# Difference between revisions of "Differential 1-form"

From Diffgeom

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===Symbol-free definition=== | ===Symbol-free definition=== | ||

− | A '''differential 1-form''' on a [[differential manifold]] is a section of its [[cotangent bundle]] | + | 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]] | 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 <math>M</math> be a [[differential manifold]]. A '''differential 1-form''' on <math>M</math> is defined in the following equivalent ways: | ||

+ | |||

+ | * It is an element of <math>\Gamma(T^*M)</math>. Here <math>T^*M</math> denotes the [[cotangent bundle]] of <math>M</math> | ||

+ | * It associates, to every point <math>p \in M</math>, a linear functional on <math>T_p(M)</math>, in a smooth manner | ||

+ | * it is a [[smooth map]] from <math>TM</math> to <math>\R</math>, such that the restriction to any fiber <math>T_p(M)</math>, is a linear map. | ||

+ | |||

+ | ===Related notions=== | ||

+ | |||

+ | 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. |

## Revision as of 22:51, 12 April 2008

This article defines a basic construct that makes sense on any differential manifold

View a complete list of basic constructs on differential manifolds

## Definition

### Symbol-free definition

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.

### Related notions

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.