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2008-05-18T20:13:12Z

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Vipul: New page: ==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 fr...

2008-04-12T23:06:22Z

New page: ==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 fr...

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==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 <math>M</math> be a [[differential manifold]] and <math>V</math> a finite-dimensional real vector space. A <math>V</math>-valued differential 1-form on <math>M</math> is defined as follows:

* A [[smooth map]] from <math>TM</math> to <math>V</math>, whose restriction to each fiber <math>T_pM</math>, is a linear map from <math>T_pM</math> to <math>V</math>
* A smoothly varying collection of linear maps from <math>T_pM</math> to <math>V</math> for every <math>p \in M</math>
* A section of the bundle <math>T^*M \otimes V</math>

A particular case of this is a [[Lie algebra-valued differential 1-form]]. Here, a [[Lie group]] <math>G</math> is acting on the differential manifold <matH>M</math>, and the vector space is the Lie algebra <math>\mathfrak{g}</math> of <math>G</math>. Examples are the [[Maurer-Cartan form]] and [[connection form]].

Vector space-valued differential 1-form - Revision history

Vipul: 1 revision

Vipul: New page: ==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 fr...