Connection form
Encyclopedia
In mathematics
, and specifically differential geometry, a connection form is a manner of organizing the data of a connection
using the language of moving frame
s and differential form
s.
Historically, connection forms were introduced by Élie Cartan
in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of frame, and so is not a tensor
ial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle
, a principal connection
is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold
, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics
, connection forms are also used broadly in the context of gauge theory
, through the gauge covariant derivative
.
A connection form associates to each basis of a vector bundle
a matrix
of differential forms. The connection form is not tensorial because under a change of basis
, the connection form transforms in a manner that involves the exterior derivative
of the transition functions, in much the same way as the Christoffel symbols
for the Levi-Civita connection
. The main tensorial invariant of a connection form is its curvature form
. In the presence of a solder form
identifying the vector bundle with the tangent bundle
, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle
with a structure group
.
of fibre dimension k over a differentiable manifold
M. A local frame for E is an ordered basis of local sections
of E.
Let e=(eα)α=1,2,...,k be a local frame on E. This frame can be used to express locally any section of E. For suppose that ξ is a local section, defined over the same open set as the frame e, then
where ξα(e) denotes the components of ξ in the frame e. As a matrix equation, this reads
in E is a type of differential operator
where Γ denotes the sheaf
of local sections of a vector bundle, and Ω1M is the bundle of differential 1-forms on M. For D to be a connection, it must be correctly coupled to the exterior derivative
. Specifically, if v is a local section of E, and f is a smooth function, then
where df is the exterior derivative of f.
Sometimes it is convenient to extend the definition of D to arbitrary E-valued forms, thus regarding it as a differential operator on the tensor product of E with the full exterior algebra
of differential forms. Given an exterior connection D satisfying this compatibility property, there exists a unique extension of D:
such that
where v is homogeneous of degree deg v. In other words, D is a derivation
on the sheaf of graded modules Γ(E ⊗ Ω*M).
s on M such that
In terms of the connection form, the exterior connection of any section of E can now be expressed, for suppose that ξ = Σα eαξα. Then
Taking components on both sides,
where it is understood that d and ω refer to the exterior derivative and a matrix of 1-forms, respectively, acting on the components of ξ. Conversely, a matrix of 1-forms ω is a priori sufficient to completely determine the connection locally on the open set over which the basis of sections e is defined.
Suppose that e′ is a different choice of local basis. Then there is an invertible k × k matrix of functions g such that
Applying the exterior connection to both sides gives the transformation law for ω:
Note in particular that ω fails to transform in a tensor
ial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix g.
This compatibility condition ensures in particular that the exterior connection of a section of E, when regarded abstractly as a section of E ⊗ Ω1M, does not depend on the choice of basis section used to define the connection.
Unlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using the Poincaré lemma. Specifically, if e → e g is a change of frame, then the curvature two-form transforms by
One interpretation of this transformation law is as follows. Let e* be the dual basis corresponding to the frame e. Then the 2-form
is independent of the choice of frame. In particular, Ω is a vector-valued two-form on M with values in the endomorphism ring
Hom(E,E). Symbolically,
In terms of the exterior connection D, the curvature endomorphism is given by
for v ∈ E. Thus the curvature measures the failure of the sequence
to be a chain complex
(in the sense of de Rham cohomology
).
. A solder form is a globally defined vector-valued one-form
θ ∈ Γ(Ω1(M,E)) such that the mapping
is a linear isomorphism for all x ∈ M. If a solder form is given, then it is possible to define the torsion of the connection (in terms of the exterior connection) as
The torsion Θ is an E-valued 2-form on M.
A solder form and the associated torsion may both be described in terms of a local frame e of E. If θ is a solder form, then it decomposes into the frame components
The components of the torsion are then
Much like the curvature, it can be shown that Θ behaves as a contravariant tensor under a change in frame:
The frame-independent torsion may also be recovered from the frame components:
on the tangent bundle
of M. A local frame on the tangent bundle is an ordered list of vector fields e = (ei | i = 1,2,...,n=dim M) defined on an open subset of M that are linearly independent at every point of their domain. The Christoffel symbols define the Levi-Civita connection by
If θ = (θi | i=1,2,...,n), denotes the dual basis of the cotangent bundle
, such that θi(ej) = δij (the Kronecker delta), then the connection form is
In terms of the connection form, the exterior connection on a vector field v = Σieivi is given by
One can recover the Levi-Civita connection, in the usual sense, from this by contracting with ei:
For simplicity, suppose that the frame e is holonomic
, so that dθi=0. Then, employing now the summation convention on repeated indices,
where R is the Riemann curvature tensor
.
in the tangent bundle with zero torsion. To describe the torsion, note that the vector bundle E is the tangent bundle. This carries a canonical solder form (sometimes called the canonical one-form) that is the section θ of Hom(TM,TM) = T*M ⊗ TM corresponding to the identity endomorphism of the tangent spaces. In the frame e, the solder form is θ = Σi ei ⊗ θi, where again θi is the dual basis.
The torsion of the connection is given by Θ = D θ, or in terms of the frame components of the solder form by
Assuming again for simplicity that e is holonomic, this expression reduces to
,
which vanishes if and only if Γikj is symmetric on its lower indices.
. This amounts to a preferred class of frames e on E, which are related by a Lie group
G. For example, in the presence of a metric
in E, one works with frames that form an orthonormal basis
at each point. The structure group is then the orthogonal group
, since this group preserves the orthonormality of frames. Other examples include:
In general, let E be a given vector bundle of fibre dimension k and G ⊂ GL(k) a given Lie subgroup of the general linear group of Rk. If (eα) is a local frame of E, then a matrix-valued function (gij): M → G may act on the eα to produce a new frame
Two such frames are G-related. Informally, the vector bundle E has the structure of a G-bundle if a preferred class of frames is specified, all of which are locally G-related to each other. In formal terms, E is a fibre bundle with structure group G whose typical fibre is Rk with the natural action of G as a subgroup of GL(k).
maps always send one G-frame to another. Formally, along a curve γ, the following must hold locally (that is, for sufficiently small values of t):
for some matrix gαβ (which may also depend on t). Differentiation at t=0 gives
where the coefficients ωαβ are in the Lie algebra
g of the Lie group G.
With this observation, the connection form ωαβ defined by
is compatible with the structure if the matrix of one-forms ωαβ(e) takes its values in g.
The curvature form of a compatible connection is, moreover, a g-valued two-form.
where g is a G-valued function defined on an open subset of M, the connection form transforms via
Or, using matrix products:
To interpret each of these terms, recall that g : M → G is a G-valued (locally defined) function. With this in mind,
where ωg is the Maurer-Cartan form
for the group G, here pulled back to M along the function g, and Ad is the adjoint representation
of G on its Lie algebra.
in the 1940s, provides a manner of organizing these many connection forms and the transformation laws connecting them into a single intrinsic form with a single rule for transformation. The disadvantage to this approach is that the forms are no longer defined on the manifold itself, but rather on a larger principal bundle.
for some G-valued function hUV defined on U ∩ V.
Let FGE be the set of all G-frames taken over each point of M. This is a principal G-bundle over M. In detail, using the fact that the G-frames are all G-related, FGE can be realized in terms of gluing data among the sets of the open cover:
where the equivalence relation
~ is defined by
On FGE, define a principal G-connection
as follows, by specifying a g-valued one-form on each product U × G, which respects the equivalence relation on the overlap regions. First let
be the projection maps. Now, for a point (x,g) ∈ U × G, set
The 1-form ω constructed in this way respects the transitions between overlapping sets, and therefore descends to give a globally defined 1-form on the principal bundle FGE. It can be shown that ω is a principal connection in the sense that it reproduces the generators of the right G action on FGE, and equivariantly intertwines the right action on T(FGE) with the adjoint representation of G.
Changing frames by a G-valued function g, one sees that ω(e) transforms in the required manner by using the Leibniz rule, and the adjunction:
where X is a vector on M, and d denotes the pushforward.
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...
, and specifically differential geometry, a connection form is a manner of organizing the data of a connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
using the language of moving frame
Moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.-Introduction:...
s and differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s.
Historically, connection forms were introduced 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...
in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of frame, and so is not a tensor
Tensor
Tensors 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...
ial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle
Principal bundle
In 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...
, 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...
is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold
Differentiable manifold
A 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...
, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, connection forms are also used broadly in the context 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...
, through the 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...
.
A connection form associates to each basis of a 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...
a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
of differential forms. The connection form is not tensorial because under a change of basis
Change of basis
In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...
, the connection form transforms in a manner that involves the exterior derivative
Exterior derivative
In 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....
of the transition functions, in much the same way as the Christoffel symbols
Christoffel symbols
In 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...
for 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 main tensorial invariant of a connection form is its curvature form
Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
. In the presence of a solder form
Solder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
identifying the vector bundle with 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...
, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
with a structure group
Lie group
In 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...
.
Frames on a vector bundle
Let E be a vector bundleVector 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...
of fibre dimension k over a differentiable manifold
Differentiable manifold
A 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...
M. A local frame for E is an ordered basis of local 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 E.
Let e=(eα)α=1,2,...,k be a local frame on E. This frame can be used to express locally any section of E. For suppose that ξ is a local section, defined over the same open set as the frame e, then
where ξα(e) denotes the components of ξ in the frame e. As a matrix equation, this reads
Exterior connections
A connectionConnection (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...
in E is a type of differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
where Γ denotes the sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of local sections of a vector bundle, and Ω1M is the bundle of differential 1-forms on M. For D to be a connection, it must be correctly coupled to the exterior derivative
Exterior derivative
In 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....
. Specifically, if v is a local section of E, and f is a smooth function, then
where df is the exterior derivative of f.
Sometimes it is convenient to extend the definition of D to arbitrary E-valued forms, thus regarding it as a differential operator on the tensor product of E with the full exterior algebra
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
of differential forms. Given an exterior connection D satisfying this compatibility property, there exists a unique extension of D:
such that
where v is homogeneous of degree deg v. In other words, D is a derivation
Derivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
on the sheaf of graded modules Γ(E ⊗ Ω*M).
Connection forms
The connection form arises when applying the exterior connection to a particular frame e. Upon applying the exterior connection to the eα, it is the unique k × k matrix (ωαβ) of one-formOne-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
s on M such that
In terms of the connection form, the exterior connection of any section of E can now be expressed, for suppose that ξ = Σα eαξα. Then
Taking components on both sides,
where it is understood that d and ω refer to the exterior derivative and a matrix of 1-forms, respectively, acting on the components of ξ. Conversely, a matrix of 1-forms ω is a priori sufficient to completely determine the connection locally on the open set over which the basis of sections e is defined.
Change of frame
In order to extend ω to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of E is chosen. Write ωαβ = ωαβ(e) to indicate the dependence on the choice of e.Suppose that e′ is a different choice of local basis. Then there is an invertible k × k matrix of functions g such that
Applying the exterior connection to both sides gives the transformation law for ω:
Note in particular that ω fails to transform in a tensor
Tensor
Tensors 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...
ial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix g.
Global connection forms
If {Up} is an open covering of M, and each Up is equipped with a trivialization ep of E, then it is possible to define a global connection form in terms of the patching data between the local connection forms on the overlap regions. In detail, a connection form on M is a system of matrices ω(ep) of 1-forms defined on each Up that satisfy the following compatibility conditionThis compatibility condition ensures in particular that the exterior connection of a section of E, when regarded abstractly as a section of E ⊗ Ω1M, does not depend on the choice of basis section used to define the connection.
Curvature
The curvature two-form of a connection form in E is defined byUnlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using the Poincaré lemma. Specifically, if e → e g is a change of frame, then the curvature two-form transforms by
One interpretation of this transformation law is as follows. Let e* be the dual basis corresponding to the frame e. Then the 2-form
is independent of the choice of frame. In particular, Ω is a vector-valued two-form on M with values in the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
Hom(E,E). Symbolically,
In terms of the exterior connection D, the curvature endomorphism is given by
for v ∈ E. Thus the curvature measures the failure of the sequence
to be a chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
(in the sense of de Rham cohomology
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
).
Soldering and torsion
Suppose that the fibre dimension k of E is equal to the dimension of the manifold M. In this case, the vector bundle E is sometimes equipped with an additional piece of data besides its connection: a solder formSolder form
In mathematics, more precisely in differential geometry, a soldering of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent...
. A solder form is a globally defined vector-valued one-form
Vector-valued form
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms...
θ ∈ Γ(Ω1(M,E)) such that the mapping
is a linear isomorphism for all x ∈ M. If a solder form is given, then it is possible to define the torsion of the connection (in terms of the exterior connection) as
The torsion Θ is an E-valued 2-form on M.
A solder form and the associated torsion may both be described in terms of a local frame e of E. If θ is a solder form, then it decomposes into the frame components
The components of the torsion are then
Much like the curvature, it can be shown that Θ behaves as a contravariant tensor under a change in frame:
The frame-independent torsion may also be recovered from the frame components:
Example: The Levi-Civita connection
As an example, suppose that M carries a Riemannian metric, and consider the Levi-Civita connectionLevi-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...
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...
of M. A local frame on the tangent bundle is an ordered list of vector fields e = (ei | i = 1,2,...,n=dim M) defined on an open subset of M that are linearly independent at every point of their domain. The Christoffel symbols define the Levi-Civita connection by
If θ = (θi | i=1,2,...,n), denotes the dual basis of the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
, such that θi(ej) = δij (the Kronecker delta), then the connection form is
In terms of the connection form, the exterior connection on a vector field v = Σieivi is given by
One can recover the Levi-Civita connection, in the usual sense, from this by contracting with ei:
Curvature
The curvature 2-form of the Levi-Civita connection is the matrix (Ωij) given byFor simplicity, suppose that the frame e is holonomic
Holonomic
In mathematics and physics, the term holonomic may occur with several different meanings.-Holonomic basis:A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish:[e_j,e_k]=0 \,...
, so that dθi=0. Then, employing now the summation convention on repeated indices,
where R is the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
.
Torsion
The Levi-Civita connection is characterized as the unique metric connectionMetric connection
In 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 tangent bundle with zero torsion. To describe the torsion, note that the vector bundle E is the tangent bundle. This carries a canonical solder form (sometimes called the canonical one-form) that is the section θ of Hom(TM,TM) = T*M ⊗ TM corresponding to the identity endomorphism of the tangent spaces. In the frame e, the solder form is θ = Σi ei ⊗ θi, where again θi is the dual basis.
The torsion of the connection is given by Θ = D θ, or in terms of the frame components of the solder form by
Assuming again for simplicity that e is holonomic, this expression reduces to
,which vanishes if and only if Γikj is symmetric on its lower indices.
Structure groups
A more specific type of connection form can be constructed when the vector bundle E carries a structure groupAssociated bundle
In 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...
. This amounts to a preferred class of frames e on E, which are related by a Lie group
Lie group
In 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...
G. For example, in the presence of a metric
Metric (vector bundle)
In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle. Specifically, if M is a topological manifold and E → M a vector bundle on M, then a metric on E is a bundle map g : E ×M E → M × R from the fiber product of E with itself...
in E, one works with frames that form an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
at each point. The structure group is then the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, since this group preserves the orthonormality of frames. Other examples include:
- The usual frames, considered in the preceding section, have structural group GL(k) where k is the fibre dimension of E.
- The holomorphic tangent bundle of a complex manifoldComplex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
(or almost complex manifoldAlmost complex manifoldIn mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...
). Here the structure group is GLn(C) ⊂ GL2n(R). In case a hermitian metric is given, then the structure group reduces to the unitary groupUnitary groupIn mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
acting on unitary frames. - SpinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s on a manifold equipped with a spin structureSpin structureIn differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
. The frames are unitary with respect to an invariant inner product on the spin space, and the group reduces to the spin group. - Holomorphic tangent bundles on CR manifoldCR manifoldIn mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge....
s.
In general, let E be a given vector bundle of fibre dimension k and G ⊂ GL(k) a given Lie subgroup of the general linear group of Rk. If (eα) is a local frame of E, then a matrix-valued function (gij): M → G may act on the eα to produce a new frame
Two such frames are G-related. Informally, the vector bundle E has the structure of a G-bundle if a preferred class of frames is specified, all of which are locally G-related to each other. In formal terms, E is a fibre bundle with structure group G whose typical fibre is Rk with the natural action of G as a subgroup of GL(k).
Compatible connections
A connection is compatible with the structure of a G-bundle on E provided that the associated parallel transportParallel 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...
maps always send one G-frame to another. Formally, along a curve γ, the following must hold locally (that is, for sufficiently small values of t):
for some matrix gαβ (which may also depend on t). Differentiation at t=0 gives
where the coefficients ωαβ are in the Lie algebra
Lie algebra
In 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...
g of the Lie group G.
With this observation, the connection form ωαβ defined by
is compatible with the structure if the matrix of one-forms ωαβ(e) takes its values in g.
The curvature form of a compatible connection is, moreover, a g-valued two-form.
Change of frame
Under a change of framewhere g is a G-valued function defined on an open subset of M, the connection form transforms via
Or, using matrix products:
To interpret each of these terms, recall that g : M → G is a G-valued (locally defined) function. With this in mind,
where ωg is the Maurer-Cartan form
Maurer-Cartan form
In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G...
for the group G, here pulled back to M along the function g, and Ad is the adjoint representation
Adjoint representation
In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
of G on its Lie algebra.
Principal bundles
The connection form, as introduced thus far, depends on a particular choice of frame. In the first definition, the frame is just a local basis of sections. To each frame, a connection form is given with a transformation law for passing from one frame to another. In the second definition, the frames themselves carry some additional structure provided by a Lie group, and changes of frame are constrained to those that take their values in it. The language of principal bundles, pioneered by Charles EhresmannCharles Ehresmann
Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...
in the 1940s, provides a manner of organizing these many connection forms and the transformation laws connecting them into a single intrinsic form with a single rule for transformation. The disadvantage to this approach is that the forms are no longer defined on the manifold itself, but rather on a larger principal bundle.
The principal connection for a connection form
Suppose that E → M is a vector bundle with structure group G. Let {U} be an open cover of M, along with G-frames on each U, denoted by eU. These are related on the intersections of overlapping open sets byfor some G-valued function hUV defined on U ∩ V.
Let FGE be the set of all G-frames taken over each point of M. This is a principal G-bundle over M. In detail, using the fact that the G-frames are all G-related, FGE can be realized in terms of gluing data among the sets of the open cover:
where the equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
~ is defined by
On FGE, define a principal G-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...
as follows, by specifying a g-valued one-form on each product U × G, which respects the equivalence relation on the overlap regions. First let
be the projection maps. Now, for a point (x,g) ∈ U × G, set
The 1-form ω constructed in this way respects the transitions between overlapping sets, and therefore descends to give a globally defined 1-form on the principal bundle FGE. It can be shown that ω is a principal connection in the sense that it reproduces the generators of the right G action on FGE, and equivariantly intertwines the right action on T(FGE) with the adjoint representation of G.
Connection forms associated to a principal connection
Conversely, a principal G-connection ω in a principal G-bundle P→M gives rise to a collection of connection forms on M. Suppose that e : M → P is a local section of P. Then the pullback of ω along e defines a g-valued one-form on M:Changing frames by a G-valued function g, one sees that ω(e) transforms in the required manner by using the Leibniz rule, and the adjunction:
where X is a vector on M, and d denotes the pushforward.
See also
- 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...
- Cartan connectionCartan connectionIn 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...
- Affine connectionAffine connectionIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
- Curvature formCurvature formIn differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...