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 $T_{u}P=H_{u}\oplus V_{u}$ of each tangent space 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 that ρ : G → GL(V) is a representation of G on a vector space V. If ϕ is equivariant in the sense that

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

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).)

By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ:

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

Let $\omega$ be the connection one-form and $\rho (\omega )$ the representation of the connection in ${\mathfrak {gl}}(V).$ That is, $\rho (\omega )$ is a ${\mathfrak {gl}}(V)$-valued form, vanishing on the horizontal subspace. If ϕ is a tensorial k-form of type ρ, then

$D\phi =d\phi +\rho (\omega )\cdot \phi ,$^[1]

where, following the notation in Lie algebra-valued differential form § Operations, we wrote

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

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,

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

where F = ρ(Ω) is the representation in ${\mathfrak {gl}}(V)$ 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

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

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 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:

$\nabla :\Gamma (M,E)\to \Gamma (M,T^{*}M\otimes E)$

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 ${\displaystyle \Omega ^{k}(M;E)=\Gamma (\Lambda ^{k}(T^{*}M)\otimes E)}$ of $E$-valued k-forms by

${\displaystyle \nabla (\omega \otimes s)=(d\omega )\otimes s+(-1)^{k}\omega \wedge \nabla s\in \Omega ^{k+1}(M;E)}$.

For a section s of E, we also set

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

where $i_{X}$ 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

${\displaystyle -2F(X,Y)s=\left([\nabla _{X},\nabla _{Y}]-\nabla _{[X,Y]}\right)s}$

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: ${\displaystyle d\Omega +\operatorname {ad} (\omega )\cdot \Omega =d\Omega +[\omega \wedge \Omega ]=0}$.

Notes

1. If k = 0, then, writing $X^{{\#}}$ for the fundamental vector field (i.e., vertical vector field) generated by X in ${\mathfrak {g}}$ on P, we have:

   ${\displaystyle d\phi (X_{u}^{\#})=\left.{d \over dt}\right\vert _{0}\phi (u\operatorname {exp} (tX))=-\rho (X)\phi (u)=-\rho (\omega (X_{u}^{\#}))\phi (u)}$,

   since ϕ(gu) = ρ(g⁻¹)ϕ(u). On the other hand, Dϕ(X^#) = 0. If X is a horizontal tangent vector, then $D\phi (X)=d\phi (X)$ and $\omega (X)=0$. 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 πX_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:

   ${\displaystyle D\phi (X_{0},\dots ,X_{k})-d\phi (X_{0},\dots ,X_{k})={1 \over k+1}\sum _{0}^{k}(-1)^{i}\rho (\omega (X_{i}))\phi (X_{0},\dots ,{\widehat {X_{i}}},\dots ,X_{k})}$,

   which is $(\rho (\omega )\cdot \phi )(X_{0},\cdots ,X_{k})$.
2. Proof: Since ρ acts on the constant part of ω, it commutes with d and thus

   ${\displaystyle d(\rho (\omega )\cdot \phi )=d(\rho (\omega ))\cdot \phi -\rho (\omega )\cdot d\phi =\rho (d\omega )\cdot \phi -\rho (\omega )\cdot d\phi }$.

   Then, according to the example at Lie algebra-valued differential form § Operations,

   $D^{2}\phi =\rho (d\omega )\cdot \phi +\rho (\omega )\cdot (\rho (\omega )\cdot \phi )=\rho (d\omega )\cdot \phi +{1 \over 2}\rho ([\omega \wedge \omega ])\cdot \phi ,$

   which is $\rho (\Omega )\cdot \phi$ by E. Cartan's structure equation.

References

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