Metric connection

"metric compatible" redirects here. For identically-sized typefaces, seemetric-compatible.

Inmathematics,ametric connectionis aconnectionin avector bundleEequipped with abundle metric;that is, a metric for which theinner productof any two vectors will remain the same when those vectors areparallel transportedalong any curve.This is equivalent to:

• A connection for which thecovariant derivativesof the metric onEvanish.
• Aprincipal connectionon the bundle oforthonormal framesofE.

A special case of a metric connection is aRiemannian connection;there exists a unique such connection which istorsion free,theLevi-Civita connection.In this case, the bundleEis thetangent bundleTMof a manifold, and the metric onEis induced by a Riemannian metric onM.

Another special case of a metric connection is aYang–Mills connection,which satisfies theYang–Mills equationsof motion. Most of the machinery of defining a connection and its curvature can be worked through without requiring any compatibility with the bundle metric. However, once one does require compatibility, this metric connection defines an inner product,Hodge star(which additionally needs a choice of orientation), andLaplacian,which are required to formulate the Yang–Mills equations.

Definition

Let$\sigma,\tau$be anylocal sectionsof the vector bundleE,and letXbe a vector field on the base spaceMof the bundle. Let$\langle \cdot,\cdot \rangle$define abundle metric,that is, a metric on the vector fibers ofE.Then, aconnectionDonEis a metric connection if:

$d\langle \sigma,\tau \rangle =\langle D\sigma,\tau \rangle +\langle \sigma,D\tau \rangle.$

Heredis the ordinarydifferentialof a scalar function. The covariant derivative can be extended so that it acts as a map onE-valueddifferential formson the base space:

$D:\Gamma (E)\otimes \Omega ^{p}(M)\to \Gamma (E)\otimes \Omega ^{p+1}(M).$

One defines$D_{X}f=d_{X}f\equiv Xf$for a function$f\in \Omega ^{0}(M)$,and

$D(\sigma \otimes \omega )=D\sigma \wedge \omega +\sigma \otimes d\omega$

where$\sigma \in \Gamma (E)$is a local smooth section for the vector bundle and$\omega \in \Omega ^{p}(M)$is a (scalar-valued)p-form. The above definitions also apply tolocal smooth framesas well as local sections.

Metric versus dual pairing

The bundle metric$\langle \cdot,\cdot \rangle$imposed onEshould not be confused with the natural pairing$(\cdot,\cdot )$of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle ofendomorphisms${\mbox{End}}(E)=E\otimes E^{*},$so that

$(\cdot,\cdot ):E\otimes E^{*}\to M\times \mathbb {R}$

pairs vectors with dual vectors (functionals) above each point ofM.That is, if$\{e_{i}\}$is any local coordinate frame onE,then one naturally obtains a dual coordinate frame$\{e_{i}^{*}\}$onE* satisfying$(e_{i},e_{j}^{*})=\delta _{ij}$.

By contrast, the bundle metric$\langle \cdot,\cdot \rangle$is a function on$E\otimes E,$

$\langle \cdot,\cdot \rangle:E\otimes E\to M\times \mathbb {R}$

giving an inner product on each vector space fiber ofE.The bundle metric allows one to define anorthonormalcoordinate frame by the equation$\langle e_{i},e_{j}\rangle =\delta _{ij}.$

Given a vector bundle, it is always possible to define a bundle metric on it.

Following standard practice,one can define aconnection form,theChristoffel symbolsand theRiemann curvaturewithout reference to the bundle metric, using only the pairing$(\cdot,\cdot ).$They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy thesecond Bianchi identity.However, to define theHodge star,theLaplacian,the first Bianchi identity, and the Yang–Mills functional, one needs the bundle metric. The Hodge star additionally needs a choice of orientation, and produces the Hodge dual of its argument.

Connection form

Main article:Connection form

Given alocal bundle chart,the covariant derivative can be written in the form

$D=d+A$

whereAis theconnection one-form.

A bit of notational machinery is in order. Let$\Gamma (E)$denote the space of differentiable sections onE,let$\Omega ^{p}(M)$denote the space ofp-formsonM,and let${\mbox{End}}(E)=E\otimes E^{*}$be the endomorphisms onE.The covariant derivative, as defined here, is a map

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

One may express the connection form in terms of theconnection coefficientsas

$A_{j}{}^{k}\ =\ \Gamma ^{k}{}_{ij}\,dx^{i}.$

The point of the notation is to distinguish the indicesj,k,which run over thendimensions of the fiber, from the indexi,which runs over them-dimensional base space. For the case of a Riemann connection below, the vector spaceEis taken to be the tangent bundleTM,andn=m.

The notation ofAfor the connection form comes fromphysics,in historical reference to thevector potential fieldofelectromagnetismandgauge theory.In mathematics, the notation$\omega$is often used in place ofA,as in the article on theconnection form;unfortunately, the use of$\omega$for the connection form collides with the use of$\omega$to denote a genericalternating formon the vector bundle.

Skew symmetry

The connection isskew-symmetricin the vector-space (fiber) indices; that is, for a given vector field$X\in TM$,the matrix$A(X)$is skew-symmetric; equivalently, it is an element of theLie algebra${\mathfrak {o}}(n)$.

This can be seen as follows. Let the fiber ben-dimensional, so that the bundleEcan be given an orthonormallocal frame$\{e_{i}\}$withi= 1, 2,...,n.One then has, by definition, that$de_{i}\equiv 0$,so that:

$De_{i}=Ae_{i}=A_{i}{}^{j}e_{j}.$

In addition, for each point$x\in U\subset M$of the bundle chart, the local frame is orthonormal:

$\langle e_{i}(x),e_{j}(x)\rangle =\delta _{ij}.$

It follows that, for every vector$X\in T_{x}M$,that

${\begin{aligned}0&=X\langle e_{i}(x),e_{j}(x)\rangle \\&=\langle A(X)e_{i}(x),e_{j}(x)\rangle +\langle e_{i}(x),A(X)e_{j}(x)\rangle \\&=A_{i}{}^{j}(X)+A_{j}{}^{i}(X)\\\end{aligned}}$

That is,$A=-A^{\text{T}}$is skew-symmetric.

This is arrived at by explicitly using the bundle metric; without making use of this, and using only the pairing$(\cdot,\cdot )$,one can only relate the connection formAonEto its dualA^∗onE^∗,as$A^{*}=-A^{\text{T}}.$This follows from thedefinitionof the dual connection as$d(\sigma,\tau ^{*})=(D\sigma,\tau ^{*})+(\sigma,D^{*}\tau ^{*}).$

Curvature

There are several notations in use for the curvature of a connection, including a modern one usingFto denote thefield strength tensor,a classical one usingRas thecurvature tensor,and the classical notation for theRiemann curvature tensor,most of which can be extended naturally to the case of vector bundles.Noneof these definitions require either a metric tensor, or a bundle metric, and can be defined quite concretely without reference to these. The definitions do, however, require a clear idea of the endomorphisms ofE,as described above.

Compact style

The most compact definition of the curvatureFis to define it as the 2-form taking values in${\mbox{End}}(E)$,given by the amount by which the connection fails to be exact; that is, as

$F=D\circ D$

which is an element of

$F\in \Omega ^{2}(M)\otimes {\mbox{End}}(E),$

or equivalently,

$F:\Gamma (E)\to \Gamma (E)\otimes \Omega ^{2}(M)$

To relate this to other common definitions and notations, let$\sigma \in \Gamma (E)$be a section onE.Inserting into the above and expanding, one finds

$F\sigma =(D\circ D)\sigma =(d+A)\circ (d+A)\sigma =(dA+A\wedge A)\sigma$

or equivalently, dropping the section

$F=dA+A\wedge A$

as a terse definition.

Component style

In terms of components, let$A=A_{i}dx^{i},$where$dx^{i}$is the standardone-formcoordinate bases on thecotangent bundleT^*M.Inserting into the above, and expanding, one obtains (using thesummation convention):

$F={\frac {1}{2}}\left({\frac {\partial A_{j}}{\partial x^{i}}}-{\frac {\partial A_{i}}{\partial x^{j}}}+[A_{i},A_{j}]\right)dx^{i}\wedge dx^{j}.$

Keep in mind that for ann-dimensional vector space, each$A_{i}$is ann×nmatrix, the indices of which have been suppressed, whereas the indicesiandjrun over 1,...,m,withmbeing the dimension of the underlying manifold. Both of these indices can be made simultaneously manifest, as shown in the next section.

The notation presented here is that which is commonly used in physics; for example, it can be immediately recognizable as thegluon field strength tensor.For the abelian case,n=1, and the vector bundle is one-dimensional; the commutator vanishes, and the above can then be recognized as theelectromagnetic tensorin more or less standard physics notation.

Relativity style

All of the indices can be made explicit by providing asmooth frame$\{e_{i}\}$,i= 1,...,non$\Gamma (E)$.A given section$\sigma \in \Gamma (E)$then may be written as

$\sigma =\sigma ^{i}e_{i}$

In thislocal frame,the connection form becomes

$(A_{i}dx^{i})_{j}{}^{k}=\Gamma ^{k}{}_{ij}dx^{i}$

with$\Gamma ^{k}{}_{ij}$being theChristoffel symbol;again, the indexiruns over1,...,m(the dimension of the underlying manifoldM) whilejandkrun over1,...,n,the dimension of the fiber. Inserting and turning the crank, one obtains

${\begin{aligned}F\sigma &={\frac {1}{2}}\left({\frac {\partial \Gamma ^{k}{}_{jr}}{\partial x^{i}}}-{\frac {\partial \Gamma ^{k}{}_{ir}}{\partial x^{j}}}+\Gamma ^{k}{}_{is}\Gamma ^{s}{}_{jr}-\Gamma ^{k}{}_{js}\Gamma ^{s}{}_{ir}\right)\sigma ^{r}dx^{i}\wedge dx^{j}\otimes e_{k}\\&=R^{k}{}_{rij}\sigma ^{r}dx^{i}\wedge dx^{j}\otimes e_{k}\\\end{aligned}}$

where$R^{k}{}_{rij}$now identifiable as theRiemann curvature tensor.This is written in the style commonly employed in many textbooks ongeneral relativityfrom the middle-20th century (with several notable exceptions, such asMTW,that pushed early on for an index-free notation). Again, the indicesiandjrun over the dimensions of the manifoldM,whilerandkrun over the dimension of the fibers.

Tangent-bundle style

The above can be back-ported to the vector-field style, by writing$\partial /\partial x^{i}$as the standard basis elements for thetangent bundleTM.One then defines the curvature tensor as

$R\left({\frac {\partial }{\partial x^{i}}},{\frac {\partial }{\partial x^{j}}}\right)\sigma =\sigma ^{r}R_{\;rij}^{k}e_{k}$

so that the spatial directions are re-absorbed, resulting in the notation

$F\sigma =R(\cdot,\cdot )\sigma$

Alternately, the spatial directions can be made manifest, while hiding the indices, by writing the expressions in terms of vector fieldsXandYonTM.In the standard basis,Xis

$X=X^{i}{\frac {\partial }{\partial x^{i}}}$

and likewise forY.After a bit ofplug and chug,one obtains

$R(X,Y)\sigma =D_{X}D_{Y}\sigma -D_{Y}D_{X}\sigma -D_{[X,Y]}\sigma$

where

$[X,Y]={\mathcal {L}}_{Y}X$

is theLie derivativeof the vector fieldYwith respect toX.

To recap, the curvature tensor maps fibers to fibers:

$R(X,Y):\Gamma (E)\to \Gamma (E)$

so that

$R(\cdot,\cdot ):\Omega ^{2}(M)\otimes \Gamma (E)\to \Gamma (E)$

To be very clear,$F=R(\cdot,\cdot )$are alternative notations for the same thing. Observe that none of the above manipulations ever actually required the bundle metric to go through. One can also demonstrate the second Bianchi identity

$DF=0$

without having to make any use of the bundle metric.

Yang–Mills connection

Main article:Yang–Mills equations

The above development of the curvature tensor did not make any appeals to the bundle metric. That is, they did not need to assume thatDorAwere metric connections: simply having a connection on a vector bundle is sufficient to obtain the above forms. All of the different notational variants follow directly only from consideration of the endomorphisms of the fibers of the bundle.

The bundle metric is required to define theHodge starand theHodge dual;that is needed, in turn, to define the Laplacian, and to demonstrate that

$D{\star }F=0$

Any connection that satisfies this identity is referred to as aYang–Mills connection.It can be shown that this connection is acritical pointof theEuler–Lagrange equationsapplied to theYang–Mills action

$YM_{D}=\int _{M}(F,F){\star }(1)$

where${\star }(1)$is thevolume element,theHodge dualof the constant 1. Note that three different inner products are required to construct this action: the metric connection onE,an inner product on End(E), equivalent to the quadraticCasimir operator(the trace of a pair of matricies), and the Hodge dual.

Riemannian connection

An important special case of a metric connection is aRiemannian connection.This is a connection$\nabla$on thetangent bundleof apseudo-Riemannian manifold(M,g) such that$\nabla _{X}g=0$for all vector fieldsXonM.Equivalently,$\nabla$is Riemannian if theparallel transportit defines preserves the metricg.

A given connection$\nabla$is Riemannian if and only if

$\partial _{X}(g(Y,Z))=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)$

for all vector fieldsX,YandZonM,where$\partial _{X}(g(Y,Z))$denotes the derivative of the function$g(Y,Z)$along this vector field$X$.

TheLevi-Civita connectionis thetorsion-freeRiemannian connection on a manifold. It is unique by thefundamental theorem of Riemannian geometry.For every Riemannian connection, one may write a (unique) corresponding Levi-Civita connection. The difference between the two is given by thecontorsion tensor.

In component notation, thecovariant derivative$\nabla$is compatible with themetric tensor$g_{ab}$if

$\nabla _{\!c}\,g_{ab}=0.$

Although other covariant derivatives may be defined, usually one only considers the metric-compatible one. This is because given two covariant derivatives,$\nabla$and$\nabla '$,there exists a tensor for transforming from one to the other:

$\nabla _{a}x_{b}=\nabla _{a}'x_{b}-{C_{ab}}^{c}x_{c}.$

If the space is alsotorsion-free,then the tensor${C_{ab}}^{c}$is symmetric in its first two indices.

A word about notation

It is conventional to change notation and use the nabla symbol ∇ in place ofDin this setting; in other respects, these two are the same thing. That is, ∇ =Dfrom the previous sections above.

Likewise, the inner product$\langle \cdot,\cdot \rangle$onEis replaced by the metric tensorgonTM.This is consistent with historic usage, but also avoids confusion: for the general case of a vector bundleE,the underlying manifoldMisnotassumed to be endowed with a metric. The special case of manifolds with both a metricgonTMin addition to a bundle metric$\langle \cdot,\cdot \rangle$onEleads toKaluza–Klein theory.

See also

• Vertical and horizontal bundles

References

• Rodrigues, W. A.; Fernández, V. V.; Moya, A. M. (2005). "Metric compatible covariant derivatives".arXiv:math/0501561.
• Wald, Robert M. (1984),General Relativity,University of Chicago Press,ISBN0-226-87033-2
• Schmidt, B. G. (1973)."Conditions on a Connection to be a Metric Connection".Commun. Math. Phys.29(1): 55–59.Bibcode:1973CMaPh..29...55S.doi:10.1007/bf01661152.hdl:10338.dmlcz/127117.S2CID121543450.