth coordinate curve. The Шаблон:Math are a natural basis for the tangent space at point Шаблон:Mvar, and the Шаблон:Mvar the corresponding coordinates for the vector field Шаблон: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 Шаблон:Math is a Riemannian manifold then there is a unique affine connection Шаблон:Math on Шаблон:Mvar with the following two properties:

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 Шаблон:Mvar is parallel: Шаблон:Math. For a torsion-free connection, the condition is equivalent to the identity Шаблон:Math = Шаблон:Math + Шаблон:Math, "compatibility with the metric".[6] 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 Шаблон: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 Шаблон:Math is a straight line if its tangent vector remains parallel and equipollent with itself when it is transported along Шаблон:Mvar. From the linear point of view, an affine connection Шаблон:Mvar distinguishes the affine geodesics in the following way: a smooth curve Шаблон:Math is an affine geodesic if <math>\dot\gamma</math> is parallel transported along Шаблон:Mvar, that is

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

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

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

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

for all Шаблон:Math.

Conversely, any solution of this differential equation yields a curve whose tangent vector is parallel transported along the curve. For every Шаблон:Math and every Шаблон:Math, there exists a unique affine geodesic Шаблон:Math with Шаблон:Math and Шаблон:Math and where Шаблон:Mvar is the maximal open interval in Шаблон: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 Шаблон:Mvar is a (pseudo-)Riemannian manifold and Шаблон: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 Шаблон:Mvar determines a parametric curve Шаблон:Mvar through the line up to a choice of affine reparametrization Шаблон:Math, where Шаблон:Mvar and Шаблон: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 Шаблон:Mvar defined along Шаблон: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 Шаблон:Mvar is a curve in Шаблон:Mvar, then the affine tangent space at Шаблон: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 Шаблон:Mvar in this affine space: the development of Шаблон:Mvar.

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

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

In particular, Шаблон:Mvar is a geodesic if and only if its development is an affinely parametrized straight line in Шаблон:Math.[7]

Surface theory revisited

If Шаблон:Mvar is a surface in Шаблон:Math, it is easy to see that Шаблон: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 Шаблон:Mvar to Шаблон:Math, and then projecting the result orthogonally back onto the tangent spaces of Шаблон: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 Шаблон:Mvar induced by the inner product on Шаблон:Math, hence it is the Levi-Civita connection of this metric.

Example: the unit sphere in Euclidean space

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

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

Denote as Шаблон: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 Шаблон:Math with vanishing torsion.
Proof. It is straightforward to prove that Шаблон:Math satisfies the Leibniz identity and is Шаблон: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 Шаблон:Mvar in Шаблон: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

Notes

Шаблон:Notelist


Citations

Шаблон:Reflist

References

Шаблон:Refbegin

Шаблон:Refend

Bibliography

Primary historical references

Шаблон:Refbegin

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.
A more mathematically motivated account of affine connections.
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.

Шаблон:Refend

Secondary references

Шаблон:Refbegin

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.
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.
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.

Шаблон:Refend

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  1. ↑ Шаблон:Harvnb, 5 editions to 1922.
  2. ↑ Шаблон:Harvnb.
  3. ↑ Шаблон:Harvnb.
  4. ↑ For details, see Шаблон:Harvtxt. The following intuitive treatment is that of Шаблон:Harvtxt and Шаблон:Harvtxt.
  5. ↑ Cf. R. Hermann (1983), Appendix 1–3 to Шаблон:Harvtxt, and also Шаблон:Harvtxt.
  6. ↑ Шаблон:Harvnb, Vol. I
  7. ↑ This treatment of development is from Шаблон:Harvtxt; see section III.3 for a more geometrical treatment. See also Шаблон:Harvtxt for a thorough discussion of development in other geometrical situations.
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