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

Contents

DefinitionEdit

Let G be a Lie group and PM 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 is a form defined by

 

where vi 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 is a tensorial (k + 1)-form on P of the type ρ: it is equivariant and horizontal (a form ψ is horizontal if ψ(v0, ..., vk) = ψ(hv0, ..., hvk).)

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

 [1]

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,

 [2]

where F = ρ(Ω) is the representation[clarification needed] 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 (i.e. when Ω = 0).

If ρ : G → GL(Rn), 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 bundlesEdit

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.

ExamplesEdit

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

NotesEdit

  1. ^ If k = 0, then, writing   for the fundamental vector field (i.e., vertical vector field) generated by X in   on P, we have:
     ,
    since ϕ(gu) = ρ(g−1)ϕ(u). On the other hand, (X#) = 0. If X is a horizontal tangent vector, then   and  . For the general case, let Xi's be tangent vectors to P at some point such that some of Xi's are horizontal and the rest vertical. If Xi is vertical, we think of it as a Lie algebra element and then identify it with the fundamental vector field generated by it. If Xi is horizontal, we replace it with the horizontal lift of the vector field extending the pushforward πXi. This way, we have extended Xi's to vector fields. Note the extension is such that we have: [Xi, Xj] = 0 if Xi is horizontal and Xj is vertical. Finally, by the invariant formula for exterior derivative, we have:
     ,
    which is  .
  2. ^ Proof: Since ρ acts on the constant part of ω, it commutes with d and thus
     .
    Then, according to the example at Lie algebra-valued differential form § Operations,
     
    which is   by E. Cartan's structure equation.

ReferencesEdit

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