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In [[mathematics]], a '''metric connection''' is a [[connection (vector bundle)|connection]] in a [[vector bundle]] ''E'' equipped with a [[bundle metric]]; that is, a metric for which the [[inner product]] of any two vectors will remain the same when those vectors are [[parallel transport]]ed along any curve.<ref name=jost>
{{citation
 | last = Jost | first = Jürgen
 | doi = 10.1007/978-3-642-21298-7
 | edition = Sixth
 | isbn = 978-3-642-21297-0
 | mr = 2829653
 | publisher = Springer, Heidelberg
 | series = Universitext
 | title = Riemannian geometry and geometric analysis
 | year = 2011
| url = https://cds.cern.ch/record/1666885/files/9783540773405_TOC.pdf
 }}.(''Third edition: see chapter 3; Sixth edition: see chapter 4.'')
</ref>  This is equivalent to:
* A connection for which the [[connection (vector bundle)|covariant derivative]]s of the metric on ''E'' vanish.
* A [[connection (principal bundle)|principal connection]] on the bundle of [[orthonormal frame]]s of ''E''.

A special case of a metric connection is a [[#Riemannian connection|Riemannian connection]]; there exists a unique such connection which is [[torsion tensor|torsion free]], the [[Levi-Civita connection]].  In this case, the bundle ''E'' is the [[tangent bundle]] ''TM'' of a manifold, and the metric on ''E'' is induced by a Riemannian metric on ''M''.

Another special case of a metric connection is a [[#Yang–Mills connection|Yang–Mills connection]], which satisfies the [[Yang–Mills equations]] of 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), and [[Laplacian]], which are required to formulate the Yang–Mills equations.

== Definition ==
Let <math>\sigma,\tau</math> be any [[Section (fiber bundle)|local sections]] of the vector bundle ''E'', and let ''X'' be a vector field on the base space ''M'' of the bundle. Let <math>\langle\cdot ,\cdot\rangle</math> define a [[bundle metric]], that is, a metric on the vector fibers of ''E''. Then, a [[Connection (vector bundle)|connection]] ''D'' on ''E'' is a metric connection if:

:<math>d \langle \sigma,\tau\rangle = \langle D\sigma,\tau\rangle + \langle \sigma,D \tau\rangle.</math>

Here ''d'' is the ordinary [[Differential form|differential]] of a scalar function. The covariant derivative can be extended so that it acts as a map on ''E''-valued [[differential form]]s on the base space:

:<math>D: \Gamma(E)\otimes \Omega^p(M)\to \Gamma(E)\otimes \Omega^{p+1}(M).</math>

One defines <math>D_X f = d_X f \equiv Xf</math> for a function <math>f\in \Omega^0(M)</math>, and

:<math>D(\sigma\otimes\omega)= D\sigma\wedge\omega + \sigma\otimes d\omega</math>

where <math>\sigma\in\Gamma(E)</math> is a local smooth section for the vector bundle and <math>\omega\in \Omega^p(M)</math> is a (scalar-valued) ''p''-form. The above definitions also apply to [[local frame|local smooth frames]] as well as local sections.

===Metric versus dual pairing===
The bundle metric <math>\langle\cdot ,\cdot\rangle</math> imposed on ''E'' should not be confused with the natural pairing <math>(\cdot ,\cdot)</math> of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle of [[endomorphism]]s <math>\mbox{End} (E) = E\otimes E^*,</math> so that
:<math>(\cdot ,\cdot):E\otimes E^* \to M\times\mathbb{R}</math>

pairs vectors with dual vectors (functionals) above each point of ''M''.  That is, if <math>\{e_i\}</math> is any local coordinate frame on ''E'', then one naturally obtains a dual coordinate frame <math>\{e^*_i\}</math> on ''E''* satisfying <math>(e_i ,e^*_j)=\delta_{ij}</math>.

By contrast, the bundle metric <math>\langle\cdot ,\cdot\rangle</math> is a function on <math>E\otimes E,</math>

:<math>\langle\cdot ,\cdot\rangle:E\otimes E \to M\times \mathbb{R}</math>

giving an inner product on each vector space fiber of ''E''. The bundle metric allows one to define an ''orthonormal'' coordinate frame by the equation <math>\langle e_i ,e_j\rangle=\delta_{ij}.</math>

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

Following standard practice,<ref name=jost/> one can define a [[connection form]], the [[Christoffel symbols]] and the [[Riemann curvature]] without reference to the bundle metric, using only the pairing <math>(\cdot ,\cdot).</math> They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the [[Bianchi's second identity|second Bianchi identity]].  However, to define the [[Hodge star]], the [[Laplacian]], 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|Connection form}}

Given a [[atlas (topology)|local bundle chart]], the covariant derivative can be written in the form

:<math>D = d + A</math>
where ''A'' is the [[connection one-form]].

A bit of notational machinery is in order.  Let <math>\Gamma(E)</math> denote the space of differentiable sections on ''E'', let <math>\Omega^p(M)</math> denote the space of [[alternating form|''p''-forms]] on ''M'', and let <math>\mbox{End}(E)=E\otimes E^*</math> be the endomorphisms on ''E''. The covariant derivative, as defined here, is a map
:<math>D: \Gamma(E) \to \Gamma(E)\otimes \Omega^1(M)</math>

One may express the connection form in terms of the [[connection coefficient]]s as

:<math>A_j{}^{k} \ =\ \Gamma^k{}_{ij}\, dx^i.</math>

The point of the notation is to distinguish the indices ''j'', ''k'', which run over the ''n'' dimensions of the fiber, from the index ''i'', which runs over the ''m''-dimensional base space. For the case of a Riemann connection below, the vector space ''E'' is taken to be the tangent bundle ''TM'', and {{nowrap|1=''n'' = ''m''}}.

The notation of ''A'' for the connection form comes from [[physics]], in historical reference to the [[vector potential|vector potential field]] of [[electromagnetism]] and [[gauge theory]]. In mathematics, the notation <math>\omega</math> is often used in place of ''A'', as in the article on the [[connection form]]; unfortunately, the use of <math>\omega</math> for the connection form collides with the use of <math>\omega</math> to denote a generic [[alternating form]] on the vector bundle.

===Skew symmetry===
The connection is [[skew-symmetric matrix|skew-symmetric]] in the vector-space (fiber) indices; that is, for a given vector field <math>X\in TM</math>, the matrix <math>A(X)</math> is skew-symmetric; equivalently, it is an element of the [[Lie algebra]] [[Orthogonal group|<math>\mathfrak{o}(n)</math>]].

This can be seen as follows. Let the fiber be ''n''-dimensional, so that the bundle ''E'' can be given an orthonormal [[local frame]] <math>\{e_i\}</math> with {{nowrap|1=''i'' = 1, 2, ..., ''n''}}. One then has, by definition, that <math>de_i \equiv 0</math>, so that:

:<math>De_i = Ae_i = A_i{}^{j} e_j.</math>

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

:<math>\langle e_i(x), e_j(x)\rangle = \delta_{ij}.</math>

It follows that, for every vector <math>X\in T_xM</math>, that

:<math>
\begin{align}
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{align}</math>

That is, <math>A=-A^\text{T}</math> is skew-symmetric.

This is arrived at by explicitly using the bundle metric; without making use of this, and using only the pairing <math>(\cdot,\cdot)</math>, one can only relate the connection form ''A'' on ''E'' to its dual ''A''{{sup|∗}} on ''E''{{sup|∗}}, as <math>A^*=-A^\text{T}.</math>  This follows from the ''definition'' of the dual connection as <math>d(\sigma,\tau^*)=(D\sigma,\tau^*)+(\sigma,D^*\tau^*).</math>

==Curvature==
There are several notations in use for the curvature of a connection, including a modern one using ''F'' to denote the [[field strength tensor]], a classical one using ''R'' as the [[Riemann curvature tensor|curvature tensor]], and the classical notation for the [[Riemann curvature tensor]], most of which can be extended naturally to the case of vector bundles. ''None'' of 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 of ''E'', as described above.

===Compact style===
The most compact definition of the curvature ''F'' is to define it as the 2-form taking values in <math>\mbox{End}(E)</math>, given by the amount by which the connection fails to be exact; that is, as
:<math>F=D\circ D</math> 
which is an element of
:<math>F\in \Omega^2(M)\otimes \mbox{End}(E),</math>
or equivalently,
:<math>F:\Gamma(E) \to \Gamma(E)\otimes \Omega^2(M)</math>
To relate this to other common definitions and notations, let <math>\sigma\in\Gamma(E)</math> be a section on ''E''. Inserting into the above and expanding, one finds
:<math>F\sigma = (D\circ D)\sigma = (d+A)\circ(d+A)\sigma = (dA+A\wedge A)\sigma</math>

or equivalently, dropping the section
:<math>F = dA+A\wedge A</math>
as a terse definition.

===Component style===
In terms of components, let <math>A=A_idx^i,</math> where <math>dx^i</math> is the standard [[one-form]] coordinate bases on the [[cotangent bundle]] ''T''<sup>*</sup>''M''. Inserting into the above, and expanding, one obtains (using the [[Einstein notation|summation convention]]):
:<math>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.</math>

Keep in mind that for an ''n''-dimensional vector space, each <math>A_i</math> is an ''n''&times;''n'' matrix, the indices of which have been suppressed, whereas the indices ''i'' and ''j'' run over 1,...,''m'', with ''m'' being 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 the [[gluon 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 the [[electromagnetic tensor]] in more or less standard physics notation.

===Relativity style===
All of the indices can be made explicit by providing a [[smooth frame]] <math>\{e_i\}</math>, {{nowrap|1=''i'' = 1, ..., ''n''}} on <math>\Gamma(E)</math>. A given section <math>\sigma \in \Gamma(E)</math> then may be written as

:<math>\sigma=\sigma^i e_i</math>

In this [[local frame]], the connection form becomes
:<math>(A_i dx^i)_j{}^{k} = \Gamma^k{}_{ij} dx^i</math>
with <math>\Gamma^k{}_{ij}</math> being the [[Christoffel symbol]]; again, the index ''i'' runs over {{nowrap|1, ..., ''m''}} (the dimension of the underlying manifold ''M'') while ''j'' and ''k'' run over {{nowrap|1, ..., ''n''}}, the dimension of the fiber.  Inserting and turning the crank, one obtains
:<math>
\begin{align} 
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{align}</math>

where <math>R^k{}_{rij}</math> now identifiable as the [[Riemann curvature tensor]]. This is written in the style commonly employed in many textbooks on [[general relativity]] from the middle-20th century (with several notable exceptions, such as [[Gravitation (book)|MTW]], that pushed early on for an index-free notation). Again, the indices ''i'' and ''j'' run over the dimensions of the manifold ''M'', while ''r'' and ''k'' run over the dimension of the fibers.

===Tangent-bundle style===
The above can be back-ported to the vector-field style, by writing <math>\partial/\partial x^i</math> as the standard basis elements for the [[tangent bundle]] ''TM''. One then defines the curvature tensor as
:<math>R\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) \sigma 
 = \sigma^r R^k_{\;rij} e_k</math>

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

:<math>F\sigma = R(\cdot,\cdot)\sigma</math>

Alternately, the spatial directions can be made manifest, while hiding the indices, by writing the expressions in terms of vector fields ''X'' and ''Y'' on ''TM''.  In the standard basis, ''X'' is
:<math>X=X^i\frac{\partial}{\partial x^i}</math>
and likewise for ''Y''. After a bit of [[plug and chug]], one obtains
:<math>R(X,Y)\sigma = D_X D_Y\sigma - D_Y D_X\sigma - D_{[X,Y]}\sigma</math>
where
:<math>[X,Y]=\mathcal{L}_YX</math>
is the [[Lie derivative]] of the vector field ''Y'' with respect to ''X''.

To recap, the curvature tensor maps fibers to fibers:
:<math>R(X,Y) : \Gamma(E)\to\Gamma(E)</math>
so that
:<math>R(\cdot,\cdot): \Omega^2(M) \otimes \Gamma(E)\to\Gamma(E)</math>

To be very clear, <math>F=R(\cdot,\cdot)</math> 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
:<math>DF=0</math>
without having to make any use of the bundle metric.

==Yang–Mills connection==
{{Main|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 that ''D'' or ''A'' were 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 the [[Hodge star]] and the [[Hodge dual]]; that is needed, in turn, to define the Laplacian, and to demonstrate that
:<math>D{\star}F=0</math>

Any connection that satisfies this identity is referred to as a '''Yang–Mills connection'''<!--boldface per WP:R#PLA-->. It can be shown that this connection is a [[critical point (mathematics)|critical point]] of the [[Euler–Lagrange equation]]s applied to the [[Yang–Mills action]]

:<math>YM_D = \int_M (F,F) {\star}(1)</math>

where <math>{\star}(1)</math> is the [[volume element]], the [[Hodge dual]] of the constant 1.  Note that three different inner products are required to construct this action: the metric connection on ''E'', an inner product on End(''E''), equivalent to the quadratic [[Casimir operator]] (the trace of a pair of matricies), and the Hodge dual.

==Riemannian connection<!--'Riemannian connection' redirects here-->==
An important special case of a metric connection is a '''Riemannian connection'''<!--boldface per WP:R#PLA-->.  This is a connection <math>\nabla</math> on the [[tangent bundle]] of a [[pseudo-Riemannian manifold]] (''M'', ''g'') such that <math>\nabla_X g = 0</math> for all vector fields ''X'' on ''M''.  Equivalently, <math>\nabla</math> is Riemannian if the [[parallel transport]] it defines preserves the metric ''g''.

A given connection <math>\nabla</math> is Riemannian if and only if

:<math>\partial_X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla_XZ) </math>

for all vector fields ''X'', ''Y'' and ''Z'' on ''M'', where <math>\partial_X(g(Y,Z))</math> denotes the derivative of the function <math>g(Y,Z)</math> along this vector field <math>X</math>.

The [[Levi-Civita connection]] is the [[torsion tensor|torsion-free]] Riemannian connection on a manifold.  It is unique by the [[fundamental 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 the [[contorsion tensor]].

In component notation, the [[covariant derivative]] <math>\nabla</math> is compatible with the [[metric tensor]] <math>g_{ab}</math> if

:<math>\nabla_{\!c} \, g_{ab} = 0.</math>

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

:<math>\nabla_a x_b = \nabla_a' x_b - {C_{ab}}^c x_c.</math>

If the space is also [[torsion tensor|torsion-free]], then the tensor <math>{C_{ab}}^c</math> is symmetric in its first two indices.

=== A word about notation ===
It is conventional to change notation and use the nabla symbol ∇ in place of ''D'' in this setting; in other respects, these two are the same thing. That is, ∇ = ''D'' from the previous sections above.

Likewise, the inner product <math>\langle\cdot,\cdot\rangle</math> on ''E'' is replaced by the metric tensor ''g'' on ''TM''. This is consistent with historic usage, but also avoids confusion: for the general case of a vector bundle ''E'', the underlying manifold ''M'' is ''not'' assumed to be endowed with a metric. The special case of manifolds with both a metric ''g'' on ''TM'' in addition to a bundle metric <math>\langle\cdot,\cdot\rangle</math> on ''E'' leads to [[Kaluza–Klein theory]].

== See also ==
* [[Vertical and horizontal bundles]]

==References==
{{reflist}}

*{{cite arXiv
|last1=Rodrigues
|first1=W. A.
|last2=Fernández
|first2=V. V.
|last3=Moya
|first3=A. M.
|year=2005
|title= Metric compatible covariant derivatives
|eprint=math/0501561
}}
*{{Citation
|last=Wald
|first=Robert M.
|title=General Relativity
|publisher=University of Chicago Press
|year=1984
|isbn=0-226-87033-2
|title-link=General Relativity (book)
}}
* {{cite journal | last1 = Schmidt | first1 = B. G. | year = 1973 | title = Conditions on a Connection to be a Metric Connection | url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103858479| journal = Commun. Math. Phys. | volume = 29 | issue = 1| pages = 55–59 | doi=10.1007/bf01661152| bibcode = 1973CMaPh..29...55S| hdl = 10338.dmlcz/127117 | s2cid = 121543450 | hdl-access = free }}

{{Riemannian geometry}}
{{Manifolds}}
{{tensors}}

[[Category:Connection (mathematics)]]
[[Category:Riemannian geometry]]