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:
Contents
- Definition
- A word about notation
- Connection form
- Skew symmetry
- On alternating forms
- Curvature
- Compact style
- Component style
- Relativity style
- Tangent bundle style
- Yang Mills connections
- Riemannian connections
- Metric compatibility
- References
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
Here, d just the ordinary differential. The above can be written with the capital D as well; this is because
Here, the notation
A word about notation
The bundle metric
pairs vectors to their duals. That is, if the
By contrast, the bundle 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, 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
Connection form
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
Skew symmetry
The connection is skew-symmetric in the vector-space (fiber) indexes; that is, for a given vector field
This can be seen as follows. Let the fiber be n-dimensional, so that the bundle E can be given a local frame
since the ei are constant on the bundle chart. That is,
In addition, for each point
It follows that, for every vector
That is,
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
On alternating forms
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
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
which is an element of
or equivalently,
To to relate this to other common definitions and notations, let
or equivalently, dropping the section
as a terse definition.
Component style
In terms of components, let
Keep in mind that for n-dimensional vector space, each
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
In this local frame, the connection form becomes
with
where
Tangent-bundle style
The above can be back-ported to the vector-field style, by writing
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,
without having to make any use of the bundle metric.
Yang-Mills connections
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
An important special case of a metric connection is a Riemannian connection. This is a connection
A given connection
for all vector fields X, Y and Z on M, where
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
Metric compatibility
In mathematics, given a metric tensor
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,
If the space is also torsion-free, then the tensor