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
- ^{[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 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 sheaf 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, DΩ = 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:
- ,
- ,
- ↑ 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.