Metric connection

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

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

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

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

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

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

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

Following standard practice,[1] one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing ${\displaystyle (\cdot ,\cdot ).}$ They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the 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.

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

This can be seen as follows. Let the fiber be n-dimensional, so that the bundle E can be given an orthonormal local frame ${\displaystyle \{e_{i}\}}$ with i = 1, 2, ..., n. One then has, by definition, that ${\displaystyle de_{i}\equiv 0}$, so that:

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

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

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

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

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

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

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

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

${\displaystyle F=D\circ D}$

which is an element of

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

or equivalently,

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

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

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

or equivalently, dropping the section

${\displaystyle F=dA+A\wedge A}$

as a terse definition.

In terms of components, let ${\displaystyle A=A_{i}dx^{i},}$ where ${\displaystyle dx^{i}}$ is the standard one-form coordinate bases on the cotangent bundle T^*M. Inserting into the above, and expanding, one obtains (using the summation convention):

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

Keep in mind that for an n-dimensional vector space, each ${\displaystyle A_{i}}$ is an n×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.

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

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

In this local frame, the connection form becomes

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

with ${\displaystyle \Gamma ^{k}{}_{ij}}$ 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

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

where ${\displaystyle R^{k}{}_{rij}}$ 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 indices i and j run over the dimensions of the manifold M, while r and k run over the dimension of the fibers.

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

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

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

${\displaystyle F\sigma =R(\cdot ,\cdot )\sigma }$

Alternately, the spatial directions can be made manifest, while hiding the indices, by writing the expressions in terms of vector fields X and Y on TM. In the standard basis, X is

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

and likewise for Y. After a bit of plug and chug, one obtains

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

where

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

is the Lie derivative of the vector field Y with respect to X.

To recap, the curvature tensor maps fibers to fibers:

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

so that

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

To be very clear, ${\displaystyle F=R(\cdot ,\cdot )}$ 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

${\displaystyle DF=0}$

without having to make any use of the bundle metric.

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 ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 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 ${\displaystyle \langle \cdot ,\cdot \rangle }$ on E leads to Kaluza–Klein theory.