Affine connection
Encyclopedia
In the branch of mathematics
called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent space
s, and so permits tangent vector fields
to be differentiated
as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan
(as part of his general theory of connections
) and Hermann Weyl
(who used the notion as a part of his foundations for general relativity
). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space
Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space
.
On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection
. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule
). This yields a possible definition of an affine connection as a covariant derivative
or (linear) connection
on the tangent bundle
. A choice of affine connection is also equivalent to a notion of parallel transport
, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle
. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection
for the affine group
or as a principal connection
on the frame bundle.
The main invariants of an affine connection are its torsion and its curvature
. The torsion measures how closely the Lie bracket
of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry
; the main differences are encapsulated in the curvature of the connection.
s and vector field
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 p can be identified naturally (by translation) with the tangent space at a nearby point q. On a general manifold, there is no such natural identification between nearby tangent spaces, and so tangent vectors at nearby points cannot be compared in a 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 which one surface was rolled along another, without slipping or twisting. In particular, the tangent plane to a point of S can be rolled on S: this should be easy to imagine when S is a surface like the 2-sphere, which is the smooth boundary of a convex
region. As the tangent plane is rolled on S, the point of contact traces out a curve on S. Conversely, given a curve on S, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve. These identifications are always given by affine transformation
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
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
's Erlangen programme. More generally, an n-dimensional affine space is a Klein geometry
for the affine group
Aff(n), the stabilizer of a point being the general linear group
GL(n). An affine n-manifold is then a manifold which looks infinitesimally like n-dimensional affine space.
of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields using their components in coordinate charts. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates. Correction terms were introduced by Elwin Bruno Christoffel
(following ideas of Bernhard Riemann
) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly
under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols. This idea was developed into the theory of the absolute differential calculus (now known as tensor calculus) by Gregorio Ricci-Curbastro
and his student Tullio Levi-Civita
between 1880 and the turn of the 20th century.
The tensor calculus really came to life, however, with the advent of Albert Einstein
's theory of general relativity
in 1915. A few years after this, Levi-Civita formalized the unique connection associated to a Riemannian metric, now known as the Levi-Civita connection
. More general affine connections were then studied around 1920, by Hermann Weyl
, who developed a detailed mathematical foundation for general relativity, and Élie Cartan
, who made the link with the geometrical ideas coming from surface theory.
The most popular approach is probably the definition motivated by covariant derivatives. On the one hand, the ideas of Weyl were taken up by physicists in the form of gauge theory
and gauge covariant derivative
s. On the other hand, the notion of covariant differentiation was abstracted by Jean-Louis Koszul
, who defined (linear or Koszul) connections
on vector bundle
s. In this language, an affine connection is simply a covariant derivative
or (linear) connection
on the tangent bundle
.
However, this approach does not explain the geometry behind affine connections nor how they acquired their name. The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean n-space is an affine space
. (Alternatively, Euclidean space is a principal homogeneous space
or torsor under the group of translations, which is a subgroup of the affine group.) As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of parallel transport
of vector fields along a curve. This also defines a parallel transport on the frame bundle
. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group Aff(n) or as a principal GL(n) connection on the frame bundle.
and let C∞(M,TM) be the space of vector fields on M, that is, the space of smooth sections
of the tangent bundle
TM. Then an affine connection on M is a bilinear map
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
s, and so permits tangent vector fields
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
to be differentiated
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
(as part of his general theory of connections
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 geometry of the principal bundle is tied to the...
) and Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
(who used the notion as a part of his foundations for general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It 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
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn 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
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
.
On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
). This yields a possible definition of an affine connection as a covariant derivative
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 differential operator, to be contrasted with the approach given...
or (linear) connection
Connection (vector bundle)
In mathematics, a connection on a fiber bundle 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. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
. A choice of affine connection is also equivalent to a notion of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection
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 geometry of the principal bundle is tied to the...
for the affine group
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
or as a principal connection
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
on the frame bundle.
The main invariants of an affine connection are its torsion and its curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
. The torsion measures how closely the Lie bracket
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
; the main differences are encapsulated in the curvature of the connection.
Motivation and history
A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space Rn: for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane. Smooth functionSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s and vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s can be defined on manifolds, just as they can on Euclidean space, and scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point p can be identified naturally (by translation) with the tangent space at a nearby point q. On a general manifold, there is no such natural identification between nearby tangent spaces, and so tangent vectors at nearby points cannot be compared in a 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
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....
and tensor calculus.
Motivation from surface theory
Consider a smooth surface S in 3-dimensional Euclidean space. Near to any point, S can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. Differential geometers in the 19th century were interested in the notion of developmentDevelopment (differential geometry)
In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface at a point can be rolled around the surface to obtain the tangent-plane at other points.The tangential contact...
in which one surface was rolled along another, without slipping or twisting. In particular, the tangent plane to a point of S can be rolled on S: this should be easy to imagine when S is a surface like the 2-sphere, which is the smooth boundary of a convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
region. As the tangent plane is rolled on S, the point of contact traces out a curve on S. Conversely, given a curve on S, the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve. These identifications are always given by affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
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
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 geometry of the principal bundle is tied to 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
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's Erlangen programme. More generally, an n-dimensional affine space is a Klein geometry
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 the symmetry group of the geometry.For background and motivation...
for the affine group
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
Aff(n), the stabilizer of a point being the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(n). An affine n-manifold is then a manifold which looks infinitesimally like n-dimensional affine space.
Motivation from tensor calculus
The second motivation for affine connections comes from the notion of a covariant derivativeCovariant 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 differential operator, to be contrasted with the approach given...
of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields using their components in coordinate charts. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates. Correction terms were introduced by Elwin Bruno Christoffel
Elwin Bruno Christoffel
Elwin Bruno Christoffel was a German mathematician and physicist.-Life:...
(following ideas of Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly
Covariant transformation
In physics, a covariant transformation is a rule , that describes how certain physical entities change under a change of coordinate system....
under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols. This idea was developed into the theory of the absolute differential calculus (now known as tensor calculus) by Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....
and his student Tullio Levi-Civita
Tullio Levi-Civita
Tullio Levi-Civita, FRS was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus...
between 1880 and the turn of the 20th century.
The tensor calculus really came to life, however, with the advent of Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It 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
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
. More general affine connections were then studied around 1920, by Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, who developed a detailed mathematical foundation for general relativity, and Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
, 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
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
and gauge covariant derivative
Gauge covariant derivative
The gauge covariant derivative is like a generalization of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations...
s. On the other hand, the notion of covariant differentiation was abstracted by Jean-Louis Koszul
Jean-Louis Koszul
Jean-Louis Koszul is a mathematician best known for studying geometry and discovering the Koszul complex.He was educated at the Lycée Fustel-de-Coulanges in Strasbourg before studying at the Faculty of Science in Strasbourg and the Faculty of Science in Paris...
, who defined (linear or Koszul) connections
Connection (vector bundle)
In mathematics, a connection on a fiber bundle 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. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
on vector bundle
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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s. In this language, an affine connection is simply a covariant derivative
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 differential operator, to be contrasted with the approach given...
or (linear) connection
Connection (vector bundle)
In mathematics, a connection on a fiber bundle 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. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
.
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 n-space is an affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
. (Alternatively, Euclidean space is a principal homogeneous space
Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
or torsor under the group of translations, which is a subgroup of the affine group.) As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
of vector fields along a curve. This also defines a parallel transport on the frame bundle
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group Aff(n) or as a principal GL(n) connection on the frame bundle.
Formal definition as a differential operator
Let M be a smooth manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
and let C∞(M,TM) be the space of vector fields on M, that is, the space of smooth sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
TM. Then an affine connection on M is a bilinear map
-
such that for all smooth functions f ∈ C∞(M,R) and all vector fields X, Y on M:-
, that is, ∇ is C∞(M,R)-linear in the first variable; -
, that is, ∇ satisfies Leibniz rule in the second variable.
Elementary properties
- It follows from the property (1) above that the value of ∇XY at a point x ∈ M depends only on the value of X at x and not on the value of X on M−{x}. It also follows from property (2) above that the value of ∇XY at a point x ∈ M depends only on the value of Y on a neighbourhood of x.
- If ∇1 and ∇2 are affine connections then the value at x of ∇1XY - ∇2XY may be written Γx(Xx,Yx) where
-
- Γx: TxM × TxM → TxM
- is bilinear and depends smoothly on x (i.e., it defines a smooth bundle homomorphism). Conversely if ∇ is an affine connection and Γ is such a smooth bilinear bundle homomorphism (called a connection formConnection formIn 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....
on M) then ∇+Γ is an affine connection.
- If M is an open subset of Rn, then the tangent bundle of M is the trivial bundle M×Rn. In this situation there is a canonical affine connection d on M: any vector field Y is given by a smooth function V from M to Rn; then dXY is the vector field corresponding to the smooth function dV(X)=∂XY from M to Rn. Any other affine connection ∇ on M may therefore be written ∇ = d +Γ, where Γ is a connection form on M.
- More generally, a local trivialization of the tangent bundle is a bundle isomorphismBundle mapIn mathematics, a bundle map 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 are also several variations on the basic theme, depending on precisely...
between the restriction of TM to an open subset U of M, and M×Rn. The restriction of an affine connection ∇ to U may then be written in the form d + Γ where Γ is a connection form on U.
Parallel transport for affine connections
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 transportParallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
, and indeed this can be used to give a definition of an affine connection.
Let M be a manifold with an affine connection ∇. Then a vector field X is said to be parallel if ∇X = 0
in the sense that for any vector field Y, ∇Y X=0. Intuitively speaking, parallel vectors have all their derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s equal to zero and are therefore in some sense constant. By evaluating a parallel vector field at two points x and y, an identification between a tangent vector at x and one at y is obtained. Such tangent vectors are said to be parallel transports of each other.
Unfortunately, nonzero parallel vector fields do not, in general, exist, because the equation ∇X = 0 is a partial differential equationPartial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
which is overdeterminedOverdetermined systemIn mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constraints. Each unknown can be seen as an available degree of freedom...
: the integrability condition for this equation is the vanishing of the curvature of ∇ (see below). However, if this equation is restricted to a curveCurveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
from x to y it becomes an ordinary differential equationOrdinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
. There is then a unique solution for any initial value of X at x.
More precisely, if γ: I→M a smooth curveCurveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
parametrized by an interval [a,b] and ξ ∈ TxM, where x=γ(a), then a vector fieldVector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
X along γ (and in particular, the value of this vector field at y=γ(b)) is called the parallel transport of ξ along γ if
, for all t ∈ [a,b]
Formally, the first condition means that X is parallel with respect to the pullback connection on the pullback bundlePullback bundleIn mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
γ*TM. However, in a local trivialization it is a first-order system of linear ordinary differential equationsLinear differential equationLinear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
, which has a unique solution for any initial condition given by the second condition (for instance, by the Picard–Lindelöf theoremPicard–Lindelöf theoremIn mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.The theorem is named after Charles...
).
Thus parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve. The isomorphism obtained in this way will in general depend on the choice of the curve: if it does not, then parallel transport along every curve can be used to define parallel vector fields on M, which can only happen if the curvature of ∇ is zero.
A linear isomorphism is determined by its action on an ordered basis or frame. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent) frame bundleFrame bundleIn mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
GL(M) along a curve. In other words, the affine connection provides a lift of any curve γ in M to a curve
in GL(M).
Formal definition on the frame bundle
An affine connection may also be defined as a principal GL(n) connectionConnection (principal bundle)In mathematics, 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...
ω on the frame bundleFrame bundleIn mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
FM or GL(M) of a manifold M. In more detail, ω is a smooth map from the tangent bundle T(FM) of the frame bundle to the space of n×n matrices (which is the Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
gl(n) of the Lie groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
GL(n) of invertible n×n matrices) satisfying two properties:- ω is equivariantEquivariantIn mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant iffor all g ∈ G and all x in X...
with respect to the action of GL(n) on T(FM) and gl(n); - ω(Xξ) = ξ for any ξ in gl(n), where Xξ is the vector field on FM corresponding to ξ.
Such a connection ω immediately defines a covariant derivativeCovariant derivativeIn 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 differential operator, to be contrasted with the approach given...
not only on the tangent bundle, but on vector bundleVector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s associatedAssociated bundleIn mathematics, the theory of fiber bundles with a structure group G 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 with a group action of G...
to any group representationGroup representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of GL(n), including bundles of tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s and tensor densitiesTensor densityIn differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the...
. Conversely, an affine connection on the tangent bundle determines an affine connection on the frame bundle, for instance, by requiring that ω vanishes on tangent vectors to the lifts of curves to the frame bundle defined by parallel transport.
The frame bundle also comes equipped with a solder form θ: T(FM) → Rn which is horizontal in the sense that it vanishes on vertical vectorsVertical bundleThe vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers...
such as the point values of the vector fields Xξ: indeed θ is defined first by projecting a tangent vector (to FM at a frame f) to M, then by taking the components of this tangent vector on M with respect to the frame f. Note that θ is also GL(n)-equivariant (where GL(n) acts on Rn by matrix multiplication).
The pair (θ,ω) define a bundle isomorphism of T(FM) with the trivial bundle FM × aff(n), where aff(n) is the cartesian productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of Rn and gl(n) (viewed as the Lie algebra of the affine group, which is actually a semidirect productSemidirect productIn mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
— see below).
Affine connections as Cartan connections
Affine connections can be defined within Cartan's general framework. In the modern approach, this is closely related to the definition of affine connections on the frame bundle. Indeed, in one formulation, a Cartan connection is an absolute parallelism of a principal bundle satisfying suitable properties. From this point of view the aff(n)-valued 1-form (θ,ω): T(FM) → aff(n) on the frame bundle (of an affine manifold) is a Cartan connection. However, Cartan's original approach was different from this in a number of ways:- the concept of frame bundles or principal bundles did not exist;
- a connection was viewed in terms of parallel transport between infinitesimally nearby points;
- 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 spaceAffine spaceIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
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 spaceTangent spaceIn mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
is really an infinitesimal notion, whereas the planes, as affine subspaces of R3, 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 affineAffine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
rather than linear; the linear parallel transport can be recovered by applying a translation.
Abstracting this idea, an affine manifold should therefore be an n-manifold M with an affine space Ax, of dimension n, attached to each x ∈ M at a marked point ax ∈ Ax, together with a method for transporting elements of these affine spaces along any curve C in M. This method is required to satisfy several properties:- for any two points x, y on C, parallel transport is an affine transformationAffine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
from Ax to Ay; - parallel transport is a defined infinitesimally in the sense that it is differentiable at any point on C and depends only on the tangent vector to C at that point;
- the derivative of the parallel transport at x determines a linear isomorphism from TxM 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 referenceFrames of ReferenceFrames of Reference is a 1960 educational film by Physical Sciences Study Committee.The film was made to be shown in high school physics courses. In the film University of Toronto physics professors Patterson Hume and Donald Ivey explain the distinction between inertial and nonintertial frames of...
transform infinitesimally with respect to parallel transport. (This is the origin of Cartan's method of moving frames.) An affine frame at a point consists of a list (p, e1, ..., en), where p ∈ Ax and the ei form a basis of Tp(Ax). The affine connection is then given symbolically by a first order differential system
defined by a collection of one-forms (θj, ωij). Geometrically, an affine frame undergoes a displacement travelling along a curve γ from γ(t) to γ(t + δt) given (approximately, or infinitesimally) by
Furthermore, the affine spaces Ax are required to be tangent to M in the informal sense that the displacement of ax along γ can be identified (approximately or infinitesimally) with the tangent vector γ(t) to γ at x=γ(t) (which is the infinitesimal displacement of x). Since- ax(γ(t+δt)) - ax(γ(t)) = θ(γ(t))δt,
where θ is defined by θ(X) = θ1(X)e1 + ... + θn(X)en, this identification is given by θ, so the requirement is that θ should be a linear isomorphism at each point.
The tangential affine space Ax is thus identified intuitively with an infinitesimal affine neighborhood of x.
The modern point of view makes all this intuition more precise using principal bundles (the essential idea is to replace a frame or a variable frame by the space of all frames and functions on this space). It also draws on the inspiration of Felix KleinFelix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's Erlangen programme, in which a geometry is defined to be a homogeneous spaceHomogeneous spaceIn 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 continuously by symmetry in a transitive way. A special case of this is when the topological group,...
. 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.
Definition of an affine space
Informally, an affine spaceAffine spaceIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
is a vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
without a fixed choice of originOrigin (mathematics)In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
. It describes the geometry of points and free vectors in space. As a consequence of the lack of origin, points in affine space cannot be added together as this requires a choice of origin with which to form the parallelogram law for vector addition. However, a vector v may be added to a point p by placing the initial point of the vector at p and then transporting p to the terminal point. The operation thus described p → p+v is the translation of p along v. In technical terms, affine n-space is a set An equipped with a free transitive actionGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the vector group Rn on it through this operation of translation of points: An is thus a principal homogeneous spacePrincipal homogeneous spaceIn mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
for the vector group Rn.
The general linear groupGeneral linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(n) is the group of transformations of Rn which preserve the linear structure of Rn in the sense that T(av+bw) = aT(v) + bT(w). By analogy, the affine groupAffine groupIn mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
Aff(n) is the group of transformations of An preserving the affine structure. Thus φ ∈ Aff(n) must preserve translations in the sense that
where T is a general linear transformation. The map sending φ ∈ Aff(n) to T ∈ GL(n) is a group homomorphismGroup homomorphismIn mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
. Its kernelKernel (algebra)In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
is the group of translations Rn. The stabilizer of any point p in A can thus be identified with GL(n) using this projection: this realises the affine group as a semidirect productSemidirect productIn mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of GL(n) and Rn, and affine space as the homogeneous spaceHomogeneous spaceIn 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 continuously by symmetry in a transitive way. A special case of this is when the topological group,...
Aff(n)/GL(n).
Affine frames and the flat affine connection
An affine frame for A consists of a point p ∈ A and a basis (e1,...,en) of the vector space TpA = Rn. The general linear group GL(n) acts freely on the set FA of all affine frames by fixing p and transforming the basis (e1,...,en) in the usual way, and the map π sending an affine frame (p;e1,...,en) to p is the quotient map. Thus FA is a principal GL(n)-bundlePrincipal bundleIn mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
over A. The action of GL(n) extends naturally to a free transitive action of the affine group Aff(n) on FA, so that FA is an Aff(n)-torsorPrincipal homogeneous spaceIn mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
, and the choice of a reference frame identifies FA → A with the principal bundle Aff(n) → Aff(n)/GL(n).
On FA there is a collection of n+1 functions defined by
(as before)
After choosing a basepoint for A, these are all functions with values in Rn, so it is possible to take their exterior derivativeExterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
s to obtain differential 1-forms with values in Rn. Since the functions εi yield a basis for Rn at each point of FA, these 1-forms must be expressible as sums of the form
for some collection (θi, ωjk)1≤i,j,k≤n of real-valued one-forms on Aff(n). This system of one-forms on the principal bundle FA → A defines the affine connection on A.
Taking the exterior derivative a second time, and using the fact that d2=0 as well as the linear independence of the εi, the following relations are obtained:
These are the Maurer-Cartan equations for the Lie group Aff(n) (identified with FA by the choice of a reference frame). Furthermore:- the Pfaffian system θj=0 (for all j) is integrable, and its integral manifolds are the fibres of the principal bundle Aff(n) → A.
- the Pfaffian system ωij=0 (for all i, j) is also integrable, and its integral manifolds define parallel transport in FA.
Thus the forms (ωij) define a flat principal connectionConnection (principal bundle)In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
on FA → A.
For a strict comparison with the motivation, one should actually define parallel transport in a principal Aff(n)-bundle over A. This can be done by pulling backPullback bundleIn mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
FA by the smooth map φ : Rn × A → A defined by translation.
Then the composite φ*FA→FA → A is a principal Aff(n)-bundle over A, and the forms (θi,ωjk) pull back to give a flat principal Aff(n)-connection on this bundle.
General affine geometries: formal definitions
An affine space, as with essentially any smooth Klein geometryKlein geometryIn 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 the symmetry group of the geometry.For background and motivation...
, 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 (θi,ωjk) in the flat model fit together to give a 1-form with values in the Lie algebra aff(n) of the affine group Aff(n).
In these definitions, M is a smooth n-manifold and A = Aff(n)/GL(n) is an affine space of the same dimension.
Definition via absolute parallelism
Let M be a manifold, and P a principal GL(n)-bundle over M. Then an affine connection is a 1-form η on P with values
in aff(n) satisfying the following properties- η is equivariant with respect to the action of GL(n) on P and aff(n);
- η(Xξ) = ξ for all ξ in the Lie algebra gl(n) of all n×n matrices;
- η is a linear isomorphism of each tangent space of P with aff(n).
The last condition means that η is an absolute parallelism on P, i.e., it identifies the tangent bundle of P with a trivial bundle (in this case P × aff(n)). The pair (P,η) defines the structure of an affine geometry on M, making it into an affine manifold.
The affine Lie algebra aff(n) splits as a semidirect product of Rn and gl(n) and so η may be written as a pair (θ,ω) where θ takes values in Rn and ω takes values in gl(n). The conditions (1) and (2) are equivalent to ω being a principal GL(n)-connection and θ being a horizontal equivariant 1-form, which induces a bundle homomorphism from TM to the associated bundleAssociated bundleIn mathematics, the theory of fiber bundles with a structure group G 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 with a group action of G...
P ×GL(n) Rn. The condition (3) is equivalent to the fact that this bundle homomorphism is an isomorphism. (However, this decomposition is a consequence of the rather special structure of the affine group.) Since P is the frame bundleFrame bundleIn mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
of P ×GL(n) Rn, it follows that θ provides a bundle isomorphism between P and the frame bundle FM of M; this recovers the definition of an affine connection as a principal GL(n)-connection on FM.
The 1-forms arising in the flat model are just the components of θ and ω.
Definition as a principal affine connection
An affine connection on M is a principal Aff(n)-bundle Q over M, together with a principal GL(n)-subbundle P of Q
and a principal Aff(n)-connection α (a 1-form on Q with values in aff(n)) which satisfies the following (generic) Cartan condition. The Rn component of pullback of α to P is a horizontal equivariant 1-form and so defines a bundle homomorphism from TM to P ×GL(n) Rn: this is required to be an isomorphism.
Relation to the motivation
Since Aff(n) acts on A, there is, associated to the principal bundle Q, a bundle A = Q ×Aff(n) A, which is a fiber bundle over M whose fiber at x in M is an affine space Ax. A sectionSection (fiber bundle)In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
a of A (defining a marked point ax in Ax for each x ∈ M) determines a principal GL(n)-subbundle P of Q (as the bundle of stabilizers of these marked points) and vice versa. The principal connection α defines an Ehresmann connectionEhresmann connectionIn differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section a always moves under parallel transport.
Curvature and torsion
Curvature and torsion are the main invariants of an affine connection. As there are many equivalent ways to define the notion of an affine connection, so there are many different ways to define curvature and torsion.
From the Cartan connection point of view, the curvature is the failure of the affine connection η to satisfy the Maurer-Cartan equation
where the second term on the left hand side is the wedge product using the Lie bracketLie bracket of vector fieldsIn the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi–Lie bracket, or commutator of vector fields is a bilinear differential operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y]...
in aff(n) to contract the values. By expanding η into the pair (θ,ω) and using the structure of the Lie algebra aff(n), this left hand side can be expanded into the two formulae
where the wedge products are evaluated using matrix multiplication. The first expression is called the torsionTorsion tensorIn differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
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 TM as follows.
The torsion is given by the formula
If the torsion vanishes, the connection is said to be torsion-free or symmetric.
The curvature is given by the formula
When both curvature and torsion vanish, the connection defines a pre-Lie algebraPre-Lie algebraIn mathematics, a pre-Lie algebra is an algebraic structure on a vector space, that describes some properties of objects such as rooted trees and vector fields on affine space....
structure on the space of global sections of the tangent bundle.
The Levi-Civita connection
If (M,g) is a Riemannian manifoldRiemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
then there is a unique affine connection ∇ on M with the following two properties:- the connection is torsion-free, i.e., T∇ is zero;
- parallel transport is an isometry, i.e., the inner products (defined using g) between tangent vectors are preserved.
This connection is called the Levi-Civita connectionLevi-Civita connectionIn Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
.
The second condition means that the connection is a metric connectionMetric connectionIn mathematics, a metric connection is a connection in a vector bundle E equipped with a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve...
in the sense that the Riemannian metric g is parallel: ∇g = 0. In local coordinates the components of the connection form are called Christoffel symbolsChristoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
: because of the uniqueness of the Levi-Civita connection, there is a formula for these components in terms of the components of g.
Geodesics
Since straight lines are a concept in affine geometry, affine connections define a generalized notion of (parametrized) straight lines on any affine manifold, called affine geodesics. Abstractly, a parametric curve γ : I → M is a straight line if its tangent vector remains parallel and equipollent with itself when it is transported along γ. From the linear point of view, an affine connection M distinguishes the affine geodesics in the following way: a smooth curve γ: I → M is an affine geodesic if
is parallel transported along 
, that is
where τts : TγsM → TγtM is the parallel transport map defining the connection.
In terms of the infinitesimal connection ∇, the derivative of this equation implies
for all t ∈ I.
Conversely, any solution of this differential equation yields a curve whose tangent vector is parallel transported along the curve. For every x ∈ M and every X ∈ TxM, there exists a unique affine geodesic γ: I → M with γ(0) = x and
and where I is the maximal open interval in R, containing 0, on which the geodesic is defined. This follows from the Picard–Lindelöf theoremPicard–Lindelöf theoremIn mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.The theorem is named after Charles...
, and allows for the definition of an exponential mapExponential mapIn differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
associated to the affine connection.
In particular, when M is a (pseudoPseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
-)Riemannian manifoldRiemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
and ∇ is the Levi-Civita connectionLevi-Civita connectionIn Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
, then the affine geodesic are the usual geodesicGeodesicIn mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
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 M determines a parametric curve γ through the line up to a choice of affine reparametrization γ(t) → γ(at+b), where a and b are constants. The tangent vector to an affine geodesic is parallel and equipollent along itself. An unparametrized geodesic, or one which is merely parallel along itself without necessarily being equipollent, need only satisfy
for some function k defined along γ. Unparametrized geodesics are often studied from the point of view of projective connectionProjective connectionIn 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 an affine connection. Much like affine connections,...
s.
Development
An affine connection defines a notion of developmentDevelopment (differential geometry)In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface at a point can be rolled around the surface to obtain the tangent-plane at other points.The tangential contact...
of curves. Intuitively, development captures the notion that if xt is a curve in M, then the affine tangent space at x0 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 Ct in this affine space: the development of xt.
In formal terms, let τt0 : TxtM → Tx0M be the linear parallel transport map associated to the affine connection. Then the development Ct is the curve in Tx0M starts off at 0 and is parallel to the tangent of xt for all time t:
In particular, xt is a geodesic if and only if its development is an affinely parametrized straight line in Tx0M.
Surface theory revisited
If M is a surface in R3, it is easy to see that M has a natural affine connection. From the linear connection point of view, the covariant derivative of a vector field is defined by differentiating the vector field, viewed as a map from M to R3, and then projecting the result orthogonally back onto the tangent spaces of M. 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 M induced by the inner product on R3, hence it is the Levi-Civita connection of this metric.
Example: the unit sphere in Euclidean space
Let
be the usual scalar product on R3, and let 
be the unit sphere. The tangent space to 
at a point x is naturally identified with the vector sub-space of R3 consisting of all vectors orthogonal to x. It follows that a vector field 
on 
can be seen as a map
which satisfies
Denote by dY the differential of such a map. Then we have:
Lemma. The formula
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 x in 
Consider the map
The map
is constant, hence its differential vanishes. In particular
The equation (1) above follows.
See also
- Atlas (topology)Atlas (topology)In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
- Chart (topology)
- Differentiable manifoldDifferentiable manifoldA differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
- Differential geometry
- Introduction to mathematics of general relativity
- List of formulas in Riemannian geometry
- Riemannian geometryRiemannian geometryRiemannian 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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
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 geometryRiemannian geometryRiemannian 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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
. Robert Hermann's appendices discuss the motivation from surface theory, as well as the notion of affine connections in the modern sense of Koszul. He develops the basic properties of the differential operator ∇, and relates them to the classical affine connections in the sense of Cartan.
Secondary references
.-
- This is the main reference for the technical details of the article. Volume 1, chapter III gives a detailed account of affine connections from the perspective of principal bundles on a manifold, parallel transport, development, geodesics, and associated differential operators. Volume 1 chapter VI gives an account of affine transformations, torsion, and the general theory of affine geodesy. Volume 2 gives a number of applications of affine connections to homogeneous spaces and complex manifolds, as well as to other assorted topics...
- Two articles by Lumiste, giving precise conditions on parallel transport maps in order that they define affine connections. They also treat curvature, torsion, and other standard topics from a classical (non-principal bundle) perspective..
- This fills in some of the historical details, and provides a more reader-friendly elementary account of Cartan connections in general. Appendix A elucidates the relationship between the principal connection and absolute parallelism viewpoints. Appendix B bridges the gap between the classical "rolling" model of affine connections, and the modern one based on principal bundles and differential operators..
- The section 3. Cartan Connections [pages 127-130] treats conformal and projective connections in a unified manner.
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