Frames on a vector bundle

Let E be a vector bundle of fibre dimension k over a differentiable manifold M. A local frame for E is an ordered basis of local sections of E. It is always possible to construct a local frame, as vector bundles are always defined in terms of local trivializations, in analogy to the atlas of a manifold. That is, given any point x on the base manifold M, there exists an open neighborhood U ⊂ M of x for which the vector bundle over U is isomorphic to the space U × R^k: this is the local trivialization. The vector space structure on R^k can thereby be extended to the entire local trivialization, and a basis on R^k can be extended as well; this defines the local frame. (Here, R is intended to mean the real numbers ${\displaystyle \mathbb {R} }$, although much of the development here can be extended to modules over rings in general, and to vector spaces over complex numbers ${\displaystyle \mathbb {C} }$ in particular.)

Let e = (e_α)_α=1,2,...,k be a local frame on E. This frame can be used to express locally any section of E. For example, suppose that ξ is a local section, defined over the same open set as the frame e. Then

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

where ξ^α(e) denotes the components of ξ in the frame e. As a matrix equation, this reads

${\displaystyle \xi ={\mathbf {e} }{\begin{bmatrix}\xi ^{1}(\mathbf {e} )\\\xi ^{2}(\mathbf {e} )\\\vdots \\\xi ^{k}(\mathbf {e} )\end{bmatrix}}={\mathbf {e} }\,\xi (\mathbf {e} )}$

In general relativity, such frame fields are referred to as tetrads. The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold M (the coordinate system on M being established by the atlas).

Exterior connections

A connection in E is a type of differential operator

${\displaystyle D:\Gamma (E)\rightarrow \Gamma (E\otimes \Omega ^{1}M)}$

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

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

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:

${\displaystyle D:\Gamma (E\otimes \Omega ^{*}M)\rightarrow \Gamma (E\otimes \Omega ^{*}M)}$

such that

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

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).

Connection forms

The connection form arises when applying the exterior connection to a particular frame e. Upon applying the exterior connection to the e_α, it is the unique k × k matrix (ω_α^β) of one-forms on M such that

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

In terms of the connection form, the exterior connection of any section of E can now be expressed. For example, suppose that ξ = Σ_α e_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 that d and ω refer to the component-wise derivative with respect to the frame e, 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.

Curvature

The curvature two-form of a connection form in E is defined by

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

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

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

One interpretation of this transformation law is as follows. Let e^* be the dual basis corresponding to the frame e. Then the 2-form

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

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,

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

In terms of the exterior connection D, the curvature endomorphism is given by

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

for v ∈ E. Thus the curvature measures the failure of the sequence

${\displaystyle \Gamma (E)\ {\stackrel {D}{\to }}\ \Gamma (E\otimes \Omega ^{1}M)\ {\stackrel {D}{\to }}\ \Gamma (E\otimes \Omega ^{2}M)\ {\stackrel {D}{\to }}\ \dots \ {\stackrel {D}{\to }}\ \Gamma (E\otimes \Omega ^{n}(M))}$

to be a chain complex (in the sense of de Rham cohomology).

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 form. A solder form is a globally defined vector-valued one-form θ ∈ Ω¹(M,E) such that the mapping

${\displaystyle \theta _{x}:T_{x}M\rightarrow E_{x}}$

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

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

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

${\displaystyle \theta =\sum _{i}\theta ^{i}(\mathbf {e} )e_{i}.}$

The components of the torsion are then

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

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

${\displaystyle \Theta ^{i}(\mathbf {e} \,g)=\sum _{j}g_{j}^{i}\Theta ^{j}(\mathbf {e} ).}$

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

${\displaystyle \Theta =\sum _{i}e_{i}\Theta ^{i}(\mathbf {e} ).}$

Bianchi identities

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

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

while the second Bianchi identity states that

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

Example: the Levi-Civita connection

As an example, suppose that M carries a Riemannian metric. If one has a vector bundle E over M, then the metric can be extended to the entire vector bundle, as the bundle metric. One may then define a connection that is compatible with this bundle metric, this is the metric connection. For the special case of E being the tangent bundle TM, the metric connection is called the Riemannian connection. Given a Riemannian connection, one can always find a unique, equivalent connection that is torsion-free. This is the Levi-Civita connection on the tangent bundle TM of M.^[2][3]

A local frame on the tangent bundle is an ordered list of vector fields e = (e_i | i = 1, 2, ..., n), where 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

${\displaystyle \nabla _{e_{i}}e_{j}=\sum _{k=1}^{n}\Gamma _{ij}^{k}(\mathbf {e} )e_{k}.}$

If θ = {θⁱ | i = 1, 2, ..., n}, denotes the dual basis of the cotangent bundle, such that θⁱ(e_j) = δⁱ_j (the Kronecker delta), then the connection form is

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

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

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

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

${\displaystyle \nabla _{e_{i}}v=\langle Dv,e_{i}\rangle =\sum _{k}e_{k}\left(\nabla _{e_{i}}v^{k}+\sum _{j}\Gamma _{ij}^{k}(\mathbf {e} )v^{j}\right)}$

Compatible connections

A connection is compatible with the structure of a G-bundle on E provided that the associated parallel transport 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):

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

for some matrix g_α^β (which may also depend on t). Differentiation at t=0 gives

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

where the coefficients ω_α^β are in the Lie algebra g of the Lie group G.

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

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

is compatible with the structure if the matrix of one-forms ω_α^β(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 frame

${\displaystyle e_{\alpha }'=\sum _{\beta }e_{\beta }g_{\alpha }^{\beta }}$

where g is a G-valued function defined on an open subset of M, the connection form transforms via

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

Or, using matrix products:

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

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

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

where ω_g 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.

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 e_U. These are related on the intersections of overlapping open sets by

${\displaystyle {\mathbf {e} }_{V}={\mathbf {e} }_{U}\cdot h_{UV}}$

for some G-valued function h_UV defined on U ∩ V.

Let F_GE 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, F_GE can be realized in terms of gluing data among the sets of the open cover:

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

where the equivalence relation ${\displaystyle \sim }$ is defined by

${\displaystyle ((x,g_{U})\in U\times G)\sim ((x,g_{V})\in V\times G)\iff {\mathbf {e} }_{V}={\mathbf {e} }_{U}\cdot h_{UV}{\text{ and }}g_{U}=h_{UV}^{-1}(x)g_{V}.}$

On F_GE, 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

${\displaystyle \pi _{1}:U\times G\to U,\quad \pi _{2}:U\times G\to G}$

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

${\displaystyle \omega _{(x,g)}=Ad_{g^{-1}}\pi _{1}^{*}\omega (\mathbf {e} _{U})+\pi _{2}^{*}\omega _{\mathbf {g} }.}$

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 right G action on F_GE, and equivariantly intertwines the right action on T(F_GE) 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:

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

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:

${\displaystyle \langle X,({\mathbf {e} }\cdot g)^{*}\omega \rangle =\langle [d(\mathbf {e} \cdot g)](X),\omega \rangle }$

where X is a vector on M, and d denotes the pushforward.