Vector-valued Differential Form - Formal Definition

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 Λp(T*M), the p-th exterior power of the cotangent bundle of M. The space of such forms is denoted by

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

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,

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

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

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

and that because is a sub-ring of Ω0(M) via the constant functions,