# Exterior covariant derivative

In mathematics, the **exterior covariant derivative** is an analog of an exterior derivative that takes into account the presence of a connection.

## Definition

Let *G* be a Lie group and *P* → *M* 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 *v*_{i} are tangent vectors to *P* at *u*.

Suppose that *ρ* : *G* → GL(*V*) is a representation of *G* on a vector space *V*. If *ϕ* is equivariant in the sense that

where
, then *Dϕ* is a tensorial (*k* + 1)-form on *P* of the type *ρ*: it is equivariant and horizontal (a form *ψ* is horizontal if *ψ*(*v*_{0}, ..., *v*_{k}) = *ψ*(*hv*_{0}, ..., *hv*_{k}).)

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 Lie algebra-valued differential form § Operations, 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 *F* = *ρ*(Ω) 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 *D*^{2} vanishes for a flat connection (i.e. when Ω = 0).

If *ρ* : *G* → GL(**R**^{n}), then one can write

where
is the matrix with 1 at the (*i*, *j*)-th entry and zero on the other entries. The matrix
whose entries are 2-forms on *P* is called the **curvature matrix**.

## Exterior covariant derivative for vector bundles

When *ρ* : *G* → GL(*V*) is a representation, one can form the associated bundle *E* = *P* ×_{ρ} *V*. Then the exterior covariant derivative *D* given by a connection on *P* induces an exterior covariant derivative (sometimes called the exterior connection) on the associated bundle, this time using the nabla symbol:

Here, Γ denotes the space of local sections of the vector bundle. The extension is made through the correspondence between *E*-valued forms and tensorial forms of type *ρ* (see tensorial forms on principal bundles.)

Requiring ∇ to satisfy Leibniz's rule, ∇ also acts on any *E*-valued form; thus, it is given on decomposable elements of the space
of
-valued *k*-forms by

- .

For a section *s* of *E*, we also set

where
is the contraction by *X*.

Conversely, given a vector bundle *E*, one can take its frame bundle, which is a principal bundle, and so obtain an exterior covariant differentiation on *E* (depending on a connection). Identifying tensorial forms and *E*-valued forms, one may show that

which can be easily recognized as the definition of the Riemann curvature tensor on Riemannian manifolds.

## Examples

- If
*ω*is the connection form on*P*, then Ω =*Dω*is called the curvature form of*ω*. - Bianchi's second identity, which says that the exterior covariant derivative of Ω is zero (that is,
*D*Ω = 0) can be stated as: .

## Notes

- If
*k*= 0, then, writing for the fundamental vector field (i.e., vertical vector field) generated by*X*in on*P*, we have:- ,

*ϕ*(*gu*) =*ρ*(*g*^{−1})*ϕ*(*u*). On the other hand,*Dϕ*(*X*^{#}) = 0. If*X*is a horizontal tangent vector, then and . For the general case, let*X*_{i}'s be tangent vectors to*P*at some point such that some of*X*_{i}'s are horizontal and the rest vertical. If*X*_{i}is vertical, we think of it as a Lie algebra element and then identify it with the fundamental vector field generated by it. If*X*_{i}is horizontal, we replace it with the horizontal lift of the vector field extending the pushforward π*X*_{i}. This way, we have extended*X*_{i}'s to vector fields. Note the extension is such that we have: [*X*_{i},*X*_{j}] = 0 if*X*_{i}is horizontal and*X*_{j}is vertical. Finally, by the invariant formula for exterior derivative, we have:- ,

- Proof: Since
*ρ*acts on the constant part of*ω*, it commutes with*d*and thus- .

## References

- Kobayashi, Shoshichi; Nomizu, Katsumi (1996).
*Foundations of Differential Geometry, Vol. 1*(New ed.). Wiley-Interscience. ISBN 0-471-15733-3.