Vector-valued 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.

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 Λ^p(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 seventh example here). By convention, an E-valued 0-form is just a section of the bundle E. That is,

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

which is totally skew-symmetric.

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