affine connection
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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, 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 may ...
which ''connects'' nearby
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
. Connections are among the simplest methods of defining differentiation of the sections 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, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, 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. He ...
(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, logician and philosopher. Although much of his working life was spent in ZĂ¼rich, Switzerland, and then Princeton, New Jersey, ...
(who used the notion as a part of his foundations for
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
). 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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
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 relat ...
. 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 that preserves the ( pseudo-) Riemannian ...
. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (
linearity In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
and the Leibniz rule). This yields a possible definition of an affine connection as a
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
or (linear) connection on the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. A choice of affine connection is also equivalent to a notion of
parallel transport In differential 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 on ...
, 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 with 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 naturally on ...
. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the
affine group In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real nu ...
or as a principal connection on the frame bundle. The main invariants of an affine connection are its torsion and its
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
. 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 locally 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 connec ...
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, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
; 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 may ...
is a
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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 is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...
s and
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s can be defined on manifolds, just as they can on Euclidean space, and scalar functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point 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, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
.


Motivation from surface theory

Consider a smooth surface in a 3-dimensional Euclidean space. Near 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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
at that point, which is an
affine subspace In mathematics, an affine space is a geometry, geometric structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance (mathematics), distance ...
of Euclidean space. Differential geometers in the 19th century were interested in the notion of
development Development or developing may refer to: Arts *Development (music), the process by which thematic material is reshaped * Photographic development *Filmmaking, development phase, including finance and budgeting * Development hell, when a proje ...
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 polytop ...
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 general ...
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 connections. In more modern approaches, the point of contact is viewed as the ''origin'' in the tangent plane (which is then a vector space), and the movement of the origin is corrected by a translation, so that parallel transport is linear, rather than affine. In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are ''model'' surfaces — they are the simplest surfaces in Euclidean 3-space, and are homogeneous under the affine group of the plane — and every smooth surface has a unique model surface tangent to it at each point. These model surfaces are ''Klein geometries'' in the sense of
Felix Klein Felix Christian Klein (; ; 25 April 1849 â€“ 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
's Erlangen programme. 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 ...
for the
affine group In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real nu ...
, the stabilizer of a point being the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
. 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 and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
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 (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...
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. Euclidean vectors can be added and scal ...
s into an
atlas An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets. Atlases have traditio ...
. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates. Correction terms were introduced by Elwin Bruno Christoffel (following ideas of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols. This idea was developed into the theory of ''absolute differential calculus'' (now known as
tensor calculus In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
) 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 signifi ...
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 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's theory of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
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 that preserves the ( pseudo-) Riemannian ...
. 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, logician and philosopher. Although much of his working life was spent in ZĂ¼rich, Switzerland, and then Princeton, New Jersey, ...
, 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. He ...
,. 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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
and gauge covariant derivatives. On the other hand, the notion of covariant differentiation was abstracted by Jean-Louis Koszul, who defined (linear or Koszul) connections on
vector 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 eve ...
s. In this language, an affine connection is simply a
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
or (linear) connection on the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. 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 relat ...
. (Alternatively, Euclidean space is a principal homogeneous space 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 no ...
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 differential 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 on ...
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 with 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 naturally on ...
. 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 A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. 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. A bilinear map can also be defined for ...
: \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 multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vect ...
; 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 mathematics, and particularly topology, a fiber bundle ( ''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 ( ''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 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 differential 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 on ...
, 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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
which is overdetermined: the integrability condition for this equation is the vanishing of the curvature of (see below). However, if this equation is restricted to a
curve 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 ...
from to it becomes an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
. There is then a unique solution for any initial value of at . More precisely, if a 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 space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
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 . However, in a
local trivialization In mathematics, and particularly topology, a fiber bundle ( ''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, specifically the study of differential equations, 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 Cauchy– ...
). 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 pr ...
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 Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * ...
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 with 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 naturally on ...
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 with 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 naturally on ...
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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of the
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
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 and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
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 eve ...
s associated 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 ...
of , including bundles of
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s and 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 solder form 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 of with the trivial bundle , where is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
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. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...
– 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, 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 relat ...
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 is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
is really an
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
notion, whereas the planes, as
affine subspace In mathematics, an affine space is a geometry, geometric structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance (mathematics), distance ...
s of , are infinite in extent. However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point. The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point. However, the parallel transport defined by rolling does not fix this origin: it is
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...
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 general ...
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 pr ...
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 have been specified in physical space. It is based on a set of reference points, defined as geometric ...
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 :(*) \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 Felix Christian Klein (; ; 25 April 1849 â€“ 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
's Erlangen programme, in which a ''geometry'' is defined to be a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
. 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 relat ...
is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
without a fixed choice of origin. It describes the geometry of
points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
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 for the vector group . The
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
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 is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real nu ...
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) whe ...
. Its kernel 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. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...
of and , and affine space as the
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
.


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 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 no ...
, 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 re ...
s 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
linear independence In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then th ...
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 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 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 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 ...
, 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 with 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 naturally on ...
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 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 eta\wedge\eta= 0, where the second term on the left hand side is the wedge product 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 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 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 : ,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 and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
. This is independent of coordinate system choice and : \partial_i = \left(\frac\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 In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
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 that preserves the ( pseudo-) Riemannian ...
. 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 (vector bundle), 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 p ...
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 surface (topology), surfaces or other manifolds endowed with a metri ...
: 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 \dot\gamma 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, specifically the study of differential equations, 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 Cauchy– ...
, and allows for the definition of an exponential map associated to the affine connection. In particular, when is a ( pseudo-)
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
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 that preserves the ( pseudo-) Riemannian ...
, then the affine geodesics are the usual
geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...
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 connections.


Development

An affine connection defines a notion of
development Development or developing may refer to: Arts *Development (music), the process by which thematic material is reshaped * Photographic development *Filmmaking, development phase, including finance and budgeting * Development hell, when a proje ...
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 for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
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 List of Latin phrases (full), Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof ...


See also

*
Atlas (topology) In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies th ...
*
Connection (mathematics) In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as Tangent vector, tangent vectors or Tensor, tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consist ...
*
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 connecti ...
* Connection (affine bundle) *
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 ...
*
Differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
* Introduction to the mathematics of general relativity *
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 that preserves the ( pseudo-) Riemannian ...
* List of formulas in Riemannian geometry *
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...


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. * :: 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, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
. 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, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the Group action (mathematics), action of a Group (mathematics), group. Homogeneous spaces occur in th ...
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 Connection (mathematics) Differential geometry Maps of manifolds Smooth functions de:Zusammenhang (Differentialgeometrie)#Linearer Zusammenhang