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.
An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.)
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). An 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
\Omegap(M,E)=\Gamma(E ⊗ ΛpT*M).
\Gamma(E ⊗ ΛpT*M)=\Gamma(E)
⊗ | |
\Omega0(M) |
\Gamma(ΛpT*M)=\Gamma(E)
⊗ | |
\Omega0(M) |
\Omegap(M),
\Omega0(M,E)=\Gamma(E).
TM ⊗ … ⊗ TM\toE
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
\Omegap(M) ⊗ RV\to\Omegap(M,V),
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
*\omega) | |
(\varphi | |
x(v |
1, … ,vp)=\omega\varphi(x)(d\varphix(v1), … ,d\varphix(vp)).
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 (E1⊗E2)-valued (p+q)-form:
\wedge:
p(M,E | |
\Omega | |
1) |
x
q(M,E | |
\Omega | |
2) |
\to\Omegap+q(M,E1 ⊗ E2).
(\omega\wedgeη)(v1, … ,vp+q)=
1 | |
p!q! |
\sum | |
\sigma\inSp+q |
sgn(\sigma)\omega(v\sigma(1), … ,v\sigma(p)) ⊗ η(v\sigma(p+1), … ,v\sigma(p+q)).
\omega\wedgeη=(-1)pqη\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)=
\dimM | |
oplus | |
p=0 |
\Omegap(M,E)
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 is a basis for V then the differential of a V-valued p-form ω = ωαeα is given by
d\omega=
\alpha)e | |
(d\omega | |
\alpha. |
\begin{align} &d(\omega+η)=d\omega+dη\\ &d(\omega\wedgeη)=d\omega\wedgeη+(-1)p\omega\wedgedη (p=\deg\omega)\\ &d(d\omega)=0. \end{align}
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:\Omega0(M,E)\to\Omega1(M,E).
d\nabla:\Omegap(M,E)\to\Omegap+1(M,E)
d\nabla(\omega\wedgeη)=d\nabla\omega\wedgeη+(-1)p\omega\wedgedη
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 GLk(R) bundle over M. The pullback of E by π is canonically isomorphic to F(E) ×ρ Rk via the inverse of [''u'', ''v''] →u(v), where ρ is the standard representation. 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 : 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
*\omega | |
(R | |
g) |
=\rho(g-1)\omega
\omega(v1,\ldots,vp)=0
Example: If ρ is the adjoint representation of G on the Lie algebra, then the connection form ω satisfies the first condition (but not the second). The associated curvature form Ω satisfies both; hence Ω is a tensorial form of adjoint type. The "difference" of two connection forms is a tensorial form.
Given P and ρ as above one can construct the associated vector bundle E = P ×ρ V. Tensorial q-forms on P are in a natural one-to-one correspondence with E-valued q-forms on M. As in the case of the principal bundle F(E) above, given a q-form
\overline{\phi}
\phi=u-1\pi*\overline{\phi}
V\overset{\simeq}\toE\pi(u)=
*E) | |
(\pi | |
u, |
v\mapsto[u,v]
\overline{\phi}
\Gamma(M,E)\simeq\{f:P\toV|f(ug)=\rho(g)-1f(u)\},\overline{f}\leftrightarrowf
Example: Let E be the tangent bundle of M. Then identity bundle map idE: E →E is an E-valued one form on M. The tautological one-form is a unique one-form on the frame bundle of E that corresponds to idE. Denoted by θ, it is a tensorial form of standard type.
Now, suppose there is a connection on P so that there is an exterior covariant differentiation D on (various) vector-valued forms on P. Through the above correspondence, D also acts on E-valued forms: define ∇ by
\nabla\overline{\phi}=\overline{D\phi}.
In particular for zero-forms,
\nabla:\Gamma(M,E)\to\Gamma(M,T*M ⊗ E)
This is exactly the covariant derivative for the connection on the vector bundle E.[3]
Siegel modular forms arise as vector-valued differential forms on Siegel modular varieties.[4]
\Omegap(M,V)=\Omega0(M,V)
⊗ | |
\Omega0(M) |
\Omegap(M),
R
\Omega0(M,V)
⊗ | |
\Omega0(M) |
\Omegap(M)=(V ⊗ R\Omega0(M))
⊗ | |
\Omega0(M) |
\Omegap(M)=V ⊗ R(\Omega0(M)
⊗ | |
\Omega0(M) |
\Omegap(M))=V ⊗ R\Omegap(M).
D(f\phi)=Df ⊗ \phi+fD\phi