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

$D\phi(v_0, v_1,\dots, v_k)= d \phi(h v_0 ,h v_1,\dots, h v_k)$

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

$(\rho(\omega) \cdot \phi)(v_1, \dots, v_{k+1}) = {1 \over (1+k)!} \sum_{\sigma} \operatorname{sgn}(\sigma)\rho(\omega(v_{\sigma(1)})) \phi(v_{\sigma(2)}, \dots, v_{\sigma(k+1)}).$

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,

$D^2\phi=F \cdot \phi.$^[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 $\Omega^k(M;E) = \Gamma(\Lambda^k(T^*M) \otimes E)$ of $E$-valued k-forms by

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

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

$-2F(X, Y) s = \left([\nabla_X, \nabla_Y] - \nabla_{[X, Y]}\right) s$

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

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:

   $d \phi(X^{\#}_u) = \left . {d \over dt}\right\vert_0 \phi(u \operatorname{exp}(tX)) = -\rho(X)\phi(u) = -\rho(\omega(X^{\#}_u))\phi(u)$,

   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:

   $D \phi(X_0, \dots, X_k) - d \phi(X_0, \dots, X_k) = {1 \over k+1} \sum_0^k (-1)^i \rho(\omega(X_i)) \phi(X_0, \dots, \widehat{X_i}, \dots, X_k)$,

   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

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

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