Exterior covariant derivative

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 ${\displaystyle T_{u}P=H_{u}\oplus V_{u}}$ of each tangent space into the horizontal and vertical subspaces. Let ${\displaystyle h:T_{u}P\to H_{u}}$ 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

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

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

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

where ${\displaystyle R_{g}(u)=ug}$, 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:

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

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

${\displaystyle D\phi =d\phi +\rho (\omega )\cdot \phi ,}$[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 ${\displaystyle {\mathfrak {gl}}(V)}$ 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

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

where ${\displaystyle {e^{i}}_{j}}$ is the matrix with 1 at the (i, j)-th entry and zero on the other entries. The matrix ${\displaystyle {F^{i}}_{j}}$ whose entries are 2-forms on P is called the curvature matrix.

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:

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

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

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

where ${\displaystyle i_{X}}$ 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.

• Bianchi's second identity, which says that the exterior covariant derivative of Ω is zero (that is, DΩ = 0) can be stated as: ${\displaystyle d\Omega +\operatorname {ad} (\omega )\cdot \Omega =d\Omega +[\omega \wedge \Omega ]=0}$.

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