Affine connection
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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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 vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
which ''connects'' nearby
tangent space In mathematics, the tangent space of a manifold 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 ...
s, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. Connections are among the simplest methods of defining differentiation of the
sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of
vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
. The notion of an affine connection has its roots in 19th-century geometry and
tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
, but was not fully developed until the early 1920s, by
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
(as part of his general theory of connections) and
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in ZĂ¼rich, Switzerland, and then Princeton, New Jersey, he is assoc ...
(who used the notion as a part of his foundations for
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
). 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 geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
. On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
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), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
. 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 include rectilinear motion, the linear r ...
and the
Leibniz rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, al ...
). 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 on a manifold by means of a different ...
or (linear) connection on the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
. A choice of affine connection is also equivalent to a notion of
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
, 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 bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
. 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 principal connection, in which the ...
for the
affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relat ...
or as a
principal connection In mathematics, and especially differential geometry and 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. A principal ''G''-conne ...
on the frame bundle. The main invariants of an affine connection are its
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
and its
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
. The torsion measures how closely the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine)
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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 ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
; 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 vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is a
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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 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 some domain, called ''differentiability cl ...
s and vector fields 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 ...
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, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
.


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 geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
at that point, which is an
affine subspace In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
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 phase, including finance and budgeting *Development (music), the process thematic material is reshaped *Photographi ...
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 or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
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, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
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 principal connection, in which the ...
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 â€“ 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 geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen Ă¼ber neuere geometrische Forschungen.'' It is nam ...
. More generally, an -dimensional affine space is a
Klein geometry In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as t ...
for the
affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relat ...
, the stabilizer of a point being the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
. 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 on a manifold by means of a different ...
of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields by
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
their respective
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
s into an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographic ...
. 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 Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provid ...
(following ideas of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
) 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 surfaces or other manifolds endowed with a metric, allowing distan ...
s. This idea was developed into the theory of ''absolute differential calculus'' (now known as
tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
) by
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the ...
and his student
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an 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 significa ...
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 â€“ 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
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), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
. 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 mathematician, theoretical physicist and philosopher. Although much of his working life was spent in ZĂ¼rich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, who developed a detailed mathematical foundation for general relativity, and
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
,. 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 in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
and
gauge covariant derivative The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are p ...
s. On the other hand, the notion of covariant differentiation was abstracted by
Jean-Louis Koszul Jean-Louis Koszul (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in ...
, who defined (linear or Koszul) connections on
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s. 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 on a manifold by means of a different ...
or (linear) connection on the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
. 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, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
. (Alternatively, Euclidean space is a
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-em ...
or
torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
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 geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
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 bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
. 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
and let be the space of vector fields on , that is, the space of smooth sections of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
. Then an affine connection on is a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
: \begin \Gamma(\mathrmM)\times \Gamma(\mathrmM) & \rightarrow \Gamma(\mathrmM)\\ (X,Y) & \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 In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
; 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 In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
). Conversely if is an affine connection and is such a smooth bilinear bundle homomorphism (called a
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...
on ) then is an affine connection. * If is an open subset of , then the tangent bundle of is the
trivial bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. 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 In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
of the tangent bundle is a
bundle isomorphism In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
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 geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
, 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
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s 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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
which is overdetermined: the
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
for this equation is the vanishing of the curvature of (see below). However, if this equation is restricted to a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
from to it becomes an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
. There is then a unique solution for any initial value of at . More precisely, if a
smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
parametrized by an interval and , where , then a vector field 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 connection on the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
. However, in a
local trivialization In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
it is a first-order system of linear ordinary differential equations, which has a unique solution for any initial condition given by the second condition (for instance, by the
Picard–Lindelöf theorem In mathematics â€“ specifically, in differential equations â€“ the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauc ...
). 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
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
ordered basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
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 bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
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 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 bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
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 In mathematics, a Lie algebra (pronounced ) 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 \rightarrow ...
of the
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of invertible matrices) satisfying two properties: # is
equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
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 on a manifold by means of a different ...
not only on the tangent bundle, but on
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s
associated Associated may refer to: *Associated, former name of Avon, Contra Costa County, California * Associated Hebrew Schools of Toronto, a school in Canada *Associated Newspapers, former name of DMG Media, a British publishing company See also *Associati ...
to any
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
of , including bundles of
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s and
tensor densities In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it ...
. 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
solder form In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
which is horizontal in the sense that it vanishes on 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 In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
of with the trivial bundle , where is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''Ă—''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of and (viewed as the Lie algebra of the affine group, which is actually a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
– 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 In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a basis of the tangent space at p. Equiva ...
of a principal bundle satisfying suitable properties. From this point of view the -valued one-form on the frame bundle (of an
affine manifold In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of ^n, with monodromy acting by affine t ...
) 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, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
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 mathematics, the tangent space of a manifold 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 ...
is really an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
notion, whereas the planes, as
affine subspace In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
s of , are
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
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 Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
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, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
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 In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathem ...
transform infinitesimally with respect to parallel transport. (This is the origin of Cartan's
method of moving frames In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. Introduction In lay ...
.) 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 In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear differential equations, linear or non-linear differential equations, non-linear. Also, such a system can be either a sy ...
:(*) \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 one-forms . Geometrically, an affine frame undergoes a displacement travelling along a curve from to given (approximately, or infinitesimally) by :\begin p(\gamma(t+\delta t)) - p(\gamma(t)) &= \left(\theta^1\left(\gamma'(t)\right)\mathbf_1 + \cdots + \theta^n\left(\gamma'(t)\right)\mathbf_n\right)\mathrm \delta t \\ \mathbf_i(\gamma(t+\delta t)) - \mathbf_i(\gamma(t)) &= \left(\omega^1_i\left(\gamma'(t)\right)\mathbf_1 + \cdots + \omega^n_i\left(\gamma'(t)\right) \mathbf_n\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 (\gamma(t + \delta t)) - a_x (\gamma(t)) = \theta\left(\gamma'(t)\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 â€“ 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 geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen Ă¼ber neuere geometrische Forschungen.'' It is nam ...
, in which a ''geometry'' is defined to be a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
. 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, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
without a fixed choice of
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
. It describes the geometry of
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
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 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 free transitive action of the vector group on it through this operation of translation of points: is thus a
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-em ...
for the vector group . The
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
is the group of transformations of which preserve the ''linear structure'' of in the sense that . By analogy, the
affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relat ...
is the group of transformations of preserving the ''affine structure''. Thus must ''preserve translations'' in the sense that :\varphi(p+v)=\varphi(p)+T(v) where is a general linear transformation. The map sending to is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
. Its
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
is the group of translations . The stabilizer of any point in can thus be identified with using this projection: this realises the affine group as a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
of and , and affine space as the
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
.


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 In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
. Thus is a principal -bundle over . The action of extends naturally to a free transitive action of the affine group on , so that is an -
torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
, and the choice of a reference frame identifies with the principal bundle . On there is a collection of functions defined by :\pi(p;\mathbf_1, \dots ,\mathbf_n) = p (as before) and :\varepsilon_i(p;\mathbf_1,\dots , \mathbf_n) = \mathbf_i\,. After choosing a basepoint for , these are all functions with values in , so it is possible to take their
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
s to obtain
differential 1-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s 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
linear independence In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
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 In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
(for all ) is
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, and its
integral manifold In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of ...
s 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, 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. A principal ''G''-conne ...
on . For a strict comparison with the motivation, one should actually define parallel transport in a principal -bundle over . This can be done by pulling back by the smooth map defined by translation. Then the composite is a principal -bundle over , and the forms
pull back In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
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, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as t ...
, 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 In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a basis of the tangent space at p. Equiva ...
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 In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
from to the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
. 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 bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
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 Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
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 In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it d ...
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 eta\wedge\eta= 0, where the second term on the left hand side is the
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
using the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
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 Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
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 Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
is given by the formula :T^\nabla(X,Y) = \nabla_X Y - \nabla_Y X - ,Y 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 In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth m ...
: ,Y\left(X^j \partial_j Y^i - Y^j \partial_j X^i\right)\partial_i in
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
. This is independent of coordinate system choice and : \partial_i = \left(\frac\right)_p\,, the tangent vector at point of the th
coordinate curve In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
. 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 In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as Tree (graph theory), rooted trees and vector fields on affine space. The notion of pre-Lie algebra has been introduced b ...
structure on the space of global sections of the tangent bundle.


The Levi-Civita connection

If is a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
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), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
. 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 mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along a ...
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 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 surfaces or other manifolds endowed with a metric, allowing distance ...
: 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(s) = \dot\gamma(t) where is the parallel transport map defining the connection. In terms of the infinitesimal connection , the derivative of this equation implies :\nabla_\dot\gamma(t) = 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 In mathematics â€“ specifically, in differential equations â€“ the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauc ...
, and allows for the definition of an exponential map associated to the affine connection. In particular, when is a (
pseudo The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, ...
-)
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
and is the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
, then the affine geodesics are the usual
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s 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 connection In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold. The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to a ...
s.


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 phase, including finance and budgeting *Development (music), the process thematic material is reshaped *Photographi ...
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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
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 ::(\nabla_Z Y)_x = \mathrmY_x(Z_x) + \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(\nabla_Z Y)_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(Z_x) = \bigl\langle (\mathrm Y)_x(Z_x),x(\gamma'(t))\bigr\rangle + \langle Y_x, Z_x\rangle = 0\,. :::Equation 1 above follows.
Q.E.D. Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...


See also

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Atlas (topology) In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
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Connection (mathematics) In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are var ...
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Connection (fibred manifold) In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connect ...
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Connection (affine bundle) Let be an affine bundle modelled over a vector bundle . A connection (fibred manifold), connection on is called the affine connection if it as a section of the jet bundle of is an affine bundle morphism over . In particular, this is an affin ...
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Differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
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Differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
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Introduction to the mathematics of general relativity The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solve ...
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Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
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List of formulas in Riemannian geometry This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwhise. Christoffel symbols, covariant deriva ...
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Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...


Notes


Citations


References

* * *


Bibliography


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 The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from con ...
. * :: A more mathematically motivated account of affine connections. * . :: Affine connections from the point of view of
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
. 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 In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G' ...
and
complex manifolds In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
, 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 Connection (mathematics) Differential geometry Maps of manifolds Smooth functions de:Zusammenhang (Differentialgeometrie)#Linearer Zusammenhang