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 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

$Omega^p(M,E)\; =\; Gamma(EotimesLambda^pT^*M).$

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

$TMotimescdotsotimes\; TM\; to\; E$

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.

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

$(varphi^*omega)\_x(v\_1,cdots,\; v\_p)\; =\; omega\_\{varphi(x)\}(mathrm\; dvarphi\_x(v\_1),cdots,mathrm\; dvarphi\_x(v\_p)).$

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:

$wedge\; :\; Omega^p(M,E\_1)\; times\; Omega^q(M,E\_2)\; to\; Omega^\{p+q\}(M,E\_1otimes\; E\_2).$

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

$(omegawedgeeta)(v\_1,cdots,v\_\{p+q\})\; =\; frac\{1\}\{p!q!\}sum\_\{piin\; S\_\{p+q\}\}sgn(pi)omega(v\_\{pi(1)\},cdots,v\_\{pi(p)\})otimes\; eta(v\_\{pi(p+1)\},cdots,v\_\{pi(p+q)\}).$

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:

$omegawedgeeta\; =\; (-1)^\{pq\}etawedgeomega.$

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

$domega\; =\; (domega^alpha)e\_alpha.,$

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

$begin\{align\}$

&d(omega+eta) = domega + deta &d(omegawedgeeta) = domegawedgeeta + (-1)^p,omegawedge detaqquad(p=degomega) &d(domega) = 0. end{align} 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 is 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

$d\_nabla(omegawedgeeta)\; =\; d\_nablaomegawedgeeta\; +\; (-1)^p,omegawedge\; deta$

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 usually denoted [ω∧η] to indicate both operations involved. For example, if ω and η are Lie algebra-valued one forms, then one has

$[omegawedgeeta](v\_1,v\_2)\; =\; [omega(v\_1),eta(v\_2)]\; -\; [omega(v\_2),eta(v\_1)].$

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

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 right-translation by 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.