affine connection

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In
differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ...
, an affine connection is a geometric object on a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an a ...
which ''connects'' nearby
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
s, so it permits to be as if they were functions on the manifold with values in a fixed
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. The notion of an affine connection has its roots in 19th-century geometry and
tensor calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, but was not fully developed until the early 1920s, by
Élie Cartan Élie Joseph Cartan, ForMemRS Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a ...
(as part of his general theory of connections) and
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

(who used the notion as a part of his foundations for
general relativity General relativity, also known as the general theory of relativity, is the of published by in 1915 and is the current description of gravitation in . General generalizes and refines , providing a unified description of gravity as a geome ...
). The terminology is due to Cartan and has its origins in the identification of tangent spaces in
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a
metric tensor In the mathematics, mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) an ...
then there is a natural choice of affine connection, called the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory o ...
. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (
linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics Physics (from grc, φυσική ( ...

and the Leibniz rule). This yields a possible definition of an affine connection as a
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection (mathematics), connection on a manifold ...
or (linear) connection on the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

. A choice of affine connection is also equivalent to a notion of
parallel transport In , parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a . If the manifold is equipped with an (a or on the ), then this connection allows one to transport vectors of the manifold al ...

, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered basis, ordered bases, or ''frames'', for ''E'x''. The general linear gro ...
. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a
Cartan connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a connection (principal bundle), princi ...
for the
affine group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
or as a
principal connection In mathematics, and especially differential geometry and gauge theory (mathematics), gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. ...
on the frame bundle. The main invariants of an affine connection are its torsion and its
curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. The torsion measures how closely the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightar ...
of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine)
geodesics In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
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 Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a smal ...
; the main differences are encapsulated in the curvature of the connection.

# Motivation and history

A
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an a ...
is a
mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 ...
object which looks locally like a smooth deformation of Euclidean space : for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane.
Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over ...

s and
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

s can be defined on manifolds, just as they can on Euclidean space, and
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
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 can be identified naturally (by translation) with the tangent space at a nearby point . 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.

## Motivation from surface theory

Consider a smooth surface in 3-dimensional Euclidean space. Near to any point, can be approximated by its
tangent plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

at that point, which is an
affine subspace In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of Euclidean space. Differential geometers in the 19th century were interested in the notion of
development Development or developing may refer to: Arts *Development hell, when a project is stuck in development *Filmmaking#Development, Filmmaking, development phase, including finance and budgeting *Development (music), the process thematic material i ...
in which one surface was ''rolled'' along another, without ''slipping'' or ''twisting''. In particular, the tangent plane to a point of can be rolled on : this should be easy to imagine when is a surface like the 2-sphere, which is the smooth boundary of a
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optical device that focuses or disperses a light beam by means of ...

region. As the tangent plane is rolled on , the point of contact traces out a curve on . Conversely, given a curve on , 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 transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...
s 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 connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a connection (principal bundle), princi ...
s. 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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's
Erlangen programme In mathematics, the Erlangen program is a method of characterizing geometry, geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' ...
. More generally, an -dimensional affine space is a
Klein geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
for the
affine group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, the stabilizer of a point being the
general linear group A general officer is an officer of high rank in the armies, and in some nations' air forces, space force A space force is a military branch of a nation's armed forces A military, also known collectively as armed forces, is a hea ...
. An affine -manifold is then a manifold which looks infinitesimally like -dimensional affine space.

## Motivation from tensor calculus

The second motivation for affine connections comes from the notion of a
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection (mathematics), connection on a manifold ...
of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields by
embedding In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
their respective
Euclidean vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s into an
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. 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#REDIRECT Elwin Bruno Christoffel Elwin Bruno Christoffel (; November 10, 1829 – March 15, 1900) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek ...
(following ideas of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
) 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 symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metric ...
s. This idea was developed into the theory of ''absolute differential calculus'' (now known as
tensor calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) by
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ( ...
and his student
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italians, Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made s ...
between 1880 and the turn of the 20th century. Tensor calculus really came to life, however, with the advent of
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the , but he also made important contributions to the develo ...

's theory of
general relativity General relativity, also known as the general theory of relativity, is the of published by in 1915 and is the current description of gravitation in . General generalizes and refines , providing a unified description of gravity as a geome ...
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 In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory o ...
. More general affine connections were then studied around 1920, by
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

, who developed a detailed mathematical foundation for general relativity, and
Élie Cartan Élie Joseph Cartan, ForMemRS Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a ...
,. 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 In physics, a gauge theory is a type of Field theory (physics), field theory in which the Lagrangian (field theory), Lagrangian (and hence the dynamics of the system itself) does not change (is Invariant (physics), invariant) under local symmetry, ...
and gauge covariant derivatives. On the other hand, the notion of covariant differentiation was abstracted by Jean-Louis Koszul, who defined (linear or Koszul) connection (vector bundle), connections on vector bundles. In this language, an affine connection is simply a
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection (mathematics), connection on a manifold ...
or (linear) connection on the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

. However, this approach does not explain the geometry behind affine connections nor how they acquired their name. The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean -space is an
affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. (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 In , parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a . If the manifold is equipped with an (a or on the ), then this connection allows one to transport vectors of the manifold al ...

of vector fields along a curve. This also defines a parallel transport on the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered basis, ordered bases, or ''frames'', for ''E'x''. The general linear gro ...
. 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.

# Formal definition as a differential operator

Let be a smooth manifold and let be the space of vector fields on , that is, the space of section (fiber bundle), smooth sections of the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

. Then an affine connection on is a bilinear map : $\begin \Gamma\left(\mathrmM\right)\times \Gamma\left(\mathrmM\right) & \rightarrow \Gamma\left(\mathrmM\right)\\ \left(X,Y\right) & \mapsto \nabla_X Y\,,\end$ such that for all in the set of smooth functions on , written , and all vector fields on : # , that is, is -''linear'' in the first variable; # , where denotes the directional derivative; that is, satisfies ''Leibniz rule'' in the second variable.

## Elementary properties

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

# Parallel transport for affine connections

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 In , parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a . If the manifold is equipped with an (a or on the ), then this connection allows one to transport vectors of the manifold al ...

, and indeed this can be used to give a definition of an affine connection. Let be a manifold with an affine connection . Then a vector field is said to be parallel if in the sense that for any vector field , . 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 and , an identification between a tangent vector at and one at 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 is a partial differential equation which is overdetermined system, 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 to it becomes an ordinary differential equation. There is then a unique solution for any initial value of at . More precisely, if a curve, smooth curve parametrized by an interval and , where , then a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

along (and in particular, the value of this vector field at ) is called the parallel transport of along if #, for all #. Formally, the first condition means that is parallel with respect to the pullback (differential geometry), pullback connection on the pullback bundle . However, in a local trivialization it is a first-order system of linear differential equation, 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 , which can only happen if the curvature of is zero. A linear isomorphism is determined by its action on an Basis (linear algebra)#Ordered bases and coordinates, ordered basis or frame. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent)
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered basis, ordered bases, or ''frames'', for ''E'x''. The general linear gro ...
along a curve. In other words, the affine connection provides a lift of any curve in to a curve in .

# Formal definition on the frame bundle

An affine connection may also be defined as a connection (principal bundle), principal connection on the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered basis, ordered bases, or ''frames'', for ''E'x''. The general linear gro ...
or of a manifold . In more detail, is a smooth map from the tangent bundle of the frame bundle to the space of matrices (which is the Lie algebra of the Lie group of invertible matrices) satisfying two properties: # is equivariant with respect to the action of on and ; # for any in , where is the vector field on corresponding to . Such a connection immediately defines a
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection (mathematics), connection on a manifold ...
not only on the tangent bundle, but on vector bundles associated bundle, associated to any group representation of , including bundles of tensors and tensor density, tensor densities. Conversely, an affine connection on the tangent bundle determines an affine connection on the frame bundle, for instance, by requiring that 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 frame bundle#Solder form, solder form which is horizontal in the sense that it vanishes on vertical bundle, vertical vectors such as the point values of the vector fields : indeed is defined first by projecting a tangent vector (to at a frame ) to , then by taking the components of this tangent vector on with respect to the frame . Note that is also -equivariant (where acts on by matrix multiplication). The pair defines a bundle isomorphism of with the trivial bundle , where is the Cartesian product of and (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. 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 -valued one-form 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; * 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
is really an differential (infinitesimal), infinitesimal notion, whereas the planes, as
affine subspace In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of , are Infinity, 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 transformation, 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 -manifold with an affine space , of dimension , ''attached'' to each at a marked point , together with a method for transporting elements of these affine spaces along any curve in . This method is required to satisfy several properties: # for any two points on , parallel transport is an
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...
from to ; # parallel transport is defined infinitesimally in the sense that it is differentiable at any point on and depends only on the tangent vector to at that point; # the derivative of the parallel transport at determines a linear isomorphism from to . These last two points are quite hard to make precise, 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 , where and the form a basis of . The affine connection is then given symbolically by a first order differential system :$\left(*\right) \begin \mathrm &= \theta^1\mathbf_1 + \cdots + \theta^n\mathbf_n \\ \mathrm\mathbf_i &= \omega^1_i\mathbf_1 + \cdots + \omega^n_i\mathbf_n \end \quad i=1,2,\ldots,n$ defined by a collection of differential forms, one-forms . Geometrically, an affine frame undergoes a displacement travelling along a curve from to given (approximately, or infinitesimally) by :$\begin p\left(\gamma\left(t+\delta t\right)\right) - p\left(\gamma\left(t\right)\right) &= \left\left(\theta^1\left\left(\gamma\text{'}\left(t\right)\right\right)\mathbf_1 + \cdots + \theta^n\left\left(\gamma\text{'}\left(t\right)\right\right)\mathbf_n\right\right)\mathrm \delta t \\ \mathbf_i\left(\gamma\left(t+\delta t\right)\right) - \mathbf_i\left(\gamma\left(t\right)\right) &= \left\left(\omega^1_i\left\left(\gamma\text{'}\left(t\right)\right\right)\mathbf_1 + \cdots + \omega^n_i\left\left(\gamma\text{'}\left(t\right)\right\right) \mathbf_n\right\right)\delta t\,. \end$ Furthermore, the affine spaces are required to be tangent to in the informal sense that the displacement of along can be identified (approximately or infinitesimally) with the tangent vector to at (which is the infinitesimal displacement of ). Since :$a_x \left(\gamma\left(t + \delta t\right)\right) - a_x \left(\gamma\left(t\right)\right) = \theta\left\left(\gamma\text{'}\left(t\right)\right\right) \delta t \,,$ where is defined by , this identification is given by , so the requirement is that should be a linear isomorphism at each point. The tangential affine space is thus identified intuitively with an ''infinitesimal affine neighborhood'' of . 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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's
Erlangen programme In mathematics, the Erlangen program is a method of characterizing geometry, geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
is a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
without a fixed choice of origin (mathematics), origin. It describes the geometry of point (mathematics), points and Vector (geometric), 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 may be added to a point by placing the initial point of the vector at and then transporting to the terminal point. The operation thus described is the translation of along . In technical terms, affine -space is a set equipped with a Group action (mathematics), free transitive action of the vector group on it through this operation of translation of points: is thus a principal homogeneous space for the vector group . The
general linear group A general officer is an officer of high rank in the armies, and in some nations' air forces, space force A space force is a military branch of a nation's armed forces A military, also known collectively as armed forces, is a hea ...
is the transformation group, group of transformations of which preserve the ''linear structure'' of in the sense that . By analogy, the
affine group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is the group of transformations of preserving the ''affine structure''. Thus must ''preserve translations'' in the sense that :$\varphi\left(p+v\right)=\varphi\left(p\right)+T\left(v\right)$ where is a general linear transformation. The map sending to is a group homomorphism. Its kernel (algebra), kernel is the group of translations . The Group action (mathematics)#Orbits and stabilizers, stabilizer of any point in can thus be identified with using this projection: this realises the affine group as a semidirect product of and , and affine space as the homogeneous space .

### Affine frames and the flat affine connection

An ''affine frame'' for consists of a point and a basis of the vector space . The general linear group acts freely on the set of all affine frames by fixing and transforming the basis in the usual way, and the map sending an affine frame to is the quotient map. Thus is a principal bundle, principal -bundle over . The action of extends naturally to a free transitive action of the affine group on , so that is an -principal homogeneous space, torsor, and the choice of a reference frame identifies with the principal bundle . On there is a collection of functions defined by :$\pi\left(p;\mathbf_1, \dots ,\mathbf_n\right) = p$ (as before) and :$\varepsilon_i\left(p;\mathbf_1,\dots , \mathbf_n\right) = \mathbf_i\,.$ After choosing a basepoint for , these are all functions with values in , so it is possible to take their exterior derivatives to obtain differential 1-forms with values in . Since the functions yield a basis for at each point of , these 1-forms must be expressible as sums of the form :$\begin \mathrm\pi &= \theta^1\varepsilon_1+\cdots+\theta^n\varepsilon_n\\ \mathrm\varepsilon_i &= \omega^1_i\varepsilon_1+\cdots+\omega^n_i\varepsilon_n \end$ for some collection of real-valued one-forms on . This system of one-forms on the principal bundle defines the affine connection on . Taking the exterior derivative a second time, and using the fact that as well as the linearly independent, linear independence of the , the following relations are obtained: :$\begin \mathrm\theta^j - \sum_i\omega^j_i\wedge\theta^i &=0\\ \mathrm\omega^j_i - \sum_k \omega^j_k\wedge\omega^k_i &=0\,. \end$ These are the Maurer–Cartan equations for the Lie group (identified with by the choice of a reference frame). Furthermore: * the Pfaffian system (for all ) is integrability condition, integrable, and its integral manifolds are the fibres of the principal bundle . * the Pfaffian system (for all ) is also integrable, and its integral manifolds define parallel transport in . Thus the forms define a flat
principal connection In mathematics, and especially differential geometry and gauge theory (mathematics), gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. ...
on . For a strict comparison with the motivation, one should actually define parallel transport in a principal -bundle over . This can be done by pullback bundle, pulling back by the smooth map defined by translation. Then the composite is a principal -bundle over , and the forms pullback (differential geometry), pull back to give a flat principal -connection on this bundle.

## General affine geometries: formal definitions

An affine space, as with essentially any smooth
Klein geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 in the flat model fit together to give a 1-form with values in the Lie algebra of the affine group . In these definitions, is a smooth -manifold and is an affine space of the same dimension.

### Definition via absolute parallelism

Let be a manifold, and a principal -bundle over . Then an affine connection is a 1-form on with values in satisfying the following properties # is equivariant with respect to the action of on and ; # for all in the Lie algebra of all matrices; # is a linear isomorphism of each tangent space of with . The last condition means that is an absolute parallelism on , i.e., it identifies the tangent bundle of with a trivial bundle (in this case ). The pair defines the structure of an affine geometry on , making it into an affine manifold. The affine Lie algebra splits as a semidirect product of and and so may be written as a pair where takes values in and takes values in . Conditions 1 and 2 are equivalent to being a principal -connection and being a horizontal equivariant 1-form, which induces a bundle homomorphism from to the associated bundle . 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 is the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered basis, ordered bases, or ''frames'', for ''E'x''. The general linear gro ...
of , it follows that provides a bundle isomorphism between and the frame bundle of ; this recovers the definition of an affine connection as a principal -connection on . The 1-forms arising in the flat model are just the components of and .

### Definition as a principal affine connection

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

### Relation to the motivation

Since acts on , there is, associated to the principal bundle , a bundle , which is a fiber bundle over whose fiber at in is an affine space . A section (fiber bundle), section of (defining a marked point in for each ) determines a principal -subbundle of (as the bundle of stabilizers of these marked points) and vice versa. The principal connection defines an Ehresmann connection on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section 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 to satisfy the Maurer–Cartan equation :$\mathrm\eta + \tfrac12\left[\eta\wedge\eta\right] = 0,$ where the second term on the left hand side is the wedge product using the Lie bracket of vector fields, Lie bracket in to contract the values. By expanding into the pair and using the structure of the Lie algebra , this left hand side can be expanded into the two formulae :$\mathrm\theta + \omega\wedge\theta \quad \text \quad \mathrm\omega + \omega\wedge\omega\,,$ 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 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 on as follows. The Torsion of connection, torsion is given by the formula :$T^\nabla\left(X,Y\right) = \nabla_X Y - \nabla_Y X - \left[X,Y\right].$ If the torsion vanishes, the connection is said to be ''torsion-free'' or ''symmetric''. The curvature is given by the formula :$R^\nabla_Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_Z.$ Note that is the Lie bracket of vector fields :$\left[X,Y\right]=\left\left(X^j \partial_j Y^i - Y^j \partial_j X^i\right\right)\partial_i$ in Einstein notation. This is independent of coordinate system choice and : $\partial_i = \left\left(\frac\right\right)_p\,,$ the tangent vector at point of the th coordinate curve. The are a natural basis for the tangent space at point , and the the corresponding coordinates for the vector field . 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 is a Riemannian manifold then there is a unique affine connection on with the following two properties: * the connection is torsion-free, i.e., is zero, so that ; * parallel transport is an isometry, i.e., the inner products (defined using ) between tangent vectors are preserved. This connection is called the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory o ...
. 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 is parallel: . For a torsion-free connection, the condition is equivalent to the identity + , "compatibility with the metric". 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 .

## 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 is a straight line if its tangent vector remains parallel and equipollent with itself when it is transported along . From the linear point of view, an affine connection distinguishes the affine geodesics in the following way: a smooth curve is an affine geodesic if is parallel transported along , that is :$\tau_t^s\dot\gamma\left(s\right) = \dot\gamma\left(t\right)$ where is the parallel transport map defining the connection. In terms of the infinitesimal connection , the derivative of this equation implies :$\nabla_\dot\gamma\left(t\right) = 0$ for all . Conversely, any solution of this differential equation yields a curve whose tangent vector is parallel transported along the curve. For every and every , there exists a unique affine geodesic with and and where is the maximal open interval in , containing 0, on which the geodesic is defined. This follows from the Picard–Lindelöf theorem, and allows for the definition of an exponential map (Riemannian geometry), exponential map associated to the affine connection. In particular, when is a (pseudo-Riemannian manifold, pseudo-)Riemannian manifold and is the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory o ...
, 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 determines a parametric curve through the line up to a choice of affine reparametrization , where and 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 :$\nabla_\dot = k\dot$ for some function defined along . Unparametrized geodesics are often studied from the point of view of projective connections.

## Development

An affine connection defines a notion of
development Development or developing may refer to: Arts *Development hell, when a project is stuck in development *Filmmaking#Development, Filmmaking, development phase, including finance and budgeting *Development (music), the process thematic material i ...
of curves. Intuitively, development captures the notion that if is a curve in , then the affine tangent space at 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 in this affine space: the development of . In formal terms, let be the linear parallel transport map associated to the affine connection. Then the development is the curve in starts off at 0 and is parallel to the tangent of for all time : :$\dot_t = \tau_t^0\dot_t\,,\quad C_0 = 0.$ In particular, is a ''geodesic'' if and only if its development is an affinely parametrized straight line in .This treatment of development is from ; see section III.3 for a more geometrical treatment. See also for a thorough discussion of development in other geometrical situations.

# Surface theory revisited

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

## Example: the unit sphere in Euclidean space

Let be the usual scalar product on , and let be the unit sphere. The tangent space to at a point is naturally identified with the vector subspace of consisting of all vectors orthogonal to . It follows that a vector field on can be seen as a map which satisfies : $\langle Y_x, x\rangle = 0\,, \quad \forall x\in \mathbf^2.$ Denote as the differential (Jacobian matrix) of such a map. Then we have: :Lemma. The formula ::$\left(\nabla_Z Y\right)_x = \mathrmY_x\left(Z_x\right) + \langle Z_x,Y_x\rangle x$ :defines an affine connection on with vanishing torsion. :::Proof. It is straightforward to prove that satisfies the Leibniz identity and is 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 in ::::$\bigl\langle\left(\nabla_Z Y\right)_x,x\bigr\rangle = 0\,.\qquad \text$ :::Consider the map ::::$\begin f: \mathbf^2&\to \mathbf\\ x &\mapsto \langle Y_x, x\rangle\,.\end$ :::The map ''f'' is constant, hence its differential vanishes. In particular ::::$\mathrmf_x\left(Z_x\right) = \bigl\langle \left(\mathrm Y\right)_x\left(Z_x\right),x\left(\gamma\text{'}\left(t\right)\right)\bigr\rangle + \langle Y_x, Z_x\rangle = 0\,.$ :::Equation 1 above follows. Q.E.D.

*Atlas (topology) *Connection (mathematics) *Connection (fibred manifold) *Connection (affine bundle) *Differentiable manifold *Differential geometry *Introduction to the mathematics of general relativity *
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory o ...
*List of formulas in Riemannian geometry *Riemannian geometry

# 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

* . :: 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. {{tensors Differential geometry Connection (mathematics) de:Zusammenhang (Differentialgeometrie)#Linearer Zusammenhang