Vector-valued differential form

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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[edit]

Let M be a smooth manifold and EM 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 Λp(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^pT^*M).

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

\Gamma(E\otimes\Lambda^pT^*M) = \Gamma(E) \otimes_{\Omega^0(M)} \Gamma(\Lambda^pT^*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 Ωp(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[edit]

Pullback[edit]

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 : MN 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[edit]

Just as for ordinary differential forms, one can define a wedge product of vector-valued forms. The wedge product of an E1-valued p-form with an E2-valued q-form is naturally an (E1E2)-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}}\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 ∈ Ωp(M) and η ∈ Ωq(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 (EE)-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[edit]

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 {eα} is a basis for V then the differential of a V-valued p-form ω = ωαeα 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{align}
&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{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 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 d2 = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishing curvature).

Lie algebra-valued forms[edit]

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[edit]

Let EM 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 GLk(R) bundle over M. The pullback of E by π is isomorphic to the trivial bundle F(E) × Rk. Therefore, the pullback by π of an E-valued form on M determines an Rk-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 GLk(R) on F(E) × Rk 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 : PM 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 gG, and
  2. \omega(v_1, \ldots, v_p) = 0 whenever at least one of the vi are vertical (i.e., dπ(vi) = 0).

Here Rg denotes the right action of G on P for some gG. 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.