Inmathematics, and specificallydifferential geometry, aconnection form is a manner of organizing the data of aconnection using the language ofmoving frames anddifferential forms.

Historically, connection forms were introduced byÉlie 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 acoordinate frame, and so is not atensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on aprincipal bundle, aprincipal 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 thedifferentiable 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.^[1] Inphysics, connection forms are also used broadly in the context ofgauge theory, through thegauge covariant derivative.

A connection form associates to eachbasis of avector bundle amatrix of differential forms. The connection form is not tensorial because under achange of basis, the connection form transforms in a manner that involves theexterior derivative of thetransition functions, in much the same way as theChristoffel symbols for theLevi-Civita connection. The maintensorial invariant of a connection form is itscurvature form. In the presence of asolder form identifying the vector bundle with thetangent bundle, there is an additional invariant: thetorsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of afiber bundle with astructure group.

Let${\displaystyle E}$ be avector bundle of fibre dimension${\displaystyle k}$ over adifferentiable manifold${\displaystyle M}$. Alocal frame for${\displaystyle E}$ is an orderedbasis oflocal sections of${\displaystyle E}$. It is always possible to construct a local frame, as vector bundles are always defined in terms oflocal trivializations, in analogy to theatlas of a manifold. That is, given any point${\displaystyle x}$ on the base manifold${\displaystyle M}$, there exists an open neighborhood${\displaystyle U\subseteq M}$ of${\displaystyle x}$ for which the vector bundle over${\displaystyle U}$ is locally trivial, that is isomorphic to${\displaystyle U\times {\mathbb{R}}^k}$ projecting to${\displaystyle U}$. The vector space structure on${\displaystyle {\mathbb{R}}^k}$ can thereby be extended to the entire local trivialization, and a basis on${\displaystyle {\mathbb{R}}^k}$ can be extended as well; this defines the local frame. (Here the real numbers are used, although much of the development can be extended to modules over rings in general, and to vector spaces over complex numbers${\displaystyle \mathbb{C}}$ in particular.)

Let${\displaystyle \mathbf{e}=(e_{\alpha }{)}_{\alpha =1,2,\dots ,k}}$ be a local frame on${\displaystyle E}$. This frame can be used to express locally any section of${\displaystyle E}$. For example, suppose that${\displaystyle \xi }$ is a local section, defined over the same open set as the frame${\displaystyle \mathbb{e}}$. Then

${\displaystyle \xi =\sum _{\alpha =1}^k{e}_{\alpha }{\xi }^{\alpha }(\mathbf{e})}$

where${\displaystyle {\xi }^{\alpha }(\mathbf{e})}$ denotes thecomponents of${\displaystyle \xi }$ in the frame${\displaystyle \mathbf{e}}$. As a matrix equation, this reads

${\displaystyle \xi =\mathbf{e}\left[\begin{array}{c}{\xi }^1(\mathbf{e})\\ {\xi }^2(\mathbf{e})\\ ⋮\\ {\xi }^k(\mathbf{e})\end{array}\right]=\mathbf{e}\xi (\mathbf{e})}$

Ingeneral relativity, such frame fields are referred to astetrads. The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold${\displaystyle M}$ (the coordinate system on${\displaystyle M}$ being established by the atlas).

Aconnection inE is a type ofdifferential operator

${\displaystyle D:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E\otimes T^{\ast }M)=\mathrm{\Gamma }(E)\otimes {\mathrm{\Omega }}^{1}M}$

where Γ denotes thesheaf of localsections of a vector bundle, and Ω¹M is the bundle of differential 1-forms onM. ForD to be a connection, it must be correctly coupled to theexterior derivative. Specifically, ifv is a local section ofE, andf is a smooth function, then

${\displaystyle D(fv)=v\otimes (df)+fDv}$

wheredf is the exterior derivative off.

Sometimes it is convenient to extend the definition ofD to arbitraryE-valued forms, thus regarding it as a differential operator on the tensor product ofE with the fullexterior algebra of differential forms. Given an exterior connectionD satisfying this compatibility property, there exists a unique extension ofD:

${\displaystyle D:\mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^{\ast }{T}^{\ast }M)\to \mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^{\ast }{T}^{\ast }M)}$

such that

${\displaystyle D(v\wedge \alpha )=(Dv)\wedge \alpha +(-1{)}^{\text{deg}v}v\wedge d\alpha }$

wherev is homogeneous of degree degv. In other words,D is aderivation on the sheaf of graded modules Γ(E ⊗ Ω^*M).

Theconnection form arises when applying the exterior connection to a particular framee. Upon applying the exterior connection to thee_α, it is the uniquek ×k matrix (ω_α^β) ofone-forms onM such that

${\displaystyle D{e}_{\alpha }=\sum _{\beta =1}^k{e}_{\beta }\otimes {\omega }_{\alpha }^{\beta }.}$

In terms of the connection form, the exterior connection of any section ofE can now be expressed. For example, suppose thatξ = Σ_αe_αξ^α. Then

${\displaystyle D\xi =\sum _{\alpha =1}^{k}D(e_{\alpha }{\xi }^{\alpha }(\mathbf{e}))=\sum _{\alpha =1}^k{e}_{\alpha }\otimes d{\xi }^{\alpha }(\mathbf{e})+\sum _{\alpha =1}^k\sum _{\beta =1}^k{e}_{\beta }\otimes {\omega }_{\alpha }^{\beta }{\xi }^{\alpha }(\mathbf{e}).}$

Taking components on both sides,

${\displaystyle D\xi (\mathbf{e})=d\xi (\mathbf{e})+\omega \xi (\mathbf{e})=(d+\omega )\xi (\mathbf{e})}$

where it is understood thatd and ω refer to the component-wise derivative with respect to the framee, and a matrix of 1-forms, respectively, acting on the components ofξ. Conversely, a matrix of 1-formsω isa priori sufficient to completely determine the connection locally on the open set over which the basis of sectionse is defined.

In order to extendω to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections ofE is chosen. Writeω_α^β =ω_α^β(e) to indicate the dependence on the choice ofe.

Suppose thate′ is a different choice of local basis. Then there is an invertiblek ×k matrix of functionsg such that

${\displaystyle {\mathbf{e}}^{\prime }=\mathbf{e}g,\text{i.e., }{e}_{\alpha }^{\prime }=\sum _{\beta }{e}_{\beta }{g}_{\alpha }^{\beta }.}$

Applying the exterior connection to both sides gives the transformation law forω:

${\displaystyle \omega (\mathbf{e}g)=g^{-1}dg+g^{-1}\omega (\mathbf{e})g.}$

Note in particular thatω fails to transform in atensorial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrixg.

If {U_p} is an open covering ofM, and eachU_p is equipped with a trivializatione_p ofE, 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, aconnection form onM is a system of matricesω(e_p) of 1-forms defined on eachU_p that satisfy the following compatibility condition

${\displaystyle \omega ({\mathbf{e}}_q)=({\mathbf{e}}_p^{-1}{\mathbf{e}}_q{)}^{-1}d({\mathbf{e}}_p^{-1}{\mathbf{e}}_q)+({\mathbf{e}}_p^{-1}{\mathbf{e}}_q{)}^{-1}\omega ({\mathbf{e}}_p)({\mathbf{e}}_p^{-1}{\mathbf{e}}_q).}$

Thiscompatibility condition ensures in particular that the exterior connection of a section ofE, when regarded abstractly as a section ofE ⊗ Ω¹M, does not depend on the choice of basis section used to define the connection.

Thecurvature two-form of a connection form inE is defined by

${\displaystyle \mathrm{\Omega }(\mathbf{e})=d\omega (\mathbf{e})+\omega (\mathbf{e})\wedge \omega (\mathbf{e}).}$

Unlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using thePoincaré lemma. Specifically, ife →eg is a change of frame, then the curvature two-form transforms by

${\displaystyle \mathrm{\Omega }(\mathbf{e}g)=g^{-1}\mathrm{\Omega }(\mathbf{e})g.}$

One interpretation of this transformation law is as follows. Lete^* be thedual basis corresponding to the framee. Then the 2-form

${\displaystyle \mathrm{\Omega }=\mathbf{e}\mathrm{\Omega }(\mathbf{e}){\mathbf{e}}^{\ast }}$

is independent of the choice of frame. In particular, Ω is a vector-valued two-form onM with values in theendomorphism ring Hom(E,E). Symbolically,

${\displaystyle \mathrm{\Omega }\in \mathrm{\Gamma }({\mathrm{\Lambda }}^2{T}^{\ast }M\otimes \text{Hom}(E,E)).}$

In terms of the exterior connectionD, the curvature endomorphism is given by

${\displaystyle \mathrm{\Omega }(v)=D(Dv)=D^{2}v}$

forv ∈E (we can extendv to a local section to define this expression). Thus the curvature measures the failure of the sequence

${\displaystyle \mathrm{\Gamma }(E)\stackrel{D}{\to }\mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^1{T}^{\ast }M)\stackrel{D}{\to }\mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^2{T}^{\ast }M)\stackrel{D}{\to }\dots \stackrel{D}{\to }\mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^n{T}^{\ast }(M))}$

to be achain complex (in the sense ofde Rham cohomology).

Suppose that the fibre dimensionk ofE is equal to the dimension of the manifoldM. In this case, the vector bundleE is sometimes equipped with an additional piece of data besides its connection: asolder form. Asolder form is a globally definedvector-valued one-form θ ∈ Ω¹(M,E) such that the mapping

${\displaystyle {\theta }_x:T_{x}M\to E_x}$

is a linear isomorphism for allx ∈M. If a solder form is given, then it is possible to define thetorsion of the connection (in terms of the exterior connection) as

${\displaystyle \mathrm{\Theta }=D\theta .}$

The torsion Θ is anE-valued 2-form onM.

A solder form and the associated torsion may both be described in terms of a local framee ofE. If θ is a solder form, then it decomposes into the frame components

${\displaystyle \theta =\sum _i{\theta }^i(\mathbf{e})e_i.}$

The components of the torsion are then

${\displaystyle {\mathrm{\Theta }}^i(\mathbf{e})=d{\theta }^i(\mathbf{e})+\sum _j{\omega }_j^i(\mathbf{e})\wedge {\theta }^j(\mathbf{e}).}$

Much like the curvature, it can be shown that Θ behaves as acontravariant tensor under a change in frame:

${\displaystyle {\mathrm{\Theta }}^i(\mathbf{e}g)=\sum _j{g}_j^i{\mathrm{\Theta }}^j(\mathbf{e}).}$

The frame-independent torsion may also be recovered from the frame components:

${\displaystyle \mathrm{\Theta }=\sum _i{e}_i{\mathrm{\Theta }}^i(\mathbf{e}).}$

TheBianchi identities relate the torsion to the curvature. The first Bianchi identity states that

${\displaystyle D\mathrm{\Theta }=\mathrm{\Omega }\wedge \theta }$

while the second Bianchi identity states that

${\displaystyle D\mathrm{\Omega }=0.}$

As an example, suppose thatM carries aRiemannian metric. If one has avector bundleE overM, then the metric can be extended to the entire vector bundle, as thebundle metric. One may then define a connection that is compatible with this bundle metric, this is themetric connection. For the special case ofE being thetangent bundleTM, the metric connection is called theRiemannian connection. Given a Riemannian connection, one can always find a unique, equivalent connection that istorsion-free. This is theLevi-Civita connection on the tangent bundleTM ofM.^[2][3]

A local frame on the tangent bundle is an ordered list of vector fieldse = (e_i |i = 1, 2, ...,n), wheren = dimM, defined on an open subset ofM that are linearly independent at every point of their domain. TheChristoffel symbols define the Levi-Civita connection by

${\displaystyle {\mathrm{\nabla }}_{e_i}{e}_j=\sum _{k=1}^n{\mathrm{\Gamma }}_{ij}^k(\mathbf{e})e_k.}$

Ifθ = {θⁱ |i = 1, 2, ...,n}, denotes thedual basis of thecotangent bundle, such thatθⁱ(e_j) =δⁱ_j (theKronecker delta), then the connection form is

${\displaystyle {\omega }_i^j(\mathbf{e})=\sum _k{\mathrm{\Gamma }}^j{}_{ki}(\mathbf{e}){\theta }^k.}$

In terms of the connection form, the exterior connection on a vector fieldv = Σ_ie_ivⁱ is given by

${\displaystyle Dv=\sum _k{e}_k\otimes (d{v}^k)+\sum _{j,k}{e}_k\otimes {\omega }_j^k(\mathbf{e})v^j.}$

One can recover the Levi-Civita connection, in the usual sense, from this by contracting withe_i:

${\displaystyle {\mathrm{\nabla }}_{e_i}v=⟨Dv,e_i⟩=\sum _k{e}_k\left({\mathrm{\nabla }}_{e_i}{v}^k+\sum _j{\mathrm{\Gamma }}_{ij}^k(\mathbf{e})v^j\right)}$

The curvature 2-form of the Levi-Civita connection is the matrix (Ω_i^j) given by

${\displaystyle {\mathrm{\Omega }}_i{}^j(\mathbf{e})=d{\omega }_i{}^j(\mathbf{e})+\sum _k{\omega }_k{}^j(\mathbf{e})\wedge {\omega }_i{}^k(\mathbf{e}).}$

For simplicity, suppose that the framee isholonomic, so thatdθⁱ = 0.^[4] Then, employing now thesummation convention on repeated indices,

whereR is theRiemann curvature tensor.

The Levi-Civita connection is characterized as the uniquemetric connection in the tangent bundle with zero torsion. To describe the torsion, note that the vector bundleE is the tangent bundle. This carries a canonical solder form (sometimes called thecanonical one-form, especially in the context ofclassical mechanics) that is the sectionθ ofHom(TM, TM) = T^∗M ⊗ TM corresponding to the identity endomorphism of the tangent spaces. In the framee, the solder form isθ = Σ_ie_i ⊗θⁱ, where againθⁱ is the dual basis.

The torsion of the connection is given byΘ =Dθ, or in terms of the frame components of the solder form by

${\displaystyle {\mathrm{\Theta }}^i(\mathbf{e})=d{\theta }^i+\sum _j{\omega }_j^i(\mathbf{e})\wedge {\theta }^j.}$

Assuming again for simplicity thate is holonomic, this expression reduces to

${\displaystyle {\mathrm{\Theta }}^i={\mathrm{\Gamma }}^i{}_{kj}{\theta }^k\wedge {\theta }^j}$,

which vanishes if and only if Γⁱ_kj is symmetric on its lower indices.

Given a metric connection with torsion, one can always find a single, unique connection that is torsion-free, this is the Levi-Civita connection. The difference between a Riemannian connection and its associated Levi-Civita connection is thecontorsion tensor.

A more specific type of connection form can be constructed when the vector bundleE carries astructure group. This amounts to a preferred class of framese onE, which are related by aLie groupG. For example, in the presence of ametric inE, one works with frames that form anorthonormal basis at each point. The structure group is then theorthogonal group, since this group preserves the orthonormality of frames. Other examples include:

• The usual frames, considered in the preceding section, have structural group GL(k) wherek is the fibre dimension ofE.
• The holomorphic tangent bundle of acomplex manifold (oralmost complex manifold).^[5] Here the structure group is GL_n(C) ⊂ GL_2n(R).^[6] In case ahermitian metric is given, then the structure group reduces to theunitary group acting on unitary frames.^[5]
• Spinors on a manifold equipped with aspin structure. The frames are unitary with respect to an invariant inner product on the spin space, and the group reduces to thespin group.
• Holomorphic tangent bundles onCR manifolds.^[7]

In general, letE be a given vector bundle of fibre dimensionk andG ⊂ GL(k) a given Lie subgroup of the general linear group ofR^k. If (e_α) is a local frame ofE, then a matrix-valued function (g_i^j):M →G may act on thee_α to produce a new frame

${\displaystyle e_{\alpha }^{\prime }=\sum _{\beta }{e}_{\beta }{g}_{\alpha }^{\beta }.}$

Two such frames areG-related. Informally, the vector bundleE has thestructure of aG-bundle if a preferred class of frames is specified, all of which are locallyG-related to each other. In formal terms,E is afibre bundle with structure groupG whose typical fibre isR^k with the natural action ofG as a subgroup of GL(k).

A connection iscompatible with the structure of aG-bundle onE provided that the associatedparallel transport maps always send oneG-frame to another. Formally, along a curve γ, the following must hold locally (that is, for sufficiently small values oft):

${\displaystyle \mathrm{\Gamma }(\gamma {)}_0^t{e}_{\alpha }(\gamma (0))=\sum _{\beta }{e}_{\beta }(\gamma (t))g_{\alpha }^{\beta }(t)}$

for some matrixg_α^β (which may also depend ont). Differentiation att=0 gives

${\displaystyle {\mathrm{\nabla }}_{\dot{\gamma }(0)}{e}_{\alpha }=\sum _{\beta }{e}_{\beta }{\omega }_{\alpha }^{\beta }(\dot{\gamma }(0))}$

where the coefficients ω_α^β are in theLie algebrag of the Lie groupG.

With this observation, the connection form ω_α^β defined by

${\displaystyle D{e}_{\alpha }=\sum _{\beta }{e}_{\beta }\otimes {\omega }_{\alpha }^{\beta }(\mathbf{e})}$

iscompatible with the structure if the matrix of one-forms ω_α^β(e) takes its values ing.

The curvature form of a compatible connection is, moreover, ag-valued two-form.

Under a change of frame

${\displaystyle e_{\alpha }^{\prime }=\sum _{\beta }{e}_{\beta }{g}_{\alpha }^{\beta }}$

whereg is aG-valued function defined on an open subset ofM, the connection form transforms via

${\displaystyle {\omega }_{\alpha }^{\beta }(\mathbf{e}\cdot g)=(g^{-1}{)}_{\gamma }^{\beta }d{g}_{\alpha }^{\gamma }+(g^{-1}{)}_{\gamma }^{\beta }{\omega }_{\delta }^{\gamma }(\mathbf{e})g_{\alpha }^{\delta }.}$

Or, using matrix products:

${\displaystyle \omega (\mathbf{e}\cdot g)=g^{-1}dg+g^{-1}\omega g.}$

To interpret each of these terms, recall thatg :M →G is aG-valued (locally defined) function. With this in mind,

${\displaystyle \omega (\mathbf{e}\cdot g)=g^{\ast }{\omega }_{\mathfrak{g}}+{\text{Ad}}_{g^{-1}}\omega (\mathbf{e})}$

where ω_g is theMaurer-Cartan form for the groupG, herepulled back toM along the functiong, and Ad is theadjoint representation ofG on its Lie algebra.

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 byCharles Ehresmann 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.

Suppose thatE →M is a vector bundle with structure groupG. Let {U} be an open cover ofM, along withG-frames on eachU, denoted bye_U. These are related on the intersections of overlapping open sets by

${\displaystyle {\mathbf{e}}_V={\mathbf{e}}_U\cdot h_{UV}}$

for someG-valued functionh_UV defined onU ∩V.

Let F_GE be the set of allG-frames taken over each point ofM. This is a principalG-bundle overM. In detail, using the fact that theG-frames are allG-related, F_GE can be realized in terms of gluing data among the sets of the open cover:

${\displaystyle F_{G}E=\coprod _{U}U\times G/\sim }$

where theequivalence relation${\displaystyle \sim }$ is defined by

${\displaystyle ((x,g_U)\in U\times G)\sim ((x,g_V)\in V\times G)⟺{\mathbf{e}}_V={\mathbf{e}}_U\cdot h_{UV}\text{ and }{g}_U=h_{UV}^{-1}(x)g_V.}$

On F_GE, define aprincipalG-connection as follows, by specifying ag-valued one-form on each productU ×G, which respects the equivalence relation on the overlap regions. First let

${\displaystyle {\pi }_1:U\times G\to U,{\pi }_2:U\times G\to G}$

be the projection maps. Now, for a point (x,g) ∈U ×G, set

${\displaystyle {\omega }_{(x,g)}=A{d}_{g^{-1}}{\pi }_1^{\ast }\omega ({\mathbf{e}}_U)+{\pi }_2^{\ast }{\omega }_{\mathbf{g}}.}$

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 F_GE. It can be shown that ω is a principal connection in the sense that it reproduces the generators of the rightG action on F_GE, and equivariantly intertwines the right action on T(F_GE) with the adjoint representation ofG.

Conversely, a principalG-connection ω in a principalG-bundleP→M gives rise to a collection of connection forms onM. Suppose thate :M →P is a local section ofP. Then the pullback of ω alonge defines ag-valued one-form onM:

${\displaystyle \omega (\mathbf{e})={\mathbf{e}}^{\ast }\omega .}$

Changing frames by aG-valued functiong, one sees that ω(e) transforms in the required manner by using the Leibniz rule, and the adjunction:

${\displaystyle ⟨X,(\mathbf{e}\cdot g{)}^{\ast }\omega ⟩=⟨[d(\mathbf{e}\cdot g)](X),\omega ⟩}$

whereX is a vector onM, andd denotes thepushforward.

• Ehresmann connection
• Cartan connection
• Affine connection
• Curvature form

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• Connection (mathematics)
• Differential geometry
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• Maps of manifolds
• Smooth functions

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