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 the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19thcentury 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 Cartan and has its origins in the identification of tangent spaces in Euclidean space R^{n} 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 Riemannian metric then there is a natural choice of affine connection, called the LeviCivita 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.
Contents

Motivation and history 1

Motivation from surface theory 1.1

Motivation from tensor calculus 1.2

Approaches 1.3

Formal definition as a differential operator 2

Elementary properties 2.1

Parallel transport for affine connections 3

Formal definition on the frame bundle 4

Affine connections as Cartan connections 5

Explanations and historical intuition 5.1

Affine space as the flat model geometry 5.2

Definition of an affine space 5.2.1

Affine frames and the flat affine connection 5.2.2

General affine geometries: formal definitions 5.3

Definition via absolute parallelism 5.3.1

Definition as a principal affine connection 5.3.2

Relation to the motivation 5.3.3

Further properties 6

Curvature and torsion 6.1

The LeviCivita connection 6.2

Geodesics 6.3

Development 6.4

Surface theory revisited 7

Example: the unit sphere in Euclidean space 7.1

See also 8

Notes 9

References 10

Primary historical references 10.1

Secondary references 10.2
Motivation and history
A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space R^{n}: 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 p can be identified naturally (by translation) with the tangent space at a nearby point q. 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 welldefined 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 S in 3dimensional Euclidean space. Near to any point, S 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 S can be rolled on S: this should be easy to imagine when S is a surface like the 2sphere, which is the smooth boundary of a convex region. As the tangent plane is rolled on S, the point of contact traces out a curve on S. Conversely, given a curve on S, 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 3space, 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 ndimensional affine space is a Klein geometry for the affine group Aff(n), the stabilizer of a point being the general linear group GL(n). An affine nmanifold is then a manifold which looks infinitesimally like ndimensional affine space.
Motivation from tensor calculus
Historically, people used covariant derivative (or LeviCivita connection given by metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2dimensional Euclidean space, the blue vector field
X sends the
1form d
r to 1 everywhere. The red vector field
Y sends the 1form dθ to
r everywhere. Endorsed by the metric d
s^{2} = d
r^{2} + dθ
^{2}, the LeviCivita connection ∇
_{Y}X is 0 everywhere, indicating
X has no change along
Y. In other words,
X parallel transports along each
concentric circle. ∇
_{X}Y sends dθ to 1 everywhere, implying
Y 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 coordinateindependent methods, it was necessary to work with vector fields using their components in coordinate charts. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates. 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 the absolute differential calculus (now known as tensor calculus) by Gregorio RicciCurbastro and his student Tullio LeviCivita between 1880 and the turn of the 20th century.
The 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, LeviCivita formalized the unique connection associated to a Riemannian metric, now known as the LeviCivita 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 JeanLouis 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.^{[3]} The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean nspace 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 Aff(n) or as a principal GL(n) connection on the frame bundle.
Formal definition as a differential operator
Let M be a smooth manifold and let C^{∞}(M,TM) be the space of vector fields on M, that is, the space of smooth sections of the tangent bundle TM. Then an affine connection on M is a bilinear map

\begin{matrix} C^\infty(M,TM)\times C^\infty(M,TM) & \rightarrow & C^\infty(M,TM)\\ (X,Y) & \mapsto & \nabla_X Y, \end{matrix}
such that for all smooth functions f in C^{∞}(M,R) and all vector fields X, Y on M:

\nabla_{fX}Y = f\nabla_X Y, that is, ∇ is C^{∞}(M,R)linear in the first variable;

\nabla_X (fY) = \mathrm df(X)Y + f\nabla_XY, that is, ∇ satisfies Leibniz rule in the second variable.
Elementary properties

It follows from the property (1) above that the value of ∇_{X}Y at a point x ∈ M depends only on the value of X at x and not on the value of X on M−{x}. It also follows from property (2) above that the value of ∇_{X}Y at a point x ∈ M depends only on the value of Y on a neighbourhood of x.

If ∇^{1}, ∇^{2} are affine connections then the value at x of ∇^{1}_{X}Y − ∇^{2}_{X}Y may be written Γ_{x}(X_{x},Y_{x}) where


Γ_{x}: T_{x}M × T_{x}M → T_{x}M

is bilinear and depends smoothly on x (i.e., it defines a smooth bundle homomorphism). Conversely if ∇ is an affine connection and Γ is such a smooth bilinear bundle homomorphism (called a connection form on M) then ∇+Γ is an affine connection.

If M is an open subset of R^{n}, then the tangent bundle of M is the trivial bundle M×R^{n}. In this situation there is a canonical affine connection d on M: any vector field Y is given by a smooth function V from M to R^{n}; then d_{X}Y is the vector field corresponding to the smooth function dV(X)=∂_{X}Y from M to R^{n}. Any other affine connection ∇ on M may therefore be written ∇ = d + Γ, where Γ is a connection form on M.

More generally, a local trivialization of the tangent bundle is a bundle isomorphism between the restriction of TM to an open subset U of M, and U × R^{n}. The restriction of an affine connection ∇ to U may then be written in the form d + Γ where Γ is a connection form on U.
Parallel transport for affine connections
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 welldefined 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 M be a manifold with an affine connection ∇. Then a vector field X is said to be parallel if ∇X = 0 in the sense that for any vector field Y, ∇_{Y}X=0. 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 x and y, an identification between a tangent vector at x and one at y is obtained. Such tangent vectors are said to be parallel transports of each other.
Unfortunately, nonzero parallel vector fields do not, in general, exist, because the equation ∇X = 0 is a partial differential equation which is overdetermined: the integrability condition for this equation is the vanishing of the curvature of ∇ (see below). However, if this equation is restricted to a curve from x to y it becomes an ordinary differential equation. There is then a unique solution for any initial value of X at x.
More precisely, if γ : I → M a smooth curve parametrized by an interval [a,b] and ξ ∈ T_{x}M, where x=γ(a), then a vector field X along γ (and in particular, the value of this vector field at y=γ(b)) is called the parallel transport of ξ along γ if

\nabla_{\dot\gamma(t)}X=0, for all t ∈ [a,b]

X_{\gamma(a)} = \xi.
Formally, the first condition means that X is parallel with respect to the pullback connection on the pullback bundle γ*TM. However, in a local trivialization it is a firstorder 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 M, which can only happen if the curvature of ∇ 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 GL(M) along a curve. In other words, the affine connection provides a lift of any curve γ in M to a curve \tilde\gamma in GL(M).
Formal definition on the frame bundle
An affine connection may also be defined as a principal GL(n) connection ω on the frame bundle FM or GL(M) of a manifold M. In more detail, ω is a smooth map from the tangent bundle T(FM) of the frame bundle to the space of n×n matrices (which is the Lie algebra gl(n) of the Lie group GL(n) of invertible n×n matrices) satisfying two properties:

ω is equivariant with respect to the action of GL(n) on T(FM) and gl(n);

ω(X_{ξ}) = ξ for any ξ in gl(n), where X_{ξ} is the vector field on FM corresponding to ξ.
Such a connection ω immediately defines a covariant derivative not only on the tangent bundle, but on vector bundles associated to any group representation of GL(n), 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 ω 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 θ : T(FM) → R^{n} which is horizontal in the sense that it vanishes on vertical vectors such as the point values of the vector fields X_{ξ}: indeed θ is defined first by projecting a tangent vector (to FM at a frame f) to M, then by taking the components of this tangent vector on M with respect to the frame f. Note that θ is also GL(n)equivariant (where GL(n) acts on R^{n} by matrix multiplication).
The pair (θ,ω) define a bundle isomorphism of T(FM) with the trivial bundle FM × aff(n), where aff(n) is the cartesian product of R^{n} and gl(n) (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.^{[4]} 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 aff(n)valued 1form (θ,ω): T(FM) → aff(n) 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;^{[5]}

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,^{[6]} whereas the planes, as affine subspaces of R^{3}, 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 nmanifold M with an affine space A_{x}, of dimension n, attached to each x ∈ M at a marked point a_{x} ∈ A_{x}, together with a method for transporting elements of these affine spaces along any curve C in M. This method is required to satisfy several properties:

for any two points x, y on C, parallel transport is an affine transformation from A_{x} to A_{y};

parallel transport is defined infinitesimally in the sense that it is differentiable at any point on C and depends only on the tangent vector to C at that point;

the derivative of the parallel transport at x determines a linear isomorphism from T_{x}M to T_{a_x}A_x.
These last two points are quite hard to make precise,^{[7]} 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 (p, e_{1}, ..., e_{n}), where p ∈ A_{x}^{[8]} and the e_{i} form a basis of T_{p}(A_{x}). The affine connection is then given symbolically by a first order differential system

(*) \left\{ \begin{matrix} \mathrm d{p} &= \theta^1{\bold e}_1 + \cdots + \theta^n{\bold e}_n & \\ \mathrm d{\bold e}_i &= \omega^1_i{\bold e}_1 + \cdots + \omega^n_i{\bold e}_n, & \quad i=1,2,\ldots,n \end{matrix} \right.
defined by a collection of oneforms (θ^{j}, ω_{i}^{j}). Geometrically, an affine frame undergoes a displacement travelling along a curve γ from γ(t) to γ(t + δt) given (approximately, or infinitesimally) by

\left. \begin{matrix} p(\gamma(t+\delta t))  p(\gamma(t)) &= \bigl(\theta^1(\gamma'(t)){\bold e}_1 + \cdots + \theta^n(\gamma'(t)){\bold e}_n\bigr)\mathrm \delta t & \\ {\bold e}_i(\gamma(t+\delta t))  {\bold e}_i(\gamma(t)) &= \bigl(\omega^1_i(\gamma'(t)){\bold e}_1 + \cdots + \omega^n_i(\gamma'(t)){\bold e}_n\bigr)\delta t. \end{matrix} \right.
Furthermore, the affine spaces A_{x} are required to be tangent to M in the informal sense that the displacement of a_{x} along γ can be identified (approximately or infinitesimally) with the tangent vector γ'(t) to γ at x=γ(t) (which is the infinitesimal displacement of x). Since

a_{x}(γ(t+δt))  a_{x}(γ(t)) = θ(γ'(t))δt,
where θ is defined by θ(X) = θ^{1}(X)e_{1} + ... + θ^{n}(X)e_{n}, this identification is given by θ, so the requirement is that θ should be a linear isomorphism at each point.
The tangential affine space A_{x} is thus identified intuitively with an infinitesimal affine neighborhood of x.
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,^{[9]} 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 v may be added to a point p by placing the initial point of the vector at p and then transporting p to the terminal point. The operation thus described p → p+v is the translation of p along v. In technical terms, affine nspace is a set A^{n} equipped with a free transitive action of the vector group R^{n} on it through this operation of translation of points: A^{n} is thus a principal homogeneous space for the vector group R^{n}. The general linear group GL(n) is the group of transformations of R^{n} which preserve the linear structure of R^{n} in the sense that T(av+bw) = aT(v) + bT(w). By analogy, the affine group Aff(n) is the group of transformations of A^{n} preserving the affine structure. Thus φ ∈ Aff(n) must preserve translations in the sense that

\phi(p+v)=\phi(p)+T(v)\,
where T is a general linear transformation. The map sending φ ∈ Aff(n) to T ∈ GL(n) is a group homomorphism. Its kernel is the group of translations R^{n}. The stabilizer of any point p in A can thus be identified with GL(n) using this projection: this realises the affine group as a semidirect product of GL(n) and R^{n}, and affine space as the homogeneous space Aff(n)/GL(n).
Affine frames and the flat affine connection
An affine frame for A consists of a point p ∈ A and a basis (e_{1},...,e_{n}) of the vector space T_{p}A = R^{n}. The general linear group GL(n) acts freely on the set FA of all affine frames by fixing p and transforming the basis (e_{1},...,e_{n}) in the usual way, and the map π sending an affine frame (p;e_{1},...,e_{n}) to p is the quotient map. Thus FA is a principal GL(n)bundle over A. The action of GL(n) extends naturally to a free transitive action of the affine group Aff(n) on FA, so that FA is an Aff(n)torsor, and the choice of a reference frame identifies FA → A with the principal bundle Aff(n) → Aff(n)/GL(n).
On FA there is a collection of n+1 functions defined by

\pi(p;\mathbf{e}_1, \dots ,\mathbf{e}_n) = p (as before)

\epsilon_i(p;\mathbf{e}_1,\dots , \mathbf{e}_n) = \mathbf{e}_i.
After choosing a basepoint for A, these are all functions with values in R^{n}, so it is possible to take their exterior derivatives to obtain differential 1forms with values in R^{n}. Since the functions ε_{i} yield a basis for R^{n} at each point of FA, these 1forms must be expressible as sums of the form

\begin{matrix} \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{matrix}
for some collection (θ^{i},ω_{j}^{k})_{1≤i,j,k≤n} of realvalued oneforms on Aff(n). This system of oneforms on the principal bundle FA → A defines the affine connection on A.
Taking the exterior derivative a second time, and using the fact that d^{2}=0 as well as the linear independence of the ε_{i}, the following relations are obtained:

\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}
These are the MaurerCartan equations for the Lie group Aff(n) (identified with FA by the choice of a reference frame). Furthermore:

the Pfaffian system θ^{j}=0 (for all j) is integrable, and its integral manifolds are the fibres of the principal bundle Aff(n) → A.

the Pfaffian system ω_{i}^{j}=0 (for all i, j) is also integrable, and its integral manifolds define parallel transport in FA.
Thus the forms (ω_{i}^{j}) define a flat principal connection on FA → A.
For a strict comparison with the motivation, one should actually define parallel transport in a principal Aff(n)bundle over A. This can be done by pulling back FA by the smooth map φ : R^{n} × A → A defined by translation. Then the composite φ'*FA → FA → A is a principal Aff(n)bundle over A, and the forms (θ^{i},ω_{j}^{k}) pull back to give a flat principal Aff(n)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 MaurerCartan equations. There are several ways to approach the definition and two will be given. Both definitions are facilitated by the realisation that 1forms (θ^{i},ω_{j}^{k}) in the flat model fit together to give a 1form with values in the Lie algebra aff(n) of the affine group Aff(n).
In these definitions, M is a smooth nmanifold and A = Aff(n)/GL(n) is an affine space of the same dimension.
Definition via absolute parallelism
Let M be a manifold, and P a principal GL(n)bundle over M. Then an affine connection is a 1form η on P with values in aff(n) satisfying the following properties

η is equivariant with respect to the action of GL(n) on P and aff(n);

η(X_{ξ}) = ξ for all ξ in the Lie algebra gl(n) of all n×n matrices;

η is a linear isomorphism of each tangent space of P with aff(n).
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 pair (P,η) defines the structure of an affine geometry on M, making it into an affine manifold.
The affine Lie algebra aff(n) splits as a semidirect product of R^{n} and gl(n) and so η may be written as a pair (θ,ω) where θ takes values in R^{n} and ω takes values in gl(n). The conditions (1) and (2) are equivalent to ω being a principal GL(n)connection and θ being a horizontal equivariant 1form, which induces a bundle homomorphism from TM to the associated bundle P ×_{GL(n)} R^{n}. The 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 P is the frame bundle of P ×_{GL(n)} R^{n}, it follows that θ provides a bundle isomorphism between P and the frame bundle FM of M; this recovers the definition of an affine connection as a principal GL(n)connection on FM.
The 1forms arising in the flat model are just the components of θ and ω.
Definition as a principal affine connection
An affine connection on M is a principal Aff(n)bundle Q over M, together with a principal GL(n)subbundle P of Q and a principal Aff(n)connection α (a 1form on Q with values in aff(n)) which satisfies the following (generic) Cartan condition. The R^{n} component of pullback of α to P is a horizontal equivariant 1form and so defines a bundle homomorphism from TM to P ×_{GL(n)} R^{n}: this is required to be an isomorphism.
Relation to the motivation
Since Aff(n) acts on A, there is, associated to the principal bundle Q, a bundle A = Q ×_{Aff(n)} A, which is a fiber bundle over M whose fiber at x in M is an affine space A_{x}. A section a of A (defining a marked point a_{x} in A_{x} for each x ∈ M) determines a principal GL(n)subbundle P of Q (as the bundle of stabilizers of these marked points) and vice versa. The principal connection α defines an Ehresmann connection on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section a 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 η to satisfy the MaurerCartan equation

\mathrm d\eta + \tfrac12[\eta\wedge\eta] = 0,
where the second term on the left hand side is the wedge product using the Lie bracket in aff(n) to contract the values. By expanding η into the pair (θ,ω) and using the structure of the Lie algebra aff(n), this left hand side can be expanded into the two formulae

\mathrm d\theta + \omega\wedge\theta

\mathrm d\omega + \omega\wedge\omega,
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 2forms 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 ∇ on TM as follows.
The torsion is given by the formula

T^\nabla(X,Y) = \nabla_X Y  \nabla_Y X  [X,Y].
If the torsion vanishes, the connection is said to be torsionfree or symmetric.
The curvature is given by the formula

R^\nabla_{X,Y}Z = \nabla_X\nabla_Y Z  \nabla_Y\nabla_X Z  \nabla_Z.
Note that [X,Y] is the Lie bracket of vector fields

[X,Y]=(X^{j}\partial_{j}Y^{i}Y^{j}\partial_{j}X^{i})\partial_{i}
in Einstein notation. This is independent of coordinate system choice and

\partial_{i}=\left(\frac{\partial}{\partial\xi^{i}}\right)_{p},
the tangent vector at point p of the ith coordinate curve. The \partial_{i} are a natural basis for the tangent space at point p, and the X^{i} the corresponding coordinates for the vector field X=X^{i}\partial_{i}.
When both curvature and torsion vanish, the connection defines a preLie algebra structure on the space of global sections of the tangent bundle.
The LeviCivita connection
If (M,g) is a Riemannian manifold then there is a unique affine connection ∇ on M with the following two properties:

the connection is torsionfree, i.e., T^{∇} is zero;

parallel transport is an isometry, i.e., the inner products (defined using g) between tangent vectors are preserved.
This connection is called the LeviCivita connection.
The second condition means that the connection is a metric connection in the sense that the Riemannian metric g is parallel: ∇g = 0. In local coordinates the components of the connection form are called Christoffel symbols: because of the uniqueness of the LeviCivita connection, there is a formula for these components in terms of the components of g.
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 γ : I → M is a straight line if its tangent vector remains parallel and equipollent with itself when it is transported along γ. From the linear point of view, an affine connection M distinguishes the affine geodesics in the following way: a smooth curve γ : I → M is an affine geodesic if \dot\gamma is parallel transported along γ, that is

\tau_t^s\dot\gamma(s) = \dot\gamma(t)
where τ_{t}^{s} : T_{γs}M → T_{γt}M is the parallel transport map defining the connection.
In terms of the infinitesimal connection ∇, the derivative of this equation implies

\nabla_{\dot\gamma(t)}\dot\gamma(t) = 0 for all t ∈ I.
Conversely, any solution of this differential equation yields a curve whose tangent vector is parallel transported along the curve. For every x ∈ M and every X ∈ T_{x}M, there exists a unique affine geodesic γ : I → M with γ(0) = x and \dot\gamma(0)=X and where I is the maximal open interval in R, 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 M is a (pseudo)Riemannian manifold and ∇ is the LeviCivita connection, then the affine geodesic 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 M determines a parametric curve γ through the line up to a choice of affine reparametrization γ(t) → γ(at+b), where a and b 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

\nabla_{\dot{\gamma}}\dot{\gamma} = k\dot{\gamma}
for some function k defined along γ. 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 x_{t} is a curve in M, then the affine tangent space at x_{0} 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 C_{t} in this affine space: the development of x_{t}.
In formal terms, let τ_{t}^{0} : T_{xt}M → T_{x0}M be the linear parallel transport map associated to the affine connection. Then the development C_{t} is the curve in T_{x0}M starts off at 0 and is parallel to the tangent of x_{t} for all time t:

\dot{C}_t = \tau_t^0\dot{x}_t,\quad C_0 = 0.
In particular, x_{t} is a geodesic if and only if its development is an affinely parametrized straight line in T_{x0}M.^{[10]}
Surface theory revisited
If M is a surface in R^{3}, it is easy to see that M 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 M to R^{3}, and then projecting the result orthogonally back onto the tangent spaces of M. It is easy to see that this affine connection is torsionfree. Furthermore, it is a metric connection with respect to the Riemannian metric on M induced by the inner product on R^{3}, hence it is the LeviCivita connection of this metric.
Example: the unit sphere in Euclidean space
Let \langle, \rangle be the usual scalar product on R^{3}, and let S^{2} be the unit sphere. The tangent space to S^{2} at a point x is naturally identified with the vector subspace of R^{3} consisting of all vectors orthogonal to x. It follows that a vector field Y on S^{2} can be seen as a map Y : S^{2} → R^{3} which satisfies

\langle Y_x, x\rangle = 0, \quad \forall x\in \mathbf{S}^2.
Denote by dY the differential (Jacobian matrix) of such a map. Then we have:
Lemma. The formula

(\nabla_Z Y)_x = \mathrm dY_x(Z_x) + \langle Z_x,Y_x\rangle x\,
defines an affine connection on S^{2} with vanishing torsion.
Proof. It is straightforward to prove that ∇ satisfies the Leibniz identity and is C^{∞}(S^{2}) 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 x in S^{2}

\langle(\nabla_Z Y)_x,x\rangle = 0\qquad (1).
Consider the map

\begin{cases} f: \mathbf{S}^2\to \mathbf{R}\\ x \longmapsto \langle Y_x, x\rangle. \end{cases}
The map f is constant, hence its differential vanishes. In particular

\mathrm df_x(Z_x) = \langle (\mathrm d Y)_x(Z_x),x\rangle + \langle Y_x, Z_x\rangle = 0.\,
The equation (1) above follows. \Box
See also
Notes

^ Weyl 1918, 5 editions to 1922.

^ Cartan 1923.

^ As a result, many mathematicians use the term linear connection (instead of affine connection) for a connection on the tangent bundle, on the grounds that parallel transport is linear and not affine. However, the same property holds for any (Koszul or linear Ehresmann) connection on a vector bundle. Originally the term affine connection is short for an affine connection in the sense of Cartan, and this implies that the connection is defined on the tangent bundle, rather than an arbitrary vector bundle. The notion of a linear Cartan connection does not really make much sense, because linear representations are not transitive.

^ Cartan 1926.

^ It is difficult to make Cartan's intuition precise without invoking smooth infinitesimal analysis, but one way is to regard his points being variable, that is maps from some unseen parameter space into the manifold, which can then be differentiated.

^ Classically, the tangent space was viewed as an infinitesimal approximation, while in modern differential geometry, tangent spaces are often defined in terms of differential objects such as derivations (see Kobayashi & Nomizu 1996, Volume 1, sections 1.1–1.2).

^ For details, see Ü. Lumiste (2001b). The following intuitive treatment is that of Cartan (1923) and Cartan (1926).

^ This can be viewed as a choice of origin: actually it suffices to consider only the case p=a_{x}; Cartan implicitly identifies this with x in M.

^ Cf. R. Hermann (1983), Appendix 1–3 to Cartan (1951), and also Sharpe (1997).

^ This treatment of development is from Kobayashi & Nomizu (1996, Volume 1, Proposition III.3.1); see section III.3 for a more geometrical treatment. See also Sharpe (1997) for a thorough discussion of development in other geometrical situations.
References
Primary historical references


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.
Secondary references

Kobayashi, Shoshichi; Nomizu, Katsumi (1996), .


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.

Lumiste, Ülo (2001a), "Affine connection", in .

Lumiste, Ülo (2001b), "Connections on a manifold", in .


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 (nonprincipal bundle) perspective.

Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, SpringerVerlag, New York, .


This fills in some of the historical details, and provides a more readerfriendly 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.
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