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 so that it gives a natural direct sum decomposition of each tangent space $T_{u}P=H_{u}\oplus V_{u}$ into the horizontal and vertical subspaces. Let $h:T_{u}P\to H_{u}$ 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

$D\phi (v_{0},v_{1},\dots ,v_{k})=d\phi (hv_{0},hv_{1},\dots ,hv_{k})$

where v_i are tangent vectors to P at u.

Suppose V is a representation of G; i.e., there is a Lie group homomorphism ρ : G → GL(V). If ϕ is equivariant in the sense:

$R_{g}^{*}\phi =\rho (g)^{-1}\phi$

where $R_{g}(u)=ug$, 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(v₀, ..., v_k) = ψ(hv₀, ..., hv_k).)

• Example: if ω is the connection form on P, then Ω = Dω is called the curvature form of ω. Bianchi's second identity says the exterior covariant derivative of Ω is zero; i.e., DΩ = 0.

We also denote the differential of ρ at the identity element by ρ:

$\rho :{\mathfrak {g}}\to {\mathfrak {gl}}(V).$

If ϕ is a tensorial k-form of type ρ, then

$D\phi =d\phi +\rho (\omega )\cdot \phi ,$

where $\rho (\omega )$ is a ${\mathfrak {gl}}(V)$-valued form, and, following the notation in Lie algebra-valued differential form § Operations,

$(\rho (\omega )\cdot \phi )(v_{1},\dots ,v_{k+1})={1 \over (1+k)!}\sum _{\sigma }\operatorname {sgn} (\sigma )\rho (\omega (v_{\sigma (1)}))\phi (v_{\sigma (2)},\dots ,v_{\sigma (k+1)}).$

• Example: Bianchi's second identity (DΩ = 0) can be stated as: $d\Omega +\operatorname {ad} (\omega )\cdot \Omega =d\Omega +[\omega \wedge \Omega ]=0$.

Unlike the usual exterior derivative, which squares to 0, we have: with F = ρ(Ω), for a tensorial zero-form ϕ,

$D^{2}\phi =F\cdot \phi .$

In particular D² vanishes for a flat connection (i.e., Ω = 0).

If ρ : G → GL(Rⁿ), then one can write

$\rho (\Omega )=F=\sum {F^{i}}_{j}{e^{j}}_{i}$

where ${e^{i}}_{j}$ is the matrix with 1 at the (i, j)-th entry and zero on the other entries. The matrix ${F^{i}}_{j}$ 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 differentiation D given by a connection on P defines

$\nabla :\Gamma (M,E)\to \Gamma (M,TM\otimes E)$

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 forms. This ∇ is called the exterior covariant differentiation on E. One also sets: for a section s of E,

$\nabla _{X}s=i_{X}\nabla s$

where $i_{X}$ is the contraction by X. Explicitly,

$\nabla _{X}s=(hX)s$

since $\nabla _{X}{\overline {\phi }}={\overline {D\phi (X)}}={\overline {d\phi (hX)}}=(hX)s$ when $s={\overline {\phi }}$.

Conversely, given a vector bundle E, one can take its frame bundle, which is a principal bundle, and so gets an exterior covariant differentiation on E (depending on a connection). Identifying tensorial forms and E-valued forms, there is, for example,

$2F(X,Y)s=(-[\nabla _{X},\nabla _{Y}]+\nabla _{[X,Y]})s$.

See also

• Exterior connections

References

• Kobayashi, Shoshichi and Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley-Interscience. ISBN 0-471-15733-3.{{cite book}}: CS1 maint: multiple names: authors list (link)