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 pseudo-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

Let M be a differentiable manifold, such as Euclidean space. A vector-valued function ${\displaystyle M\to \mathbb {R} ^{n}}$ can be viewed as a section of the trivial vector bundle ${\displaystyle M\times \mathbb {R} ^{n}\to M.}$ One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on M.

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 a function ${\displaystyle X:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ 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 defined by the standard formula

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

For every ${\displaystyle x\in \mathbb {R} ^{n}}$, this defines a new vector ${\displaystyle dX(v)(x)\in \mathbb {R} ^{m}.}$

When passing to a section ${\displaystyle X}$ of a vector bundle ${\displaystyle E}$ over 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 ${\displaystyle X(\gamma (t))}$ and ${\displaystyle X(\gamma (0))}$ are elements of the distinct vector spaces ${\displaystyle E_{\gamma (t)}}$ and ${\displaystyle E_{x}.}$ This means that subtraction of these two terms is not naturally defined.

The problem is resolved by introducing the extra structure of a connection to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.

1. (Parallel transport) A connection can be viewed as assigning to every differentiable path ${\displaystyle \gamma }$ a linear isomorphism ${\displaystyle P_{t}^{\gamma }:E_{\gamma (t)}\to E_{x}}$ for all ${\displaystyle t.}$ Using this isomorphism one can transport ${\displaystyle X(\gamma (t))}$ to the fibre ${\displaystyle E_{x}}$ and then take the difference; explicitly,
   $${\displaystyle \nabla _{v}X=\lim _{t\to 0}{\frac {P_{t}^{\gamma }X(\gamma (t))-X(\gamma (0))}{t}}.}$$

   
   In order for this to depend only on ${\displaystyle v,}$ and not on the path ${\displaystyle \gamma }$ extending ${\displaystyle v,}$ it is necessary to place restrictions (in the definition) on the dependence of ${\displaystyle P_{t}^{\gamma }}$ on ${\displaystyle \gamma .}$ This is not straightforward to formulate, and so this notion of "parallel transport" is usually derived as a by-product of other ways of defining connections. In fact, the following notion of "Ehresmann connection" is nothing but an infinitesimal formulation of parallel transport.
2. (Ehresmann connection) The section ${\displaystyle X}$ may be viewed as a smooth map from the smooth manifold ${\displaystyle M}$ to the smooth manifold ${\displaystyle E.}$ As such, one may consider the pushforward ${\displaystyle dX(v),}$ which is an element of the tangent space ${\displaystyle T_{X(x)}E.}$ In Ehresmann's formulation of a connection, one chooses a way of assigning, to each ${\displaystyle x}$ and every ${\displaystyle e\in E_{x},}$ a direct sum decomposition of ${\displaystyle T_{X(x)}E}$ into two linear subspaces, one of which is the natural embedding of ${\displaystyle E_{x}.}$ With this additional data, one defines ${\displaystyle \nabla _{v}X}$ by projecting ${\displaystyle dX(v)}$ to be valued in ${\displaystyle E_{x}.}$ In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of ${\displaystyle T_{e}E}$ moves as e is varied over a fiber.
3. (Covariant derivative) The standard derivative ${\displaystyle dX(v)}$ in Euclidean contexts satisfies certain dependencies on ${\displaystyle X}$ and ${\displaystyle v,}$ the most fundamental being linearity. A covariant derivative is defined to be any operation ${\displaystyle (v,X)\mapsto \nabla _{v}X}$ which mimics these properties, together with a form of the product rule.

Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding choice of how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certain partial differential equations. In the case of the tangent bundle, any pseudo-Riemannian metric (and in particular any Riemannian metric) determines a canonical connection, called the Levi-Civita connection.

Formal definition

Let ${\displaystyle E\to M}$ be a smooth real vector bundle over a smooth manifold ${\displaystyle M}$. Denote the space of smooth sections of ${\displaystyle E\to M}$ by ${\displaystyle \Gamma (E)}$. A covariant derivative on ${\displaystyle E\to M}$ is either of the following equivalent structures:

1. an ${\displaystyle \mathbb {R} }$-linear map ${\displaystyle \nabla$ :\Gamma (E)\to \Gamma (T^{*}M\otimes E)} such that the product rule
   $${\displaystyle \nabla (fs)=df\otimes s+f\nabla s}$$

   
   holds for all smooth functions ${\displaystyle f}$ on ${\displaystyle M}$ and all smooth sections ${\displaystyle s}$ of ${\displaystyle E.}$
2. an assignment, to any smooth section s and every ${\displaystyle x\in M}$, of a ${\displaystyle \mathbb {R} }$-linear map ${\displaystyle (\nabla s)_{x}:T_{x}M\to E_{x}}$ which depends smoothly on x and such that
   $${\displaystyle \nabla (a_{1}s_{1}+a_{2}s_{2})=a_{1}\nabla s_{1}+a_{2}\nabla s_{2}}$$

   
   for any two smooth sections ${\displaystyle s_{1},s_{2}}$ and any real numbers ${\displaystyle a_{1},a_{2},}$ and such that for every smooth function ${\displaystyle f}$, ${\displaystyle \nabla (fs)}$ is related to ${\displaystyle \nabla s}$ by
   $${\displaystyle {\big (}\nabla (fs){\big )}_{x}(v)=df(v)s(x)+f(x)(\nabla s)_{x}(v)}$$

   
   for any ${\displaystyle x\in M}$ and ${\displaystyle v\in T_{x}M.}$

Beyond using the canonical identification between the vector space ${\displaystyle T_{x}^{\ast }M\otimes E_{x}}$ and the vector space of linear maps ${\displaystyle T_{x}M\to E_{x},}$ these two definitions are identical and differ only in the language used.

It is typical to denote ${\displaystyle (\nabla s)_{x}(v)}$ by ${\displaystyle \nabla _{v}s,}$ with ${\displaystyle x}$ being implicit in ${\displaystyle v.}$ With this notation, the product rule in the second version of the definition given above is written

${\displaystyle \nabla _{v}(fs)=df(v)s+f\nabla _{v}s.}$

Remark. In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "ℝ" everywhere they appear to "complex" and "${\displaystyle \mathbb {C} .}$" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.

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.

Exterior covariant derivative and vector-valued forms

Let ${\displaystyle E\to M}$ be a vector bundle. An ${\displaystyle E}$-valued differential form of degree ${\displaystyle r}$ is a section of the tensor product bundle:

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

The space of such forms is denoted by

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

where the last tensor product denotes the tensor product of modules over the ring of smooth functions on ${\displaystyle M}$.

An ${\displaystyle E}$-valued 0-form is just a section of the bundle ${\displaystyle E}$. That is,

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

In this notation a connection on ${\displaystyle E\to M}$ 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 ${\displaystyle \nabla }$ on ${\displaystyle E}$ there is a unique way to extend ${\displaystyle \nabla }$ to an exterior covariant derivative

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

This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the form ${\displaystyle \omega \otimes s}$ and extended linearly:

${\displaystyle d_{\nabla }(\omega \otimes s)=d\omega \otimes s+(-1)^{\deg \omega }\omega \wedge \nabla s}$

where ${\displaystyle \omega \in \Omega ^{r}(M)}$ so that ${\displaystyle \deg \omega =r}$, ${\displaystyle s\in \Gamma (E)}$ is a section, and ${\displaystyle \omega \wedge \nabla s}$ denotes the ${\displaystyle (r+1)}$-form with values in ${\displaystyle E}$ defined by wedging ${\displaystyle \omega }$ with the one-form part of ${\displaystyle \nabla s}$. Notice that for ${\displaystyle E}$-valued 0-forms, this recovers the normal Leibniz rule for the connection ${\displaystyle \nabla }$.

Unlike the ordinary exterior derivative, one generally has ${\displaystyle d_{\nabla }^{2}\neq 0}$. In fact, ${\displaystyle d_{\nabla }^{2}}$ is directly related to the curvature of the connection ${\displaystyle \nabla }$ (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 ${\displaystyle \nabla _{1}}$ and ${\displaystyle \nabla _{2}}$ are two connections on ${\displaystyle E\to M}$ then their difference is a ${\displaystyle C^{\infty }(M)}$-linear operator. That is,

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

for all smooth functions ${\displaystyle f}$ on ${\displaystyle M}$ and all smooth sections ${\displaystyle s}$ of ${\displaystyle E}$. It follows that the difference ${\displaystyle \nabla _{1}-\nabla _{2}}$ can be uniquely identified with a one-form on ${\displaystyle M}$ with values in the endomorphism bundle ${\displaystyle \operatorname {End} (E)=E^{*}\otimes E}$:

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

Conversely, if ${\displaystyle \nabla }$ is a connection on ${\displaystyle E}$ and ${\displaystyle A}$ is a one-form on ${\displaystyle M}$ with values in ${\displaystyle \operatorname {End} (E)}$, then ${\displaystyle \nabla +A}$ is a connection on ${\displaystyle E}$.

In other words, the space of connections on ${\displaystyle E}$ is an affine space for ${\displaystyle \Omega ^{1}(\operatorname {End} (E))}$. This affine space is commonly denoted ${\displaystyle {\mathcal {A}}}$.

Relation to principal and Ehresmann connections

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

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

where ${\displaystyle X^{H}}$ is the horizontal lift of ${\displaystyle X}$ from ${\displaystyle M}$ to ${\displaystyle {\mathcal {F}}(E)}$. (Recall that the horizontal lift is determined by the connection on ${\displaystyle {\mathcal {F}}(E)}$.)

Conversely, a connection on ${\displaystyle E}$ determines a connection on ${\displaystyle {\mathcal {F}}(E)}$, and these two constructions are mutually inverse.

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

The induced connections discussed in #Induced connections can be constructed as connections on other associated bundles to the frame bundle of ${\displaystyle E}$, using representations other than the standard representation used above. For example if ${\displaystyle \rho }$ denotes the standard representation of ${\displaystyle \operatorname {GL} (k,\mathbb {R} )}$ on ${\displaystyle \mathbb {R} ^{k}}$, then the associated bundle to the representation ${\displaystyle \rho \oplus \rho }$ of ${\displaystyle \operatorname {GL} (k,\mathbb {R} )}$ on ${\displaystyle \mathbb {R} ^{k}\oplus \mathbb {R} ^{k}}$ is the direct sum bundle ${\displaystyle E\oplus E}$, and the induced connection is precisely that which was described above.

Local expression

Let ${\displaystyle E\to M}$ be a vector bundle of rank ${\displaystyle k}$, and let ${\displaystyle U}$ be an open subset of ${\displaystyle M}$ over which ${\displaystyle E}$ trivialises. Therefore over the set ${\displaystyle U}$, ${\displaystyle E}$ admits a local smooth frame of sections

${\displaystyle \mathbf {e} =(e_{1},\dots ,e_{k});\quad e_{i}:U\to \left.E\right|_{U}.}$

Since the frame ${\displaystyle \mathbf {e} }$ defines a basis of the fibre ${\displaystyle E_{x}}$ for any ${\displaystyle x\in U}$, one can expand any local section ${\displaystyle s:U\to \left.E\right|_{U}}$ in the frame as

${\displaystyle s=\sum _{i=1}^{k}s^{i}e_{i}}$

for a collection of smooth functions ${\displaystyle s^{1},\dots ,s^{k}:U\to \mathbb {R} }$.

Given a connection ${\displaystyle \nabla }$ on ${\displaystyle E}$, it is possible to express ${\displaystyle \nabla }$ over ${\displaystyle U}$ in terms of the local frame of sections, by using the characteristic product rule for the connection. For any basis section ${\displaystyle e_{i}}$, the quantity ${\displaystyle \nabla (e_{i})\in \Omega ^{1}(U)\otimes \Gamma (U,E)}$ may be expanded in the local frame ${\displaystyle \mathbf {e} }$ as

${\displaystyle \nabla (e_{i})=\sum _{j=1}^{k}A_{i}^{\ j}\otimes e_{j},}$

where ${\displaystyle A_{i}^{\ j}\in \Omega ^{1}(U);\,j=1,\dots ,k}$ are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by

${\displaystyle A={\begin{pmatrix}A_{1}^{\ 1}&\cdots &A_{k}^{\ 1}\\\vdots &\ddots &\vdots \\A_{1}^{\ k}&\cdots &A_{k}^{\ k}\end{pmatrix}}\in \Omega ^{1}(U,\operatorname {End} (\left.E\right|_{U}))}$

called the local connection form of ${\displaystyle \nabla }$ over ${\displaystyle U}$. The action of ${\displaystyle \nabla }$ on any section ${\displaystyle s:U\to \left.E\right|_{U}}$ can be computed in terms of ${\displaystyle A}$ using the product rule as

${\displaystyle \nabla (s)=\sum _{j=1}^{k}\left(ds^{j}+\sum _{i=1}^{k}A_{i}^{\ j}s^{i}\right)\otimes e_{j}.}$

If the local section ${\displaystyle s}$ is also written in matrix notation as a column vector using the local frame ${\displaystyle \mathbf {e} }$ as a basis,

${\displaystyle s={\begin{pmatrix}s^{1}\\\vdots \\s^{k}\end{pmatrix}},}$

then using regular matrix multiplication one can write

${\displaystyle \nabla (s)=ds+As}$

where ${\displaystyle ds}$ is shorthand for applying the exterior derivative ${\displaystyle d}$ to each component of ${\displaystyle s}$ as a column vector. In this notation, one often writes locally that ${\displaystyle \left.\nabla \right|_{U}=d+A}$. In this sense a connection is locally completely specified by its connection one-form in some trivialisation.

As explained in #Affine properties of the set of connections, any connection differs from another by an endomorphism-valued one-form. From this perspective, the connection one-form ${\displaystyle A}$ is precisely the endomorphism-valued one-form such that the connection ${\displaystyle \left.\nabla \right|_{U}}$ on ${\displaystyle \left.E\right|_{U}}$ differs from the trivial connection ${\displaystyle d}$ on ${\displaystyle \left.E\right|_{U}}$, which exists because ${\displaystyle U}$ is a trivialising set for ${\displaystyle E}$.

Parallel transport and holonomy

A connection ${\displaystyle \nabla }$ on a vector bundle ${\displaystyle E\to M}$ defines a notion of parallel transport on ${\displaystyle E}$ along a curve in ${\displaystyle M}$. Let ${\displaystyle \gamma$ :[0,1]\to M} be a smooth path in ${\displaystyle M}$. A section ${\displaystyle s}$ of ${\displaystyle E}$ along ${\displaystyle \gamma }$ is said to be parallel if

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

for all ${\displaystyle t\in [0,1]}$. Equivalently, one can consider the pullback bundle ${\displaystyle \gamma ^{*}E}$ of ${\displaystyle E}$ by ${\displaystyle \gamma }$. This is a vector bundle over ${\displaystyle [0,1]}$ with fiber ${\displaystyle E_{\gamma (t)}}$ over ${\displaystyle t\in [0,1]}$. The connection ${\displaystyle \nabla }$ on ${\displaystyle E}$ pulls back to a connection on ${\displaystyle \gamma ^{*}E}$. A section ${\displaystyle s}$ of ${\displaystyle \gamma ^{*}E}$ is parallel if and only if ${\displaystyle \gamma ^{*}\nabla (s)=0}$.

Suppose ${\displaystyle \gamma }$ is a path from ${\displaystyle x}$ to ${\displaystyle y}$ in ${\displaystyle M}$. 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 ${\displaystyle v}$ in ${\displaystyle E_{x}}$ there exists a unique parallel section ${\displaystyle s}$ of ${\displaystyle \gamma ^{*}E}$ with ${\displaystyle s(0)=v}$. Define a parallel transport map

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

by ${\displaystyle \tau _{\gamma }(v)=s(1)}$. It can be shown that ${\displaystyle \tau _{\gamma }}$ is a linear isomorphism, with inverse given by following the same procedure with the reversed path ${\displaystyle \gamma ^{-}}$ from ${\displaystyle y}$ to ${\displaystyle x}$.

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 ${\displaystyle \nabla }$ based at a point ${\displaystyle x}$ in ${\displaystyle M}$. This is the subgroup of ${\displaystyle \operatorname {GL} (E_{x})}$ consisting of all parallel transport maps coming from loops based at ${\displaystyle x}$:

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

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 ${\displaystyle \nabla }$ on ${\displaystyle E\to M}$ is a 2-form ${\displaystyle F_{\nabla }}$ on ${\displaystyle M}$ with values in the endomorphism bundle ${\displaystyle \operatorname {End} (E)=E^{*}\otimes E}$. That is,

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

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 ${\displaystyle X}$ and ${\displaystyle Y}$ are tangent vector fields on ${\displaystyle M}$ and ${\displaystyle s}$ is a section of ${\displaystyle E}$. One must check that ${\displaystyle F_{\nabla }}$ is ${\displaystyle C^{\infty }(M)}$-linear in both ${\displaystyle X}$ and ${\displaystyle Y}$ and that it does in fact define a bundle endomorphism of ${\displaystyle E}$.

As mentioned above, the covariant exterior derivative ${\displaystyle d_{\nabla }}$ need not square to zero when acting on ${\displaystyle E}$-valued forms. The operator ${\displaystyle d_{\nabla }^{2}}$ is, however, strictly tensorial (i.e. ${\displaystyle C^{\infty }(M)}$-linear). This implies that it is induced from a 2-form with values in ${\displaystyle \operatorname {End} (E)}$. This 2-form is precisely the curvature form given above. For an ${\displaystyle E}$-valued form ${\displaystyle \sigma }$ we have

${\displaystyle (d_{\nabla })^{2}\sigma =F_{\nabla }\wedge \sigma .}$

A flat connection is one whose curvature form vanishes identically.

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 ${\displaystyle {\mathcal {G}}=\Gamma (\operatorname {Ad} {\mathcal {F}}(E)).}$