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.[1] This is equivalent to:

A special case of a metric connection is a Riemannian connection; there is a unique such which is 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, which satisfies the Yang–Mills equations of motion. Most of the machinery of defining a connection and its curvature can go through without requiring any compatibility with the bundle metric. However, once one does require compatibility, this metric connection defines an inner product, Hodge star, Hodge dual, and Laplacian, which are required to formulate the Yang-Mills equations.

Definition

Let be any local sections of the vector bundle E, and let X be a vector field on the base space M of the bundle. Let define a bundle metric, that is, a metric on the vector fibers of E. Then, a connection D on E is a metric connection if:

Here d is the ordinary differential of a scalar function. The covariant derivative can be extended so that it acts as a map on E-valued differential forms on the base space:

One defines for a function , and

where is a local smooth section for the vector bundle and is a (scalar-valued) p-form. The above definitions also apply to local smooth frames as well as local sections.

Metric versus dual pairing

The bundle metric imposed on E should not be confused with the natural pairing of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle of endomorphisms so that

pairs vectors with dual vectors (functionals) above each point of M. That is, if is any local coordinate frame on E, then one naturally obtains a dual coordinate frame on E* satisfying .

By contrast, the bundle metric is a function on

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

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 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.

Connection form

Main article: Connection form

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

where A is the connection one-form.

A bit of notational machinery is in order. Let denote the space of differentiable sections on E, let denote the space of p-forms on M, and let be the endomorphisms on E. The covariant derivative, as defined here, is a map

One may express the connection form in terms of the Christoffel symbols as

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 n = m.

The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.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

Skew symmetry

The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field , the matrix is skew-symmetric; equivalently, it is an element of the Lie algebra .

This can be seen as follows. Let the fiber be n-dimensional, so that the bundle E can be given an orthonormal local frame with i=1,2,...,n. One then has, by definition, that , so that:

In addition, for each point of the bundle chart, the local frame is orthonormal:

It follows that, for every vector , that

That is, is skew-symmetric.

This is arrived at by explicitly using the bundle metric; without making use of this, and using only the pairing , one can only relate the connection form A on E to its dual A* on E*, as This follows from the definition of the dual connection as

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 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 , given by the amount by which the connection fails to be exact; that is, as

which is an element of

or equivalently,

To relate this to other common definitions and notations, let be a section on E. Inserting into the above and expanding, one finds

or equivalently, dropping the section

as a terse definition.

Component style

In terms of components, let where 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):

Keep in mind that for an n-dimensional vector space, each 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.

Relativity style

All of the indices can be made explicit by providing a smooth frame , i=1,...,n on . A given section then may be written as

In this local frame, the connection form becomes

with 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

where 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.

Tangent-bundle style

The above can be back-ported to the vector-field style, by writing as the standard basis elements for the tangent bundle TM. One then defines the curvature tensor as

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

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

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

where

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

To recap, the curvature tensor maps fibers to fibers:

so that

To be very clear, 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

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

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

where 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 on the tangent bundle of a pseudo-Riemannian manifold (M, g) such that for all vector fields X on M. Equivalently, is Riemannian if the parallel transport it defines preserves the metric g.

A given connection is Riemannian if and only if

for all vector fields X, Y and Z on M, where denotes the derivative of the function along this vector field .

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.

In component notation, the covariant derivative is compatible with the metric tensor if

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

If the space is also torsion-free, then the tensor 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 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 on E leads to Kaluza–Klein theory.

References

  1. Jost, Jürgen (2011), Riemannian geometry and geometric analysis (PDF), Universitext (Sixth ed.), Springer, Heidelberg, doi:10.1007/978-3-642-21298-7, ISBN 978-3-642-21297-0, MR 2829653.(Third edition: see chapter 3; Sixth edition: see chapter 4.)
  • 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
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