Metric connection

In mathematics, a metric connection is a 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 transported along any curve. Other common equivalent formulations of a metric connection include:

• A connection for which the covariant derivatives of the metric on E vanish.
• A principal connection on the bundle of orthonormal frames of E.

A special case of a metric connection is the Riemannian connection, of which the Levi-Civita connection is a particularly important special case. For both of these, the bundle E is the tangent bundle TM of a manifold. The Levi-Civita connection is the specific Riemannian connection that is torsion free.

A special case of the metric connection is the Yang–Mills connection. That is, most of the machinery of defining a connection on a vector bundle, and then defining a curvature tensor and the like, can go through without requiring any compatibility with the bundle metric. However, once one does require compatibility with the bundle metric, one is able to define an inner product, which can then be used to construct the Hodge star, the Hodge dual and the Laplacian. The Yang-Mills equations of motion are formulated in terms of the dual; a metric connection satisfying these may be called a Yang–Mills connection.

Definition

Let $\sigma ,\tau$ be two different sections of the vector bundle E, and let X be a vector field on the base space M of the bundle. Let $\langle \cdot ,\cdot \rangle$ define the bundle metric, that is, the metric on the vector bundle E. Then, a connection D on E is a metric connection if it satisfies the equation

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

Here, d just the ordinary differential. The above can be written with the capital D as well; this is because $\langle \sigma ,\tau \rangle$ is just a scalar, and so one trivially has that

$D_{X}\langle \sigma ,\tau \rangle =d_{X}\langle \sigma ,\tau \rangle \equiv X\langle \sigma ,\tau \rangle .$

Here, the notation $X\langle \sigma ,\tau \rangle$ merely denotes the ordinary Lie derivative of a scalar function.

A word about notation

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

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

pairs vectors to their duals. That is, if the $\{e_{i}\}$ define a local coordinate frame on E, then one has a dual coordinate frame $\{e_{i}^{*}\}$ on E^* obeying the usual definition of duality:

$(e_{i},e_{j}^{*})=\delta _{ij}$

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

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

The bundle metric allows an orthonormal local coordinate frame to be defined on E, so that

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

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

Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without having to make reference to the bundle metric. These can be defined making use only of the pairing $(\cdot ,\cdot ).$ They will obey the usual symmetry properties; for example, the curvature of the connection will be anti-symmetric in the last two indexes, and will satisfy the second Bianchi identity. However, in order to define the Hodge star, the Laplacian, the first Bianchi identity and the Yang-Mills functional, one needs the bundle metric.

Connection form

Give a local bundle chart, the covariant derivative can be written in the form

$D=d+A$

where A is the connection one-form.

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

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

Skew symmetry

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

This can be seen as follows. Let the fiber be n-dimensional, so that the bundle E can be given a local frame $\{e_{i}\}$ with i=1,2,...,n. One then has, by definition, that

$de_{i}\equiv 0$

since the e_i are constant on the bundle chart. That is,

$De_{i}=Ae_{i}=A_{i}^{\;j}e_{j}$

In addition, for each point $x\in U\subset M$ of the open set $U\subset M$ 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, $A=-A^{T}$ 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 form A on E to its dual A^* on E^*, as $A^{*}=-A^{T}.$ This followed from the definition of the dual connection as

$d(\sigma ,\tau ^{*})=(D\sigma ,\tau ^{*})+(\sigma ,D^{*}\tau ^{*})$

The skew-symmetry corresponds to the anti-symmetry of the first two indexes of the Christoffel symbols. The point of the notation here, as opposed to that of the Christoffel symbols is to distinguish two of the indexes, which run over the n dimensions of the fiber (the vector space), from the third index, which runs over the m-dimensional base-space. For the case of the Riemann connection, below, the vector space E is taken to be the tangent bundle TM, and thus one has n=m.

Nonetheless, taking care not to confuse these two indexes, the Christoffel symbol notation can still be validly used to express the connection form as

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

The notation of A for the connection form comes from physics, in historical reference to the A-field of electromagnetism and gauge theory. In mathematics, the notation $\omega$ is often used in place of A, such as in the article on the connection form; unfortunately, the use of $\omega$ for the connection form collides with the use of $\omega$ to denote a generic alternating form on the vector bundle.

On alternating forms

The covariant derivative can be extended so that it acts as a map on alternating forms on the base space.

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

This extension is done in the most minimal, straightforward way possible:

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

where $\omega \in \Omega ^{p}(M)$ is a p-form, and $\sigma \in \Gamma (E)$ is a local smooth frame or local smooth section for the vector bundle.

Curvature

There are a variety of different notations in use for the curvature of the connection, including a modern one, using F to denote the field strength tensor, a classical one, using R as the 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 ${\mbox{End}}(E)$, 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 on E. 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 standard one-form coordinate bases on the cotangent bundle T^*M. Inserting into the above, and expanding, one obtains

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

Keep in mind that for n-dimensional vector space, each $A_{i}$ is an n×n matrix, the indexes of which have been suppressed, whereas the indexes i and j run over 1,...,m, with m being the dimension of the underlying manifold. Both of these indexes 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 indexes can be made explicit by providing a smooth frame $\{e_{i}\}$, i=1,...,n on $\Gamma (E)$. A given section $\sigma \in \Gamma (E)$ then may be written as

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

In this local frame, the connection form becomes

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

with $\Gamma _{\;ij}^{k}$ being the Christoffel symbol; again, the index i runs over 1,...,m (the dimension of the underlying manifold M) while j and k run over 1,...,n, the dimension of the fiber. Inserting and turning the crank, one obtains

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

where $R_{\;rij}^{k}$ 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 MTW, that pushed early-on for an index-free notation). Again, the indexes 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 $\partial /\partial x^{i}$ as the standard basis elements for the tangent bundle TM. 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 indexes, by writing the expressions in terms of vector fields X and Y on TM. In the standard basis, X is

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

and likewise for Y. After a bit of plug 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 the Lie derivative of the vector field Y with respect to X.

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

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

$D^{*}F=0$

Any connection that satisfies this identity is referred to as a Yang–Mills connection. It can be shown that this connection is a critical point of the Euler-Lagrange equations applied to the Yang-Mills action

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

where *(1) 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

An important special case of a metric connection is a Riemannian connection. This is a connection $\nabla$ on the tangent bundle of a pseudo-Riemannian manifold (M, g) such that $\nabla _{X}g=0$ for all vector fields X on M. Equivalently, $\nabla$ is Riemannian if the parallel transport it defines preserves the metric g.

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

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

for all vector fields X, Y and Z on M, where $\nabla _{X}g(Y,Z)$ denotes the derivative of the function $g(Y,Z)$ along this vector field $X$.

The Levi-Civita connection is the 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.

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 of the previous sections above.

Likewise, the inner product $\langle \cdot ,\cdot \rangle$ 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 endowed with a metric, in general; yet one can always define the inner product (the bundle metric) for any vector bundle. The special case of manifolds that have both a metric g on TM in addition to a bundle metric $\langle \cdot ,\cdot \rangle$ on E leads to Kaluza-Klein theory.

Metric compatibility

In mathematics, given a metric tensor $g_{ab}$, a covariant derivative is said to be compatible with the metric if the following condition is satisfied:

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

Although other covariant derivatives may be supported within the metric, usually one only ever 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 also torsion-free, then the tensor ${C_{ab}}^{c}$ is symmetric in its first two indices.

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, ISBN 0-226-87033-2

• Schmidt, B. G. (1973). "Conditions on a Connection to be a Metric Connection". Commun. Math. Phys. 29: 55–59. Bibcode:1973CMaPh..29...55S. doi:10.1007/bf01661152.