Vector-valued differential form

In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. Vector-valued forms are natural objects in differential geometry and have numerous applications.

Formal definition

Let M be a smooth manifold and E → M be a smooth vector bundle over M. We denote the space of smooth sections of a bundle E by Γ(E). A E-valued differential form of degree p is a smooth section of the tensor product bundle of E with Λ^p(T*M), the p-th exterior power of the cotangent bundle of M. The space of such forms is denoted by

$\Omega ^{p}(M,E)=\Gamma (E\otimes \Lambda ^{p}T^{*}M).$

Because Γ is a monoidal functor, this can also be interpreted as

$\Gamma (E\otimes \Lambda ^{p}T^{*}M)=\Gamma (E)\otimes _{\Omega ^{0}(M)}\Gamma (\Lambda ^{p}T^{*}M)=\Gamma (E)\otimes _{\Omega ^{0}(M)}\Omega ^{p}(M),$

where the latter two tensor products are the tensor product of modules over the ring Ω⁰(M) of smooth R-valued functions on M (see the fifth example here). By convention, an E-valued 0-form is just a section of the bundle E. That is,

$\Omega ^{0}(M,E)=\Gamma (E).\,$

Equivalently, a E-valued differential form can be defined as a bundle morphism

$TM\otimes \cdots \otimes TM\to E$

which is totally skew-symmetric.

Let V be a fixed vector space. A V-valued differential form of degree p is a differential form of degree p with values in the trivial bundle M × V. The space of such forms is denoted Ω^p(M, V). When V = R one recovers the definition of an ordinary differential form. If V is finite-dimensional, then one can show that the natural homomorphism

$\Omega ^{p}(M)\otimes _{\mathbb {R} }V\to \Omega ^{p}(M,V),$

where the first tensor product is of vector spaces over R, is an isomorphism. One can verify this for p=0 by turning a basis for V into a set of constant functions to V, which allows the construction of an inverse to the above homomorphism. The general case can be proved by noting that

$\Omega ^{p}(M,V)=\Omega ^{0}(M,V)\otimes _{\Omega ^{0}(M)}\Omega ^{p}(M),$

and that because $\mathbb {R}$ is a sub-ring of Ω⁰(M) via the constant functions,

$\Omega ^{0}(M,V)\otimes _{\Omega ^{0}(M)}\Omega ^{p}(M)=(V\otimes _{\mathbb {R} }\Omega ^{0}(M))\otimes _{\Omega ^{0}(M)}\Omega ^{p}(M)=V\otimes _{\mathbb {R} }(\Omega ^{0}(M)\otimes _{\Omega ^{0}(M)}\Omega ^{p}(M))=V\otimes _{\mathbb {R} }\Omega ^{p}(M).$

Operations on vector-valued forms

Pullback

One can define the pullback of vector-valued forms by smooth maps just as for ordinary forms. The pullback of an E-valued form on N by a smooth map φ : M → N is an (φ*E)-valued form on M, where φ*E is the pullback bundle of E by φ.

The formula is given just as in the ordinary case. For any E-valued p-form ω on N the pullback φ*ω is given by

$(\varphi ^{*}\omega )_{x}(v_{1},\cdots ,v_{p})=\omega _{\varphi (x)}(\mathrm {d} \varphi _{x}(v_{1}),\cdots ,\mathrm {d} \varphi _{x}(v_{p})).$

Wedge product

Just as for ordinary differential forms, one can define a wedge product of vector-valued forms. The wedge product of a E₁-valued p-form with a E₂-valued q-form is naturally a (E₁⊗E₂)-valued (p+q)-form:

$\wedge :\Omega ^{p}(M,E_{1})\times \Omega ^{q}(M,E_{2})\to \Omega ^{p+q}(M,E_{1}\otimes E_{2}).$

The definition is just as for ordinary forms with the exception that real multiplication is replaced with the tensor product:

$(\omega \wedge \eta )(v_{1},\cdots ,v_{p+q})={\frac {1}{p!q!}}\sum _{\pi \in S_{p+q}}\operatorname {sgn}(\pi )\omega (v_{\pi (1)},\cdots ,v_{\pi (p)})\otimes \eta (v_{\pi (p+1)},\cdots ,v_{\pi (p+q)}).$

In particular, the wedge product of an ordinary (R-valued) p-form with an E-valued q-form is naturally an E-valued (p+q)-form (since the tensor product of E with the trivial bundle M × R is naturally isomorphic to E). For ω ∈ Ω^p(M) and η ∈ Ω^q(M, E) one has the usual commutativity relation:

$\omega \wedge \eta =(-1)^{pq}\eta \wedge \omega .$

In general, the wedge product of two E-valued forms is not another E-valued form, but rather an (E⊗E)-valued form. However, if E is an algebra bundle (i.e. a bundle of algebras rather than just vector spaces) one can compose with multiplication in E to obtain an E-valued form. If E is a bundle of commutative, associative algebras then, with this modified wedge product, the set of all E-valued differential forms

$\Omega (M,E)=\bigoplus _{p=0}^{\dim M}\Omega ^{p}(M,E)$

becomes a graded-commutative associative algebra. If the fibers of E are not commutative then Ω(M,E) will not be graded-commutative.

Exterior derivative

For any vector space V there is a natural exterior derivative on the space of V-valued forms. This is just the ordinary exterior derivative acting component-wise relative to any basis of V. Explicitly, if {e_α} is a basis for V then the differential of a V-valued p-form ω = ω^αe_α is given by

$d\omega =(d\omega ^{\alpha })e_{\alpha }.\,$

The exterior derivative on V-valued forms is completely characterized by the usual relations:

${\begin{aligned}&d(\omega +\eta )=d\omega +d\eta \\&d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{p}\,\omega \wedge d\eta \qquad (p=\deg \omega )\\&d(d\omega )=0.\end{aligned}}$

More generally, the above remarks apply to E-valued forms where E is any flat vector bundle over M (i.e. a vector bundle whose transition functions are constant). The exterior derivative is defined as above on any local trivialization of E.

If E is not flat then there is no natural notion of an exterior derivative acting on E-valued forms. What is needed is a choice of connection on E. A connection on E is a linear differential operator taking sections of E to E-valued one forms:

$\nabla :\Omega ^{0}(M,E)\to \Omega ^{1}(M,E).$

If E is equipped with a connection ∇ then there is a unique covariant exterior derivative

$d_{\nabla }:\Omega ^{p}(M,E)\to \Omega ^{p+1}(M,E)$

extending ∇. The covariant exterior derivative is characterized by linearity and the equation

$d_{\nabla }(\omega \wedge \eta )=d_{\nabla }\omega \wedge \eta +(-1)^{p}\,\omega \wedge d\eta$

where ω is a E-valued p-form and η is an ordinary q-form. In general, one need not have d_∇² = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishing curvature).

Lie algebra-valued forms

An important case of vector-valued differential forms are Lie algebra-valued forms. These are ${\mathfrak {g}}$-valued forms where ${\mathfrak {g}}$ is a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation is denoted by $[\omega \wedge \eta ]$ to indicate both operations involved, or often just $[\omega ,\eta ]$. For example, if $\omega$ and $\eta$ are Lie algebra-valued one forms, then one has

$[\omega \wedge \eta ](v_{1},v_{2})=[\omega (v_{1}),\eta (v_{2})]-[\omega (v_{2}),\eta (v_{1})].$

With this operation the set of all Lie algebra-valued forms on a manifold M becomes a graded Lie superalgebra.

The operation $[\omega \wedge \eta ]$ can also be defined as the bilinear operation on $\Omega (M,{\mathfrak {g}})$ satisfying

$[(g\otimes \alpha )\wedge (h\otimes \beta )]=[g,h]\otimes (\alpha \wedge \beta )$

for all $g,h\in {\mathfrak {g}}$ and $\alpha ,\beta \in \Omega (M,\mathbb {R} )$.

The alternative notation $[\omega ,\eta ]$, which resembles a commutator, is justified by the fact that if the Lie algebra ${\mathfrak {g}}$ is a matrix algebra then $[\omega \wedge \eta ]$ is nothing but the graded commutator of $\omega$ and $\eta$, i. e. if $\omega \in \Omega ^{p}(M,{\mathfrak {g}})$ and $\eta \in \Omega ^{q}(M,{\mathfrak {g}})$ then

$[\omega \wedge \eta ]=\omega \wedge \eta -(-1)^{pq}\eta \wedge \omega ,$

where $\omega \wedge \eta ,\ \eta \wedge \omega \in \Omega ^{p+q}(M,{\mathfrak {g}})$ are wedge products formed using the matrix multiplication on ${\mathfrak {g}}$.

Basic or tensorial forms on principal bundles

Let E → M be a smooth vector bundle of rank k over M and let π : F(E) → M be the (associated) frame bundle of E, which is a principal GL_k(R) bundle over M. The pullback of E by π is isomorphic to the trivial bundle F(E) × R^k. Therefore, the pullback by π of an E-valued form on M determines an R^k-valued form on F(E). It is not hard to check that this pulled back form is right-equivariant with respect to the natural action of GL_k(R) on F(E) × R^k and vanishes on vertical vectors (tangent vectors to F(E) which lie in the kernel of dπ). Such vector-valued forms on F(E) are important enough to warrant special terminology: they are called basic or tensorial forms on F(E).

Let π : P → M be a (smooth) principal G-bundle and let V be a fixed vector space together with a representation ρ : G → GL(V). A basic or tensorial form on P of type ρ is a V-valued form ω on P which is equivariant and horizontal in the sense that

1. $(R_{g})^{*}\omega =\rho (g^{-1})\omega \,$ for all g ∈ G, and
2. $\omega (v_{1},\ldots ,v_{p})=0$ whenever at least one of the v_i are vertical (i.e., dπ(v_i) = 0).

Here R_g denotes the right action of G on P for some g ∈ G. Note that for 0-forms the second condition is vacuously true.

Given P and ρ as above one can construct the associated vector bundle E = P ×_ρ V. Tensorial forms on P are in one-to-one correspondence with E-valued forms on M. As in the case of the principal bundle F(E) above, E-valued forms on M pull back to V-valued forms on P. These are precisely the basic or tensorial forms on P of type ρ. Conversely given any tensorial form on P of type ρ one can construct the associated E-valued form on M in a straightforward manner.