Most of the machinery of defining a connection and its curvature can be worked through without requiring any compatibility with the bundle metric.
However, once one does require compatibility, this metric connection defines an inner product, Hodge star (which additionally needs a choice of orientation), and Laplacian, which are required to formulate the Yang–Mills equations.
The covariant derivative can be extended so that it acts as a map on E-valued differential forms on the base space: One defines
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
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.
Given a local bundle chart, the covariant derivative can be written in the form where A is the connection one-form.
be the endomorphisms on E. The covariant derivative, as defined here, is a map One may express the connection form in terms of the connection coefficients 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E is taken to be the tangent bundle TM, and n = m. The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.
The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field
of the bundle chart, the local frame is orthonormal: It follows that, for every vector
, 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
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
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
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).
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.
After a bit of plug and chug, one obtains where is the Lie derivative of the vector field Y with respect to X.
The above development of the curvature tensor did not make any appeals to the bundle metric.
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
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.
is Riemannian if the parallel transport it defines preserves the metric g. A given connection
if Although other covariant derivatives may be defined, usually one only considers the metric-compatible one.
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.
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.