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2008-05-18T19:48:43Z

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Vipul: New page: ==Definition== ===Standard definition=== Let $M$ be a differential manifold. A '''metric bundle''' on $M$ is the following data: * A vector bundle $E<...$

2008-04-06T21:24:30Z

New page: ==Definition== ===Standard definition=== Let <math>M</math> be a differential manifold. A '''metric bundle''' on <math>M</math> is the following data: * A vector bundle <math>E<...

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==Definition==

===Standard definition===

Let <math>M</math> be a [[differential manifold]]. A '''metric bundle''' on <math>M</math> is the following data:

* A [[vector bundle]] <math>E</math> over <math>M</math>
* For every point <math>p \in M</math>, a symmetric positive-definite bilinear form on the vector space <math>E(p)</math> over <math>p</math>, that varies smoothly with <math>p</math>.

===Definition as a section===

A metric on <math>E</math> is defined as a section of the bundle <math>Sym^2(E^*)</math>, with the property that the value of the section at every point is positive-definite. Note that a section of <math>Sym^2(E^*)</math> is ''precisely'' the same thing as associating, to every point of <math>M</math>, a symmetric bilinear form. The condition of positive-definiteness needs to be imposed additionally to get a metric.
==Related notions==

===Space of metrics on a bundle===

{{further|[[space of metrics on a bundle]]}}

Given any [[vector bundle]] over a [[differential manifold]], we can look at the space of metrics on it. This space lives as a subset (''not'' a vector subspace) of the space of sections of <math>Sym^2(E^*)</math>.

Metric bundle - Revision history

Vipul: 1 revision

Vipul: New page: ==Definition== ===Standard definition=== Let M be a differential manifold. A '''metric bundle''' on M is the following data: * A vector bundle E<...

Vipul: New page: ==Definition== ===Standard definition=== Let $M$ be a differential manifold. A '''metric bundle''' on $M$ is the following data: * A vector bundle $E<...$