# Vector space-valued differential 1-form

## Definition

### Symbol-free definition

A **vector space-valued differential 1-form** on a differential manifold, is defined in the following ways:

- It is a smooth map from the tangent bundle of the manifold, to a given vector space, such that the restriction to the tangent space at any point, is a linear map
- It is a smoothly varying collection of linear maps from the tangent spaces at different points on the manifold, to a fixed vector space
- It is a section of the tensor product of the cotangent bundle of the manifold, with a vector space

### Definition with symbols

Let be a differential manifold and a finite-dimensional real vector space. A -valued differential 1-form on is defined as follows:

- A smooth map from to , whose restriction to each fiber , is a linear map from to
- A smoothly varying collection of linear maps from to for every
- A section of the bundle

A particular case of this is a Lie algebra-valued differential 1-form. Here, a Lie group is acting on the differential manifold , and the vector space is the Lie algebra of . Examples are the Maurer-Cartan form and connection form.