Exterior covariant derivative


In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.

Definition

Let G be a Lie group and be a principal G-bundle on a smooth manifold M. Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let be the projection to the horizontal subspace.
If ϕ is a k-form on P with values in a vector space V, then its exterior covariant derivative Dϕ is a form defined by
where vi are tangent vectors to P at u.
Suppose that is a representation of G on a vector space V. If ϕ is equivariant in the sense that
where, then Dϕ is a tensorial -form on P of the type ρ: it is equivariant and horizontal
By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ:
Let be the connection one-form and the representation of the connection in That is, is a -valued form, vanishing on the horizontal subspace. If ϕ is a tensorial k-form of type ρ, then
where, following the notation in ', we wrote
Unlike the usual exterior derivative, which squares to 0, the exterior covariant derivative does not. In general, one has, for a tensorial zero-form ϕ,
where is the representation in of the curvature two-form Ω. The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism. Note that D2 vanishes for a flat connection.
If, then one can write
where is the matrix with 1 at the -th entry and zero on the other entries. The matrix whose entries are 2-forms on P is called the curvature matrix'''.

For vector bundles

Given a smooth real vector bundle with a connection and rank, the exterior covariant derivative is a real-linear map on the vector-valued differential forms that are valued in :
The covariant derivative is such a map for. The exterior covariant derivatives extends this map to general. There are several equivalent ways to define this object:
  • Suppose that a vector-valued differential 2-form is regarded as assigning to each a multilinear map which is completely anti-symmetric. Then the exterior covariant derivative assigns to each a multilinear map given by the formula
  • The exterior covariant derivative may be characterized by the axiomatic property of defining for each a real-linear map which for is the covariant derivative and in general satisfies the Leibniz rule
  • A vector-valued differential 2-form may be regarded as a certain collection of functions assigned to an arbitrary local frame of over a local coordinate chart of. The exterior covariant derivative is then defined as being given by the functions
In the case of the trivial real line bundle with its standard connection, vector-valued differential forms and differential forms can be naturally identified with one another, and each of the above definitions coincides with the standard exterior derivative.
Given a principal bundle, any linear representation of the structure group defines an associated bundle, and any connection on the principal bundle induces a connection on the associated vector bundle. Differential forms valued in the vector bundle may be naturally identified with fully anti-symmetric tensorial forms on the total space of the principal bundle. Under this identification, the notions of exterior covariant derivative for the principal bundle and for the vector bundle coincide with one another.
The curvature of a connection on a vector bundle may be defined as the composition of the two exterior covariant derivatives and, so that it is defined as a real-linear map. It is a fundamental but not immediately apparent fact that only depends on, and does so linearly. As such, the curvature may be regarded as an element of. Depending on how the exterior covariant derivative is formulated, various alternative but equivalent definitions of curvature can be obtained.
It is a well-known fact that the composition of the standard exterior derivative with itself is zero:. In the present context, this can be regarded as saying that the standard connection on the trivial line bundle has zero curvature.
Some equations involving covariant derivative can be locally solved using Chen's iterated integrals or using approach based on linear homotopy operator.

Example