Difference between revisions of "Differential 1-form"

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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]].
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A '''differential 1-form''' on a [[differential manifold]] is defined in the following ways:
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* It is a section of its [[cotangent bundle]]
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* 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]]
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===Definition with symbols===
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Let <math>M</math> be a [[differential manifold]]. A '''differential 1-form''' on <math>M</math> is defined in the following equivalent ways:
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* It is an element of <math>\Gamma(T^*M)</math>. Here <math>T^*M</math> denotes the [[cotangent bundle]] of <math>M</math>
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* It associates, to every point <math>p \in M</math>, a linear functional on <math>T_p(M)</math>, in a smooth manner
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* 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.
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===Related notions===
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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.

Latest revision as of 19:38, 18 May 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 M be a differential manifold. A differential 1-form on M is defined in the following equivalent ways:

  • It is an element of \Gamma(T^*M). Here T^*M denotes the cotangent bundle of M
  • It associates, to every point p \in M, a linear functional on T_p(M), in a smooth manner
  • it is a smooth map from TM to \R, such that the restriction to any fiber T_p(M), 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.