Connection (vector bundle)

In mathematics, and especially differential geometry and gauge theory, 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. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950).

This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).

Motivation

A section of a vector bundle generalises the notion of a function on a manifold, in the sense that a standard vector-valued function ${\displaystyle f:M\to \mathbb {R} ^{n}}$ can be viewed as a section of the trivial vector bundle ${\displaystyle M\times \mathbb {R} ^{n}\to M}$. It is therefore natural to ask if it is possible to differentiate a section in analogy to how one differentiates a vector field. When the vector bundle is the tangent bundle to a Riemannian manifold, this question is answered naturally by the Levi-Civita connection which is the unique torsion-free connection compatible with the Riemannian metric on the tangent bundle. In general there is no such natural choice of a way to differentiate sections.

A section of a bundle may be viewed as a generalized function from the base into the fibers of the vector bundle. This can be visualized by the graph of the section, as in the figure above.

The model case is to differentiate an ${\displaystyle m}$-component vector field ${\displaystyle X:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ on Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. In this setting the derivative ${\displaystyle dX}$ at a point ${\displaystyle x\in \mathbb {R} ^{n}}$ in the direction ${\displaystyle v\in \mathbb {R} ^{n}}$ may be simply defined by

${\displaystyle dX(v)(x)=\lim _{t\to 0}{\frac {X(x+tv)-X(x)}{t}}.}$

Notice that for every ${\displaystyle x\in \mathbb {R} ^{n}}$, we have defined a new vector ${\displaystyle dX(v)(x)\in \mathbb {R} ^{m}}$ so the derivative of ${\displaystyle X}$ in the direction of ${\displaystyle v}$ has yielded a new ${\displaystyle m}$-component vector field on ${\displaystyle \mathbb {R} ^{n}}$.

When passing to a section ${\displaystyle X\in \Gamma (E)}$ of a vector bundle ${\displaystyle E}$ on a manifold ${\displaystyle M}$, one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term ${\displaystyle x+tv}$ makes no sense on ${\displaystyle M}$. Instead one takes a path ${\displaystyle \gamma$ :(-1,1)\to M} such that ${\displaystyle \gamma (0)=x,\gamma '(0)=v}$ and computes

${\displaystyle dX(v)(x)=\lim _{t\to 0}{\frac {X(\gamma (t))-X(\gamma (0))}{t}}.}$

However this still does not make sense, because ${\displaystyle X(\gamma (t))\in E_{\gamma (t)}}$ is a vector in the fibre over ${\displaystyle \gamma (t)}$, and ${\displaystyle X(\gamma (0))=X(x)\in E_{x}}$, the fibre over ${\displaystyle x}$, which is a different vector space. This means there is no way to make sense of the subtraction of these two terms lying in different vector spaces.

The goal is to resolve the above conundrum by coming up with a way of differentiating sections of a vector bundle in the direction of vector fields, and getting back another section of the vector bundle. There are three possible resolutions to this problem. All three require making a choice of how to differentiate sections, and only in special settings like the tangent bundle on a Riemannian manifold is there a natural such choice.

1. (Parallel transport) Since the problem is that the vectors ${\displaystyle X(\gamma (t))}$ and ${\displaystyle X(x)}$ lie in different fibres of ${\displaystyle E}$, one solution is to define an isomorphism ${\displaystyle P_{t}^{\gamma }:E_{\gamma (t)}\to E_{x}}$ for all ${\displaystyle t}$ close to zero. Using this isomorphism one can transport ${\displaystyle X(\gamma (t))}$ to the fibre ${\displaystyle E_{x}}$ and then take the difference. Explicitly

${\displaystyle dX(v)(x)=\lim _{t\to 0}{\frac {P_{t}^{\gamma }X(\gamma (t))-X(\gamma (0))}{t}}.}$

This is the parallel transport, and the choice of the isomorphisms ${\displaystyle P_{t}^{\gamma }}$ for all curves ${\displaystyle \gamma }$ in ${\displaystyle M}$ can be taken as the definition of how to differentiate a section.

1. (Ehresmann connection) Use the notion of differential of a map of smooth manifolds. A section ${\displaystyle s\in \Gamma (E)}$ is by definition a smooth map ${\displaystyle s:M\to E}$ such that ${\displaystyle \pi \circ s=\operatorname {Id} }$. This has a differential ${\displaystyle ds:TM\to TE}$, with the property that ${\displaystyle ds(X)\in \Gamma (TE)}$ for a vector field ${\displaystyle X\in \Gamma (TM)}$. However, one would like instead for ${\displaystyle ds(X)}$ to be a section of ${\displaystyle E}$ itself. In fact, the vertical bundle ${\displaystyle V\subset TE}$ is the pullback of ${\displaystyle E}$ along ${\displaystyle E}$ with the same fiber as ${\displaystyle E\to M}$. If one chooses a projection ${\displaystyle \nu :TE\to V}$ of vector bundles, composing with this projection would land ${\displaystyle ds(X)}$ back in ${\displaystyle E}$. This is called a linear Ehresmann connection on the vector bundle ${\displaystyle E\to M}$. There are many choices of projection operators ${\displaystyle \nu :TE\to V}$ so in general there are many different ways of differentiating a vector field.

1. (Covariant derivative) The third solution is to abstract the properties that a derivative of a section of a vector bundle should have and take this as an axiomatic definition. This is the notion of a connection or covariant derivative described in this article. The other two approaches above can both be shown to be equivalent to this axiomatic definition of differentiation.

Formal definition

Let E → M be a smooth vector bundle over a differentiable manifold M. Denote the space of smooth sections of E by Γ(E). A connection on E is an ℝ-linear map

${\displaystyle \nabla$ :\Gamma (E)\to \Gamma (T^{*}M\otimes E)}

such that the Leibniz rule

${\displaystyle \nabla (f\sigma )=f\nabla \sigma +df\otimes \sigma }$

holds for all smooth functions f on M and all smooth sections σ of E.

If X is a tangent vector field on M (i.e. a section of the tangent bundle TM) one can define a covariant derivative along X

${\displaystyle \nabla _{X}:\Gamma (E)\to \Gamma (E)}$

by contracting X with the resulting covariant index in the connection: ∇_X σ = (∇σ)(X). The covariant derivative satisfies:

${\displaystyle {\begin{aligned}&\nabla \!_{X}(\sigma _{1}{+}\sigma _{2})\ =\ \nabla \!_{X}\sigma _{1}+\nabla \!_{X}\sigma _{2}\\&\nabla \!_{(X_{1}{+}X_{2})}\sigma \ =\ \nabla \!_{X_{1}}\sigma +\nabla \!_{X_{2}}\sigma \\&\nabla \!_{X}(f\sigma )\ =\ f\,\nabla \!_{X}\sigma +X(f)\sigma \\&\nabla \!_{fX}\sigma \ =\ f\,\nabla \!_{X}\sigma .\end{aligned}}}$

Conversely, any operator satisfying the above properties defines a connection on E and a connection in this sense is also known as a covariant derivative on E.

Induced connections

Given a vector bundle ${\displaystyle E\to M}$, there are many associated bundles to ${\displaystyle E}$ which may be constructed, for example the dual vector bundle ${\displaystyle E^{*}}$, tensor powers ${\displaystyle E^{\otimes k}}$, symmetric and antisymmetric tensor powers ${\displaystyle S^{k}E,\Lambda ^{k}E}$, and the direct sums ${\displaystyle E^{\oplus k}}$. A connection on ${\displaystyle E}$ induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory of principal bundle connections, but here we present some of the basic induced connections.

Given ${\displaystyle \nabla }$ a connection on ${\displaystyle E}$, the induced dual connection ${\displaystyle \nabla ^{*}}$ on ${\displaystyle E^{*}}$ is defined by

${\displaystyle d(\langle \xi ,s\rangle )(X)=\langle \nabla _{X}^{*}\xi ,s\rangle +\langle \xi ,\nabla _{X}s\rangle .}$

Here ${\displaystyle X\in \Gamma (TM)}$ is a smooth vector field, ${\displaystyle s\in \Gamma (E)}$ is a section of ${\displaystyle E}$, and ${\displaystyle \xi \in \Gamma (E^{*})}$ a section of the dual bundle, and ${\displaystyle \langle \cdot ,\cdot \rangle }$ the natural pairing between a vector space and its dual (occurring on each fibre between ${\displaystyle E}$ and ${\displaystyle E^{*}}$). Notice that this definition is essentially enforcing that ${\displaystyle \nabla ^{*}}$ be the connection on ${\displaystyle E^{*}}$ so that a natural product rule is satisfied for pairing ${\displaystyle \langle \cdot ,\cdot \rangle }$.

Given ${\displaystyle \nabla ^{E},\nabla ^{F}}$ connections on two vector bundles ${\displaystyle E,F\to M}$, define the tensor product connection by the formula

${\displaystyle (\nabla ^{E}\otimes \nabla ^{F})_{X}(s\otimes t)=\nabla _{X}^{E}(s)\otimes t+s\otimes \nabla _{X}^{F}(t).}$

Here we have ${\displaystyle s\in \Gamma (E),t\in \Gamma (F),X\in \Gamma (TM)}$. Notice again this is the natural way of combining ${\displaystyle \nabla ^{E},\nabla ^{F}}$ to enforce the product rule for the tensor product connection. Similarly define the direct sum connection by

${\displaystyle (\nabla ^{E}\oplus \nabla ^{F})_{X}(s\oplus t)=\nabla _{X}^{E}(s)\oplus \nabla _{X}^{F}(t),}$

where ${\displaystyle s\oplus t\in \Gamma (E\oplus F)}$.

Since the exterior power and symmetric power of a vector bundle may be viewed as subspaces of the tensor power, ${\displaystyle S^{k}E,\Lambda ^{k}E\subset E^{\otimes k}}$, the definition of the tensor product connection applies in a straightforward manner to this setting. Namely, if ${\displaystyle \nabla }$ is a connection on ${\displaystyle E}$, one has the tensor power connection by repeated applications on the tensor product connection above. We also have the symmetric product connection defined by

${\displaystyle \nabla _{X}^{\odot 2}(s\cdot t)=\nabla _{X}s\cdot t+s\cdot \nabla _{X}t}$

and the exterior product connection defined by

${\displaystyle \nabla _{X}^{\wedge 2}(s\wedge t)=\nabla _{X}s\wedge t+s\wedge \nabla _{X}t}$

for all ${\displaystyle s,t\in \Gamma (E),X\in \Gamma (TM)}$. Repeated applications of these products gives induced symmetric power and exterior power connections on ${\displaystyle S^{k}E}$ and ${\displaystyle \Lambda ^{k}E}$ respectively.

Finally, one obtains the induced connection ${\displaystyle \nabla ^{\operatorname {End} {E}}}$ on the vector bundle ${\displaystyle \operatorname {End} (E)=E^{*}\otimes E}$, the endomorphism connection. This is simply the tensor product connection of the dual connection ${\displaystyle \nabla ^{*}}$ on ${\displaystyle E^{*}}$ and ${\displaystyle \nabla }$ on ${\displaystyle E}$. If ${\displaystyle s\in \Gamma (E)}$ and ${\displaystyle u\in \Gamma (\operatorname {End} (E))}$, so that the composition ${\displaystyle u(s)\in \Gamma (E)}$ also, then the following product rule holds:

${\displaystyle \nabla _{X}(u(s))=\nabla _{X}^{\operatorname {End} (E)}(u)(s)+u(\nabla _{X}(s)).}$

Exterior covariant derivative and vector-valued forms

Let E → M be a vector bundle. An E-valued differential form of degree r is a section of the tensor product bundle:

${\displaystyle E\otimes \bigwedge ^{r}T^{*}M.}$

The space of such forms is denoted by

${\displaystyle \Omega ^{r}(E)=\Omega ^{r}(M;E)=\Gamma \left(E\otimes \bigwedge ^{r}T^{*}M\right).}$

An E-valued 0-form is just a section of the bundle E. That is,

${\displaystyle \Omega ^{0}(E)=\Gamma (E).}$

In this notation a connection on E → M is a linear map

${\displaystyle \nabla$ :\Omega ^{0}(E)\to \Omega ^{1}(E).}

A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms. In fact, given a connection ∇ on E there is a unique way to extend ∇ to an exterior covariant derivative

${\displaystyle d^{\nabla }:\Omega ^{r}(E)\to \Omega ^{r+1}(E).}$

Unlike the ordinary exterior derivative, one generally has (d^∇)² ≠ 0. In fact, (d^∇)² is directly related to the curvature of the connection ∇ (see below).

Affine properties of the set of connections

Every vector bundle over a manifold admits a connection, which can be proved using partitions of unity. However, connections are not unique. If ∇₁ and ∇₂ are two connections on E → M then their difference is a C^∞-linear operator. That is,

${\displaystyle (\nabla _{1}-\nabla _{2})(f\sigma )=f(\nabla _{1}\sigma -\nabla _{2}\sigma )}$

for all smooth functions f on M and all smooth sections σ of E. It follows that the difference ∇₁ − ∇₂ is induced by a one-form on M with values in the endomorphism bundle End(E) = E⊗E*:

${\displaystyle (\nabla _{1}-\nabla _{2})\in \Omega ^{1}(M;\mathrm {End} \,E).}$

Conversely, if ∇ is a connection on E and A is a one-form on M with values in End(E), then ∇+A is a connection on E.

In other words, the space of connections on E is an affine space for Ω¹(End E). This affine space is commonly denoted ${\displaystyle {\mathcal {A}}}$.

Relation to principal and Ehresmann connections

Let E → M be a vector bundle of rank k and let F(E) be the principal frame bundle of E. Then a (principal) connection on F(E) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F(E) → R^k. (This can be seen by considering the pullback of E over F(E) → M, which is isomorphic to the trivial bundle F(E) × R^k.) Given a section σ of E let the corresponding equivariant map be ψ(σ). The covariant derivative on E is then given by

${\displaystyle \psi (\nabla _{X}\sigma )=X^{H}(\psi (\sigma ))}$

where X^H is the horizontal lift of X from M to F(E). (Recall that the horizontal lift is determined by the connection on F(E).)

Conversely, a connection on E determines a connection on F(E), and these two constructions are mutually inverse.

A connection on E is also determined equivalently by a linear Ehresmann connection on E. This provides one method to construct the associated principal connection.

Local expression

Let E → M be a vector bundle of rank k, and let U be an open subset of M over which E is trivial. Given a local smooth frame (e₁, ..., e_k) of E over U, any section σ of E can be written as ${\displaystyle \sigma =\sigma ^{\alpha }e_{\alpha }}$ (Einstein notation assumed). A connection on E restricted to U then takes the form

${\displaystyle \nabla \sigma =e_{\alpha }\otimes \left(d\sigma ^{\alpha }+\sigma ^{\beta }{\omega ^{\alpha }}_{\beta }\right),}$

which, means, considering each component:

${\displaystyle \left(\nabla \sigma \right)^{\alpha }=d\sigma ^{\alpha }+\sigma ^{\beta }{\omega ^{\alpha }}_{\beta },}$

where

${\displaystyle e_{\alpha }\otimes {\omega ^{\alpha }}_{\beta }=\nabla e_{\beta }.}$

Here ${\displaystyle {\omega ^{\alpha }}_{\beta }}$ defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ${\displaystyle {\omega ^{\alpha }}_{\beta }}$ determines a one-form ω with values in End(E) and this expression defines ∇ to be the connection d+ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame. In this context ω is sometimes called the connection form of ∇ with respect to the local frame.

If U is a coordinate neighborhood with coordinates (xⁱ) then we can write

${\displaystyle {\omega ^{\alpha }}_{\beta }={{\omega _{i}}^{\alpha }}_{\beta }\,dx^{i}.}$

Note the mixture of coordinate indices (i) and fiber indices (α,β) in this expression.

The coefficient functions ${\displaystyle {{\omega _{i}}^{\alpha }}_{\beta }}$ are tensorial in the index i (they define a one-form) but not in the indices α and β. The transformation law for the fiber indices is more complicated. Let (f₁, ..., f_k) be another smooth local frame over U and let the change of coordinate matrix be denoted t, i.e.:

${\displaystyle f_{\alpha }\ =\ e_{\beta }\,{t^{\beta }}_{\alpha }.}$

The connection matrix with respect to frame (f_α) is then given by the matrix expression

${\displaystyle \varpi =t^{-1}\omega t+t^{-1}dt.}$

Here dt is the matrix of one-forms obtained by taking the exterior derivative of the components of t.

The covariant derivative in the local coordinates and with respect to the local frame field (e_α) is given by the expression

${\displaystyle \nabla _{X}\sigma =e_{\alpha }\otimes X^{i}\left(\partial _{i}\sigma ^{\alpha }+{{\omega _{i}}^{\alpha }}_{\beta }\sigma ^{\beta }\right).}$

For example, if we saturate the subscript argument of ∇ with the basis tangent vector ${\displaystyle \partial _{i}={\frac {\partial }{\partial x^{i}}}}$, and set ${\displaystyle \nabla _{i}:=\nabla _{\partial _{i}}}$, we have:

${\displaystyle \left(\nabla _{i}\sigma \right)^{\alpha }={\frac {\partial \sigma ^{\alpha }}{\partial x^{i}}}+\sigma ^{\beta }{{\omega _{i}}^{\alpha }}_{\beta }.}$

Parallel transport and holonomy

A connection ∇ on a vector bundle E → M defines a notion of parallel transport on E along a curve in M. Let γ : [0, 1] → M be a smooth path in M. A section σ of E along γ is said to be parallel if

${\displaystyle \nabla _{{\dot {\gamma }}(t)}\sigma =0}$

for all t ∈ [0, 1]. Equivalently, one can consider the pullback bundle γ*E of E by γ. This is a vector bundle over [0, 1] with fiber E_γ(t) over t ∈ [0, 1]. The connection ∇ on E pulls back to a connection on γ*E. A section σ of γ*E is parallel if and only if γ*∇(σ) = 0.

Suppose γ is a path from x to y in M. The above equation defining parallel sections is a first-order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector v in E_x there exists a unique parallel section σ of γ*E with σ(0) = v. Define a parallel transport map

${\displaystyle \tau _{\gamma }:E_{x}\to E_{y}\,}$

by τ_γ(v) = σ(1). It can be shown that τ_γ is a linear isomorphism.

How to recover the covariant derivative of a connection from its parallel transport. The values ${\displaystyle s(\gamma (t))}$ of a section ${\displaystyle s\in \Gamma (E)}$ are parallel transported along the path ${\displaystyle \gamma }$ back to ${\displaystyle \gamma (0)=x}$, and then the covariant derivative is taken in the fixed vector space, the fibre ${\displaystyle E_{x}}$ over ${\displaystyle x}$.

Parallel transport can be used to define the holonomy group of the connection ∇ based at a point x in M. This is the subgroup of GL(E_x) consisting of all parallel transport maps coming from loops based at x:

${\displaystyle \mathrm {Hol} _{x}=\{\tau _{\gamma }:\gamma {\text{ is a loop based at }}x\}.\,}$

The holonomy group of a connection is intimately related to the curvature of the connection (AmbroseSinger 1953).

The connection can be recovered from its parallel transport operators as follows. If ${\displaystyle X\in \Gamma (TM)}$ is a vector field and ${\displaystyle s\in \Gamma (E)}$ a section, at a point ${\displaystyle x\in M}$ pick an integral curve ${\displaystyle \gamma$ :(-\varepsilon ,\varepsilon )\to M} for ${\displaystyle X}$ at ${\displaystyle x}$. For each ${\displaystyle t\in (-\varepsilon ,\varepsilon )}$ we will write ${\displaystyle \tau _{t}:E_{\gamma (t)}\to E_{x}}$ for the parallel transport map traveling along ${\displaystyle \gamma }$ from ${\displaystyle t}$ to ${\displaystyle 0}$. In particular for every ${\displaystyle t\in (-\varepsilon ,\varepsilon )}$, we have ${\displaystyle \tau _{t}s(\gamma (t))\in E_{x}}$. Then ${\displaystyle t\mapsto \tau _{t}s(\gamma (t))}$ defines a curve in the vector space ${\displaystyle E_{x}}$, which may be differentiated. The covariant derivative is recovered as

${\displaystyle \nabla _{X}s(x)={\frac {d}{dt}}\left(\tau _{t}s(\gamma (t))\right)_{t=0}.}$

This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms ${\displaystyle \tau _{\gamma }}$ between fibres of ${\displaystyle E}$ and taking the above expression as the definition of ${\displaystyle \nabla }$.

Curvature

The curvature of a connection ∇ on E → M is a 2-form F^∇ on M with values in the endomorphism bundle End(E) = E⊗E*. That is,

${\displaystyle F^{\nabla }\in \Omega ^{2}(\mathrm {End} \,E)=\Gamma (\mathrm {End} \,E\otimes \Lambda ^{2}T^{*}M).}$

It is defined by the expression

${\displaystyle F^{\nabla }(X,Y)(s)=\nabla _{X}\nabla _{Y}s-\nabla _{Y}\nabla _{X}s-\nabla _{[X,Y]}s}$

where X and Y are tangent vector fields on M and s is a section of E. One must check that F^∇ is C^∞-linear in both X and Y and that it does in fact define a bundle endomorphism of E.

As mentioned above, the covariant exterior derivative d^∇ need not square to zero when acting on E-valued forms. The operator (d^∇)² is, however, strictly tensorial (i.e. C^∞-linear). This implies that it is induced from a 2-form with values in End(E). This 2-form is precisely the curvature form given above. For an E-valued form σ we have

${\displaystyle (d^{\nabla })^{2}\sigma =F^{\nabla }\wedge \sigma .}$

A flat connection is one whose curvature form vanishes identically.

Local form and Cartan's structure equation

The curvature form has a local description called Cartan's structure equation. If ${\displaystyle \nabla }$ has local form ${\displaystyle \omega }$ on some trivialising open subset ${\displaystyle U\subset M}$ for ${\displaystyle E}$, then

${\displaystyle F^{\nabla }=d\omega +\omega \wedge \omega }$

on ${\displaystyle U}$. To clarify, ${\displaystyle \nabla =d+\omega }$ where ${\displaystyle \omega \in \Omega ^{1}(U,\operatorname {End} (E))}$ is an endomorphism-valued one-form. For simplicity let us suppose ${\displaystyle \omega =\alpha \otimes u}$ for a one-form ${\displaystyle \alpha \in \Omega ^{1}(U)}$ and an endomorphism ${\displaystyle u\in \Gamma (U,\operatorname {End} (E))}$. Then we use the conventions

${\displaystyle d(\alpha \otimes u)=(d\alpha )\otimes u,\quad (\alpha \otimes u)\wedge (\beta \otimes v)=(\alpha \wedge \beta )\otimes uv,}$

where ${\displaystyle \beta \otimes v}$ is another endomorphism valued one-form. In general ${\displaystyle \omega }$ will be a sum of simple tensors of this form, and the operators ${\displaystyle d}$ and ${\displaystyle \wedge }$ are extended linearly.

It can be checked that if we define ${\displaystyle [\omega ,\omega ]}$ to be wedge product of forms but commutator of endomorphisms as opposed to composition, then ${\displaystyle \omega \wedge \omega ={\frac {1}{2}}[\omega ,\omega ]}$, and with this alternate notation the Cartan structure equation takes the form

${\displaystyle F^{\nabla }=d\omega +{\frac {1}{2}}[\omega ,\omega ].}$

This alternate notation is commonly used in the theory of principal bundle connections, where the connection form ${\displaystyle \omega }$ is a Lie algebra-valued one-form for which there is no notion of composition (unlike in the case of endomorphisms), but there is a notion of a Lie bracket.

In some references the Cartan structure equation may be written with a minus sign:

${\displaystyle F^{\nabla }=d\omega -\omega \wedge \omega .}$

This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.

Bianchi identity

A version of the Bianchi identity from Riemannian geometry holds for a connection on any vector bundle. Recall that a connection ${\displaystyle \nabla }$ on a vector bundle ${\displaystyle E\to M}$ induces an endomorphism connection on ${\displaystyle \operatorname {End} (E)}$. This endomorphism connection has itself an exterior covariant derivative, which we ambiguously call ${\displaystyle d^{\nabla }}$. Since the curvature is a globally defined ${\displaystyle \operatorname {End} (E)}$-valued two-form, we may apply the exterior covariant derivative to it. The Bianchi identity says that

${\displaystyle d^{\nabla }F^{\nabla }=0}$.

This succinctly captures the complicated tensor formulae of the Bianchi identity in the case of Riemannian manifolds, and one may translate from this equation to the standard Bianchi identities by expanding the connection and curvature in local coordinates.

Gauge transformations

Given two connections ${\displaystyle \nabla _{1},\nabla _{2}}$ on a vector bundle ${\displaystyle E\to M}$, it is natural to ask when they might be considered equivalent. There is a well-defined notion of an automorphism of a vector bundle ${\displaystyle E\to M}$. A section ${\displaystyle u\in \Gamma (\operatorname {End} (E))}$ is an automorphism if ${\displaystyle u(x)\in \operatorname {End} (E_{x})}$ is invertible at every point ${\displaystyle x\in M}$. Such an automorphism is called a gauge transformation of ${\displaystyle E}$, and the group of all automorphisms is called the gauge group, often denoted ${\displaystyle {\mathcal {G}}}$ or ${\displaystyle \operatorname {Aut} (E)}$. The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle ${\displaystyle \operatorname {Ad} ({\mathcal {F}}(E))}$ of the frame bundle of the vector bundle ${\displaystyle E}$. This is not to be confused with the lowercase a adjoint bundle ${\displaystyle \operatorname {ad} ({\mathcal {F}}(E))}$, which is naturally identified with ${\displaystyle \operatorname {End} (E)}$ itself. The bundle ${\displaystyle \operatorname {Ad} {\mathcal {F}}(E)}$ is the associated bundle to the principal frame bundle by the conjugation representation of ${\displaystyle G=\operatorname {GL} (r)}$ on itself, ${\displaystyle g\mapsto ghg^{-1}}$, and has fibre the same general linear group ${\displaystyle \operatorname {GL} (r)}$ where ${\displaystyle \operatorname {rank} (E)=r}$. Notice that despite having the same fibre as the frame bundle ${\displaystyle {\mathcal {F}}(E)}$ and being associated to it, ${\displaystyle \operatorname {Ad} ({\mathcal {F}}(E))}$ is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised as ${\displaystyle {\mathcal {G}}=\Gamma (\operatorname {Ad} {\mathcal {F}}(E)).}$

A gauge transformation ${\displaystyle u}$ of ${\displaystyle E}$ acts on sections ${\displaystyle s\in \Gamma (E)}$, and therefore acts on connections by conjugation. Explicitly, if ${\displaystyle \nabla }$ is a connection on ${\displaystyle E}$, then one defines ${\displaystyle u\cdot \nabla }$ by

${\displaystyle (u\cdot \nabla )_{X}(s)=u(\nabla _{X}(u^{-1}(s))}$

for ${\displaystyle s\in \Gamma (E),X\in \Gamma (TM)}$. To check that ${\displaystyle u\cdot \nabla }$ is a connection, one verifies the product rule

${\displaystyle {\begin{aligned}u\cdot \nabla (fs)&=u(\nabla (u^{-1}(fs)))\\&=u(\nabla (fu^{-1}(s)))\\&=u(df\otimes u^{-1}(s))+u(f\nabla (u^{-1}(s)))\\&=df\otimes s+fu\cdot \nabla (s).\end{aligned}}}$

It may be checked that this defines a left group action of ${\displaystyle {\mathcal {G}}}$ on the affine space of all connections ${\displaystyle {\mathcal {A}}}$.

Since ${\displaystyle {\mathcal {A}}}$ is an affine space modelled on ${\displaystyle \Omega ^{1}(M,\operatorname {End} (E))}$, there should exist some endomorphism-valued one-form ${\displaystyle A_{u}\in \Omega ^{1}(M,\operatorname {End} (E))}$ such that ${\displaystyle u\cdot \nabla =\nabla +A_{u}}$. Using the definition of the endomorphism connection ${\displaystyle \nabla ^{\operatorname {End} (E)}}$ induced by ${\displaystyle \nabla }$, it can be seen that

${\displaystyle u\cdot \nabla =\nabla -d^{\nabla }(u)u^{-1}}$

which is to say that ${\displaystyle A_{u}=-d^{\nabla }(u)u^{-1}}$.

Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space ${\displaystyle {\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}}$ is the moduli space of all connections on ${\displaystyle E}$. In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on ${\displaystyle E}$, which is of significant interest in gauge theory and physics.

Examples

• A classical covariant derivative or affine connection defines a connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor products of the tangent bundle with itself and its dual.
• A connection on ${\displaystyle \pi$ :\mathbb {R} ^{2}\times \mathbb {R} \to \mathbb {R} } can be described explicitly as the operator

${\displaystyle \nabla =d+{\begin{bmatrix}f_{11}(x)&f_{12}(x)\\f_{21}(x)&f_{22}(x)\end{bmatrix}}dx}$

where ${\displaystyle d}$ is the exterior derivative evaluated on vector-valued smooth functions and ${\displaystyle f_{ij}(x)}$ are smooth. A section ${\displaystyle a\in \Gamma (\pi )}$ may be identified with a map

${\displaystyle {\begin{cases}\mathbb {R} \to \mathbb {R} ^{2}\\x\mapsto (a_{1}(x),a_{2}(x))\end{cases}}}$

and then

${\displaystyle \nabla (a)=\nabla {\begin{bmatrix}a_{1}(x)\\a_{2}(x)\end{bmatrix}}={\begin{bmatrix}{\frac {da_{1}(x)}{dx}}+f_{11}(x)a_{1}(x)+f_{12}(x)a_{2}(x)\\{\frac {da_{2}(x)}{dx}}+f_{21}(x)a_{1}(x)+f_{22}(x)a_{2}(x)\end{bmatrix}}dx}$

• If the bundle is endowed with a bundle metric, an inner product on its vector space fibers, a metric connection is defined as a connection that is compatible with the bundle metric.
• A Yang-Mills connection is a special metric connection which satisfies the Yang-Mills equations of motion.
• A Riemannian connection is a metric connection on the tangent bundle of a Riemannian manifold.
• A Levi-Civita connection is a special Riemannian connection: the metric-compatible connection on the tangent bundle that is also torsion-free. It is unique, in the sense that given any Riemannian connection, one can always find one and only one equivalent connection that is torsion-free. "Equivalent" means it is compatible with the same metric, although the curvature tensors may be different; see teleparallelism. The difference between a Riemannian connection and the corresponding Levi-Civita connection is given by the contorsion tensor.
• The exterior derivative is a flat connection on ${\displaystyle E=M\times \mathbb {R} }$ (the trivial line bundle over M).
• More generally, there is a canonical flat connection on any flat vector bundle (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.

See also

• D-module
• Connection (mathematics)

References

• Chern, Shiing-Shen (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes
• Darling, R. W. R. (1994), Differential Forms and Connections, Cambridge, UK: Cambridge University Press, ISBN 0-521-46800-0
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996) [1963], Foundations of Differential Geometry, Vol. 1, Wiley Classics Library, New York: Wiley Interscience, ISBN 0-471-15733-3
• Koszul, J. L. (1950), "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique, 78: 65–127
• Wells, R.O. (1973), Differential analysis on complex manifolds, Springer-Verlag, ISBN 0-387-90419-0
• Ambrose, W.; Singer, I.M. (1953), "A theorem on holonomy", Transactions of the American Mathematical Society, 75: 428–443, doi:10.2307/1990721