Metric connection

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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] Other common equivalent formulations of a metric connection include:

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[edit]

Let 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 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

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

Here, the notation merely denotes the ordinary Lie derivative of a scalar function.

A word about notation[edit]

The bundle metric should not be confused with the natural pairing of a vector space to its dual. The latter is a function on the space of endomorphisms so that

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

By contrast, the bundle metric is a function on that is, a bona-fide metric

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

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 having to make reference to the bundle metric. These can be defined making use only of the pairing 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[edit]

Give 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

Skew symmetry[edit]

The connection is skew-symmetric in the vector-space (fiber) indexes; 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 a local frame with i=1,2,...,n. One then has, by definition, that

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

In addition, for each point of the open set 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 followed from the definition of the dual connection as

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

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 is often used in place of A, such 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.

On alternating forms[edit]

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

This extension is done in the most minimal, straight-forward way possible:

where is a p-form, and is a local smooth frame or local smooth section for the vector bundle.

Curvature[edit]

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[disambiguation needed], 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[edit]

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 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[edit]

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

Keep in mind that for n-dimensional vector space, each 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[edit]

All of the indexes 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 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[edit]

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 indexes, 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 connections[edit]

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 *(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 connections[edit]

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.

A word about notation[edit]

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 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 on E leads to Kaluza-Klein theory.

Metric compatibility[edit]

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

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, 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.

References[edit]

  1. ^ a b Jost, Jürgen (2011), Riemannian geometry and geometric analysis, 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.)