This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds
A differential 1-form on a differential manifold is defined in the following ways:
- It is a section of its cotangent bundle
- It associates in a smooth manner, a linear functional on the tangent space at every point
We can also talk of a differential 1-form restricted to an open subset, which is a section of the restriction of the cotangent bundle to the open subset. This gives rise to the notion of the sheaf of differential 1-forms
Definition with symbols
Let be a differential manifold. A differential 1-form on is defined in the following equivalent ways:
- It is an element of . Here denotes the cotangent bundle of
- It associates, to every point , a linear functional on , in a smooth manner
- it is a smooth map from to , such that the restriction to any fiber , is a linear map.
There is a related notion of a vector space-valued differential 1-form. This is the analog of differential 1-form, taking values in a vector space, rather than in real numbers.