connection (vector bundle)

Let {{mvar|M}} be a differentiable manifold, such as Euclidean space. A vector-valued function $M\; \backslash to\; \backslash mathbb\{R\}^n$ can be viewed as a section of the trivial vector bundle $M\backslash times\; \backslash mathbb\{R\}^n\; \backslash to\; M.$ 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 {{mvar|M}}.

File:Bundle section.svg

The model case is to differentiate a function $X:\; \backslash mathbb\{R\}^n\; \backslash to\; \backslash mathbb\{R\}^m$ on Euclidean space $\backslash mathbb\{R\}^n$. In this setting the derivative $dX$ at a point $x\backslash in\; \backslash mathbb\{R\}^n$ in the direction $v\backslash in\; \backslash mathbb\{R\}^n$ may be defined by the standard formula

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

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

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

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

However this still does not make sense, because $X(\backslash gamma(t))$ and $X(\backslash gamma(0))$ are elements of the distinct vector spaces $E\_\{\backslash gamma(t)\}$ and $E\_x.$ 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 $\backslash gamma$ a linear isomorphism $P\_t^\{\backslash gamma\}\; :\; E\_\{\backslash gamma(t)\}\; \backslash to\; E\_\{x\}$ for all $t.$ Using this isomorphism one can transport $X(\backslash gamma(t))$ to the fibre $E\_x$ and then take the difference; explicitly, $$\backslash nabla\_vX\; =\; \backslash lim\_\{t\backslash to\; 0\}\; \backslash frac\{P\_\{t\}^\{\backslash gamma\}\; X(\backslash gamma(t))\; -\; X(\backslash gamma(0))\}\{t\}.$$In order for this to depend only on $v,$ and not on the path $\backslash gamma$ extending $v,$ it is necessary to place restrictions (in the definition) on the dependence of $P\_t^\{\backslash gamma\}$ on $\backslash gamma.$ 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 $X$ may be viewed as a smooth map from the smooth manifold $M$ to the smooth manifold $E.$ As such, one may consider the pushforward $dX(v),$ which is an element of the tangent space $T\_\{X(x)\}E.$ In Ehresmann's formulation of a connection, one chooses a way of assigning, to each $x$ and every $e\backslash in\; E\_x,$ a direct sum decomposition of $T\_\{X(x)\}E$ into two linear subspaces, one of which is the natural embedding of $E\_x.$ With this additional data, one defines $\backslash nabla\_vX$ by projecting $dX(v)$ to be valued in $E\_x.$ In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of $T\_\{e\}E$ moves as {{mvar|e}} is varied over a fiber.
3. (Covariant derivative) The standard derivative $dX(v)$ in Euclidean contexts satisfies certain dependencies on $X$ and $v,$ the most fundamental being linearity. A covariant derivative is defined to be any operation $(v,X)\backslash mapsto\backslash nabla\_vX$ 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.

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

1. an $\backslash mathbb\{R\}$-linear map $\backslash nabla\; :\; \backslash Gamma(E)\; \backslash to\; \backslash Gamma(T^*M\backslash otimes\; E)$ such that the product rule $$\backslash nabla(fs)\; =\; df\backslash otimes\; s\; +\; f\backslash nabla\; s$$ holds for all smooth functions $f$ on $M$ and all smooth sections $s$ of $E.$
2. an assignment, to any smooth section {{mvar|s}} and every $x\backslash in\; M$, of a $\backslash mathbb\{R\}$-linear map $(\backslash nabla\; s)\_x:T\_xM\backslash to\; E\_x$ which depends smoothly on {{mvar|x}} and such that $$\backslash nabla(a\_1s\_1+a\_2s\_2)=a\_1\backslash nabla\; s\_1+a\_2\backslash nabla\; s\_2$$ for any two smooth sections $s\_1,s\_2$ and any real numbers $a\_1,a\_2,$ and such that for every smooth function $f$, $\backslash nabla(fs)$ is related to $\backslash nabla\; s$ by $$\backslash big(\backslash nabla(fs)\backslash big)\_x(v)=df(v)s(x)+f(x)(\backslash nabla\; s)\_x(v)$$ for any $x\backslash in\; M$ and $v\backslash in\; T\_xM.$

Beyond using the canonical identification between the vector space $T\_x^\backslash ast\; M\backslash otimes\; E\_x$ and the vector space of linear maps $T\_xM\backslash to\; E\_x,$ these two definitions are identical and differ only in the language used.

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

:$\backslash nabla\_v(fs)=df(v)s+f\backslash nabla\_vs.$

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 "$\backslash mathbb\{R\}$" everywhere they appear to "complex" and "$\backslash mathbb\{C\}.$" 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.

Given a vector bundle $E\backslash to\; M$, there are many associated bundles to $E$ which may be constructed, for example the dual vector bundle $E^*$, tensor powers $E^\{\backslash otimes\; k\}$, symmetric and antisymmetric tensor powers $S^k\; E,\; \backslash Lambda^k\; E$, and the direct sums $E^\{\backslash oplus\; k\}$. A connection on $E$ 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 $\backslash nabla$ a connection on $E$, the induced dual connection $\backslash nabla^*$ on $E^*$ is defined implicitly by

:$d(\backslash langle\; \backslash xi,\; s\; \backslash rangle)(X)\; =\; \backslash langle\; \backslash nabla\_X^*\; \backslash xi,\; s\; \backslash rangle\; +\; \backslash langle\; \backslash xi,\; \backslash nabla\_X\; s\; \backslash rangle.$

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

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

:$(\backslash nabla^E\; \backslash otimes\; \backslash nabla^F)\_X(s\backslash otimes\; t)\; =\; \backslash nabla\_X^E\; (s)\; \backslash otimes\; t\; +\; s\backslash otimes\; \backslash nabla\_X^F\; (t).$

Here we have $s\backslash in\; \backslash Gamma(E),\; t\backslash in\; \backslash Gamma(F),\; X\backslash in\; \backslash Gamma(TM)$. Notice again this is the natural way of combining $\backslash nabla^E,\; \backslash nabla^F$ to enforce the product rule for the tensor product connection. By repeated application of the above construction applied to the tensor product $E^\{\backslash otimes\; k\}\; =\; (E^\{\backslash otimes\; (k-1)\})\; \backslash otimes\; E$, one also obtains the tensor power connection on $E^\{\backslash otimes\; k\}$ for any $k\backslash ge\; 1$ and vector bundle $E$.

The direct sum connection is defined by

:$(\backslash nabla^E\; \backslash oplus\; \backslash nabla^F)\_X\; (s\backslash oplus\; t)\; =\; \backslash nabla\_X^E\; (s)\; \backslash oplus\; \backslash nabla\_X^F\; (t),$

where $s\backslash oplus\; t\backslash in\; \backslash Gamma(E\backslash oplus\; F)$.

Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power, $S^k\; E,\; \backslash Lambda^k\; E\; \backslash subset\; E^\{\backslash otimes\; k\}$, the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside the tensor algebra as direct summands, and the connection $\backslash nabla$ respects this natural splitting, one can simply restrict $\backslash nabla$ to these summands. Explicitly, define the symmetric product connection by

:$\backslash nabla^\{\backslash odot\; 2\}\_X(s\backslash cdot\; t)\; =\; \backslash nabla\_X\; s\; \backslash odot\; t\; +\; s\; \backslash odot\; \backslash nabla\_X\; t$

and the exterior product connection by

:$\backslash nabla^\{\backslash wedge\; 2\}\_X\; (s\backslash wedge\; t)\; =\; \backslash nabla\_X\; s\; \backslash wedge\; t\; +\; s\backslash wedge\; \backslash nabla\_X\; t$

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

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

:$\backslash nabla\_X(u(s))\; =\; \backslash nabla\_X^\{\backslash operatorname\{End\}(E)\}\; (u)\; (s)\; +\; u(\backslash nabla\_X\; (s)).$

By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying

:$\backslash nabla\_X^\{\backslash operatorname\{End\}(E)\}\; (u)\; (s)\; =\; \backslash nabla\_X(u(s))\; -\; u(\backslash nabla\_X(s))$

for any $u,s,X$, thus avoiding the need to first define the dual connection and tensor product connection.

{{See also|Connection (principal bundle)}}

Given a vector bundle $E$ of rank $r$, and any representation $\backslash rho:\; \backslash mathrm\{GL\}(r,\backslash mathbb\{K\})\; \backslash to\; G$ into a linear group $G\backslash subset\; \backslash mathrm\{GL\}(V)$, there is an induced connection on the associated vector bundle $F\; =\; E\backslash times\_\backslash rho\; V$. This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of $E$ and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.

{{See also|Exterior covariant derivative}}

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

:$\backslash bigwedge^rT^*M\; \backslash otimes\; E.$

The space of such forms is denoted by

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

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

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

:$\backslash Omega^0(E)\; =\; \backslash Gamma(E).$

In this notation a connection on $E\backslash to\; M$ is a linear map

:$\backslash nabla:\backslash Omega^0(E)\; \backslash to\; \backslash Omega^1(E).$

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

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

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

:$d\_\backslash nabla\; (\backslash omega\; \backslash otimes\; s)\; =\; d\backslash omega\; \backslash otimes\; s\; +\; (-1)^\{\backslash deg\; \backslash omega\}\; \backslash omega\; \backslash wedge\; \backslash nabla\; s$

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

Unlike the ordinary exterior derivative, one generally has $d\_\{\backslash nabla\}^2\; \backslash ne\; 0$. In fact, $d\_\{\backslash nabla\}^2$ is directly related to the curvature of the connection $\backslash nabla$ (see below).

Every vector bundle over a manifold admits a connection, which can be proved using partitions of unity. However, connections are not unique. If $\backslash nabla\_1$ and $\backslash nabla\_2$ are two connections on $E\backslash to\; M$ then their difference is a $C^\{\backslash infty\}(M)$-linear operator. That is,

:$(\backslash nabla\_1\; -\; \backslash nabla\_2)(fs)\; =\; f(\backslash nabla\_1s\; -\; \backslash nabla\_2s)$

for all smooth functions $f$ on $M$ and all smooth sections $s$ of $E$. It follows that the difference $\backslash nabla\_1-\backslash nabla\_2$ can be uniquely identified with a one-form on $M$ with values in the endomorphism bundle $\backslash operatorname\{End\}(E)\; =\; E^*\; \backslash otimes\; E$:

:$\backslash nabla\_1\; -\; \backslash nabla\_2\; \backslash in\; \backslash Omega^1(M;\; \backslash mathrm\{End\}\backslash ,E).$

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

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

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

:$\backslash psi(\backslash nabla\_Xs)\; =\; X^H(\backslash psi(s))$

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

Conversely, a connection on $E$ determines a connection on $\backslash mathcal\{F\}(E)$, 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.

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

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

:$\backslash mathbf\{e\}\; =\; (e\_1,\; \backslash dots,\; e\_k);\backslash quad\; e\_i:\; U\; \backslash to\; \backslash left.E\backslash right|\_U.$

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

:$s\; =\; \backslash sum\_\{i=1\}^k\; s^i\; e\_i$

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

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

:$\backslash nabla\; (e\_i)\; =\; \backslash sum\_\{j=1\}^k\; A\_i^\{\backslash \; j\}\; \backslash otimes\; e\_j,$

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

:

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

:$\backslash nabla(s)\; =\; \backslash sum\_\{j=1\}^k\; \backslash left(ds^j\; +\; \backslash sum\_\{i=1\}^k\; A\_i^\{\backslash \; j\}\; s^i\backslash right)\; \backslash otimes\; e\_j.$

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

:$s\; =\; \backslash begin\{pmatrix\}\; s^1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; s^k\backslash end\{pmatrix\},$

then using regular matrix multiplication one can write

:$\backslash nabla(s)\; =\; ds\; +\; As$

where $ds$ is shorthand for applying the exterior derivative $d$ to each component of $s$ as a column vector. In this notation, one often writes locally that $\backslash left.\backslash nabla\backslash right|\_U\; =\; d+A$. 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 $A$ is precisely the endomorphism-valued one-form such that the connection $\backslash left.\backslash nabla\backslash right|\_U$ on $\backslash left.E\backslash right|\_U$ differs from the trivial connection $d$ on $\backslash left.E\backslash right|\_U$, which exists because $U$ is a trivialising set for $E$.

In pseudo-Riemannian geometry, the Levi-Civita connection is often written in terms of the Christoffel symbols $\backslash Gamma\_\{ij\}^\{\backslash \; \backslash \; k\}$ instead of the connection one-form $A$. It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition to $U$ being a trivialising open subset for the vector bundle $E\backslash to\; M$, that $U$ is also a local chart for the manifold $M$, admitting local coordinates $\backslash mathbf\{x\}\; =\; (x^1,\backslash dots,x^n);\backslash quad\; x^i:\; U\; \backslash to\; \backslash mathbb\{R\}$.

In such a local chart, there is a distinguished local frame for the differential one-forms given by $(dx^1,\backslash dots,dx^n)$, and the local connection one-forms $A\_i^\{\; j\}$ can be expanded in this basis as

:$A\_i^\{\backslash \; j\}\; =\; \backslash sum\_\{\backslash ell=1\}^n\; \backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}\; dx^\backslash ell$

for a collection of local smooth functions $\backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}\; :\; U\; \backslash to\; \backslash mathbb\{R\}$, called the Christoffel symbols of $\backslash nabla$ over $U$. In the case where $E=TM$ and $\backslash nabla$ is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.

The expression for how $\backslash nabla$ acts in local coordinates can be further expanded in terms of the local chart $U$ and the Christoffel symbols, to be given by

:$\backslash nabla(s)\; =\; \backslash sum\_\{i,j=1\}^k\; \backslash sum\_\{\backslash ell=1\}^n\; \backslash left(\backslash frac\{\backslash partial\; s^j\}\{\backslash partial\; x^\backslash ell\}\; +\; \backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}\; s^i\backslash right)\; dx^\backslash ell\; \backslash otimes\; e\_j.$

Contracting this expression with the local coordinate tangent vector $\backslash frac\{\backslash partial\}\{\backslash partial\; x^\backslash ell\}$ leads to

:$\backslash nabla\_\{\backslash frac\{\backslash partial\}\{\backslash partial\; x^\backslash ell\}\}\; (s)\; =\; \backslash sum\_\{i,j=1\}^k\; \backslash left(\backslash frac\{\backslash partial\; s^j\}\{\backslash partial\; x^\backslash ell\}\; +\; \backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}\; s^i\backslash right)\; e\_j.$

This defines a collection of $n$ locally defined operators

:$\backslash nabla\_\backslash ell:\; \backslash Gamma(U,E)\; \backslash to\; \backslash Gamma(U,E);\backslash quad\; \backslash nabla\_\backslash ell(s)\; :=\; \backslash sum\_\{i,j=1\}^k\; \backslash left(\backslash frac\{\backslash partial\; s^j\}\{\backslash partial\; x^\backslash ell\}\; +\; \backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}\; s^i\backslash right)e\_j,$

with the property that

:$\backslash nabla(s)\; =\; \backslash sum\_\{\backslash ell=1\}^n\; dx^\backslash ell\; \backslash otimes\; \backslash nabla\_\backslash ell(s).$

Suppose $\backslash mathbf\{e\text{'}\}$ is another choice of local frame over the same trivialising set $U$, so that there is a matrix $g=(g\_i^\{\backslash \; j\})$ of smooth functions relating $\backslash mathbf\{e\}$ and $\backslash mathbf\{e\text{'}\}$, defined by

:$e\_i\; =\; \backslash sum\_\{j=1\}^k\; g\_i^\{\backslash \; j\}\; e\text{'}\_j.$

Tracing through the construction of the local connection form $A$ for the frame $\backslash mathbf\{e\}$, one finds that the connection one-form $A\text{'}$ for $\backslash mathbf\{e\text{'}\}$ is given by

:$\{A\text{'}\}\_i^\{\backslash \; j\}\; =\; \backslash sum\_\{p,q=1\}^k\; g\_p^\{\backslash \; j\}\; A\_q^\{\backslash \; p\}\; \{(g^\{-1\})\}\_i^\{\backslash \; q\}\; -\; \backslash sum\_\{p=1\}^k\; (dg)\_p^\{\backslash \; j\}\; \{(g^\{-1\})\}\_i^\{\backslash \; p\}$

where $g^\{-1\}\; =\; \backslash left(\{(g^\{-1\})\}\_i^\{\backslash \; j\}\backslash right)$ denotes the inverse matrix to $g$. In matrix notation this may be written

:$A\text{'}\; =\; g\; A\; g^\{-1\}\; -\; (dg)g^\{-1\}$

where $dg$ is the matrix of one-forms given by taking the exterior derivative of the matrix $g$ component-by-component.

In the case where $E=TM$ is the tangent bundle and $g$ is the Jacobian of a coordinate transformation of $M$, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.

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

:$\backslash nabla\_\{\backslash dot\backslash gamma(t)\}s\; =\; 0$

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

Suppose $\backslash gamma$ 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 $s$ of $\backslash gamma^*\; E$ with $s(0)\; =\; v$. Define a parallel transport map

:$\backslash tau\_\backslash gamma\; :\; E\_x\; \backslash to\; E\_y\backslash ,$

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

File:Paralleltransportcovariantderivative.png

Parallel transport can be used to define the holonomy group of the connection $\backslash nabla$ based at a point $x$ in $M$. This is the subgroup of $\backslash operatorname\{GL\}(E\_x)$ consisting of all parallel transport maps coming from loops based at $x$:

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

The holonomy group of a connection is intimately related to the curvature of the connection {{harv|AmbroseSinger|1953}}.

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

:$\backslash nabla\_X\; s(x)\; =\; \backslash frac\{d\}\{dt\}\; \backslash left(\; \backslash tau\_t\; s(\backslash gamma(t))\; \backslash right)\_\{t=0\}.$

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

{{See also|Curvature form}}

The curvature of a connection $\backslash nabla$ on $E\backslash to\; M$ is a 2-form $F\_\{\backslash nabla\}$ on $M$ with values in the endomorphism bundle $\backslash operatorname\{End\}(E)\; =\; E^*\; \backslash otimes\; E$. That is,

:$F\_\backslash nabla\; \backslash in\; \backslash Omega^2(\backslash mathrm\{End\}(E))\; =\; \backslash Gamma(\backslash Lambda^2T^*M\; \backslash otimes\; \backslash mathrm\{End\}(E)).$

It is defined by the expression

:$F\_\backslash nabla(X,Y)(s)\; =\; \backslash nabla\_X\backslash nabla\_Y\; s-\; \backslash nabla\_Y\backslash nabla\_X\; s-\; \backslash nabla\_\{[X,Y]\}s$

where $X$ and $Y$ are tangent vector fields on $M$ and $s$ is a section of $E$. One must check that $F\_\{\backslash nabla\}$ is $C^\{\backslash infty\}(M)$-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 $d\_\{\backslash nabla\}$ need not square to zero when acting on $E$-valued forms. The operator $d\_\{\backslash nabla\}^2$ is, however, strictly tensorial (i.e. $C^\{\backslash infty\}(M)$-linear). This implies that it is induced from a 2-form with values in $\backslash operatorname\{End\}(E)$. This 2-form is precisely the curvature form given above. For an $E$-valued form $\backslash sigma$ we have

:$(d\_\backslash nabla)^2\backslash sigma\; =\; F\_\backslash nabla\backslash wedge\backslash sigma.$

A flat connection is one whose curvature form vanishes identically.

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

:$F\_\{\backslash nabla\}\; =\; dA\; +\; A\; \backslash wedge\; A$

on $U$. To clarify this notation, notice that $A$ is a endomorphism-valued one-form, and so in local coordinates takes the form of a matrix of one-forms. The operation $d$ applies the exterior derivative component-wise to this matrix, and $A\backslash wedge\; A$ denotes matrix multiplication, where the components are wedged rather than multiplied.

In local coordinates $\backslash mathbf\{x\}\; =\; (x^1,\backslash dots,x^n)$ on $M$ over $U$, if the connection form is written $A=A\_\backslash ell\; dx^\backslash ell\; =\; (\backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\})\; dx^\backslash ell$ for a collection of local endomorphisms $A\_\backslash ell\; =\; (\backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\})$, then one has

:$F\_\{\backslash nabla\}\; =\; \backslash sum\_\{p,q=1\}^n\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash frac\{\backslash partial\; A\_q\}\{\backslash partial\; x^p\}\; -\; \backslash frac\{\backslash partial\; A\_p\}\{\backslash partial\; x^q\}\; +\; [A\_p,\; A\_q]\backslash right)\; dx^p\; \backslash wedge\; dx^q.$

Further expanding this in terms of the Christoffel symbols $\backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}$ produces the familiar expression from Riemannian geometry. Namely if $s=s^i\; e\_i$ is a section of $E$ over $U$, then

:$F\_\{\backslash nabla\}(s)\; =\; \backslash sum\_\{i,j=1\}^k\; \backslash sum\_\{p,q=1\}^n\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash Gamma\_\{qi\}^\{\backslash \; \backslash \; j\}\}\{\backslash partial\; x^p\}\; -\; \backslash frac\{\backslash partial\; \backslash Gamma\_\{pi\}^\{\backslash \; \backslash \; j\}\}\{\backslash partial\; x^q\}\; +\; \backslash Gamma\_\{pr\}^\{\backslash \; \backslash \; j\}\; \backslash Gamma\_\{qi\}^\{\backslash \; \backslash \; r\}\; -\; \backslash Gamma\_\{qr\}^\{\backslash \; \backslash \; j\}\; \backslash Gamma\_\{pi\}^\{\backslash \; \backslash \; r\}\; \backslash right)\; s^i\; dx^p\; \backslash wedge\; dx^q\; \backslash otimes\; e\_j\; =\; \backslash sum\_\{i,j=1\}^k\; \backslash sum\_\{p,q=1\}^n\; R\_\{pqi\}^\{\backslash \; \backslash \; \backslash \; j\}\; s^i\; dx^p\backslash wedge\; dx^q\; \backslash otimes\; e\_j.$

Here $R=(R\_\{pqi\}^\{\backslash \; \backslash \; \backslash \; j\})$ is the full curvature tensor of $F\_\{\backslash nabla\}$, and in Riemannian geometry would be identified with the Riemannian curvature tensor.

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

:$F\_\{\backslash nabla\}\; =\; dA\; +\; \backslash frac\{1\}\{2\}\; [A,\; A].$

This alternate notation is commonly used in the theory of principal bundle connections, where instead we use a connection form $\backslash omega$, 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 (see for example {{harv|MadsenTornehave1997}}) the Cartan structure equation may be written with a minus sign:

:$F\_\{\backslash nabla\}\; =\; dA\; -\; A\; \backslash wedge\; A.$

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.

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

:$d\_\{\backslash nabla\}\; F\_\{\backslash nabla\}\; =\; 0$.

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.

There is no analogue in general of the first (algebraic) Bianchi identity for a general connection, as this exploits the special symmetries of the Levi-Civita connection. Namely, one exploits that the vector bundle indices of $E=TM$ in the curvature tensor $R$ may be swapped with the cotangent bundle indices coming from $T^*M$ after using the metric to lower or raise indices. For example this allows the torsion-freeness condition $\backslash Gamma\_\{\backslash ell\; i\}^\{\backslash \; \backslash \; j\}\; =\; \backslash Gamma\_\{i\; \backslash ell\}^\{\backslash \; \backslash \; j\}$ to be defined for the Levi-Civita connection, but for a general vector bundle the $\backslash ell$-index refers to the local coordinate basis of $T^*M$, and the $i,j$-indices to the local coordinate frame of $E$ and $E^*$ coming from the splitting $\backslash mathrm\{End\}(E)=E^*\; \backslash otimes\; E$. However in special circumstance, for example when the rank of $E$ equals the dimension of $M$ and a solder form has been chosen, one can use the soldering to interchange the indices and define a notion of torsion for affine connections which are not the Levi-Civita connection.

{{See also|Gauge group (mathematics)}}

Given two connections $\backslash nabla\_1,\; \backslash nabla\_2$ on a vector bundle $E\backslash to\; M$, it is natural to ask when they might be considered equivalent. There is a well-defined notion of an automorphism of a vector bundle $E\backslash to\; M$. A section $u\backslash in\; \backslash Gamma(\backslash operatorname\{End\}(E))$ is an automorphism if $u(x)\backslash in\; \backslash operatorname\{End\}(E\_x)$ is invertible at every point $x\backslash in\; M$. Such an automorphism is called a gauge transformation of $E$, and the group of all automorphisms is called the gauge group, often denoted $\backslash mathcal\{G\}$ or $\backslash operatorname\{Aut\}(E)$. The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle $\backslash operatorname\{Ad\}(\backslash mathcal\{F\}(E))$ of the frame bundle of the vector bundle $E$. This is not to be confused with the lowercase a adjoint bundle $\backslash operatorname\{ad\}(\backslash mathcal\{F\}(E))$, which is naturally identified with $\backslash operatorname\{End\}(E)$ itself. The bundle $\backslash operatorname\{Ad\}\; \backslash mathcal\{F\}(E)$ is the associated bundle to the principal frame bundle by the conjugation representation of $G=\backslash operatorname\{GL\}(r)$ on itself, $g\backslash mapsto\; ghg^\{-1\}$, and has fibre the same general linear group $\backslash operatorname\{GL\}(r)$ where $\backslash operatorname\{rank\}\; (E)\; =\; r$. Notice that despite having the same fibre as the frame bundle $\backslash mathcal\{F\}(E)$ and being associated to it, $\backslash operatorname\{Ad\}(\backslash mathcal\{F\}(E))$ is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised as $\backslash mathcal\{G\}\; =\; \backslash Gamma(\backslash operatorname\{Ad\}\; \backslash mathcal\{F\}(E)).$

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

:$(u\backslash cdot\; \backslash nabla)\_X(s)\; =\; u(\backslash nabla\_X\; (u^\{-1\}(s))$

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

:

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

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

:$u\backslash cdot\; \backslash nabla\; =\; \backslash nabla\; -\; d^\{\backslash nabla\}(u)\; u^\{-1\}$

which is to say that $A\_u\; =\; -\; d^\{\backslash nabla\}(u)\; u^\{-1\}$.

Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space $\backslash mathcal\{B\}\; =\; \backslash mathcal\{A\}/\backslash mathcal\{G\}$ is the moduli space of all connections on $E$. 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 $E$, which is of significant interest in gauge theory and physics.

• 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 $\backslash pi:\; \backslash R^2\backslash times\; \backslash R\; \backslash to\; \backslash R$ can be described explicitly as the operator

::

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

::$\backslash begin\{cases\}\; \backslash R\; \backslash to\; \backslash R^2\; \backslash \backslash \; x\; \backslash mapsto\; (a\_1(x),a\_2(x))\; \backslash end\{cases\}$

:and then

::$\backslash nabla(a)=\; \backslash nabla\; \backslash begin\{bmatrix\}\; a\_1(x)\; \backslash \backslash \; a\_2(x)\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; \backslash frac\{da\_1(x)\}\{dx\}\; +\; f\_\{11\}(x)a\_1(x)\; +\; f\_\{12\}(x)a\_2(x)\; \backslash \backslash \; \backslash frac\{da\_2(x)\}\{dx\}\; +\; f\_\{21\}(x)a\_1(x)\; +\; f\_\{22\}(x)a\_2(x)\backslash end\{bmatrix\}dx$

• 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 $E=M\; \backslash times\; \backslash R$ (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.

• D-module
• Connection (mathematics)

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Category:Connection (mathematics)

Category:Vector bundles