Horizontal bundle: Difference between revisions
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In [[mathematics]], in the field of [[differential topology]], given |
In [[mathematics]], in the field of [[differential topology]], given |
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:''π'':''E''→''M'', |
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a smooth [[fiber bundle]] over a [[smooth manifold]] ''M'', then the '''[[vertical bundle]]''' V''E'' of ''E'' is the subbundle of the [[tangent bundle]] T''E'' consisting of the vectors which are tangent to the fibers of ''E'' over ''M''. A '''horizontal bundle''' is then a particular choice of a subbundle of T''E'' which is complementary to V''E'', in other words provides a [[complementary subspace]] in each fiber. |
a smooth [[fiber bundle]] over a [[smooth manifold]] ''M'', then the '''[[vertical bundle]]''' V''E'' of ''E'' is the subbundle of the [[tangent bundle]] T''E'' consisting of the vectors which are tangent to the fibers of ''E'' over ''M''. A '''horizontal bundle''' is then a particular choice of a subbundle of T''E'' which is complementary to V''E'', in other words provides a [[complementary subspace]] in each fiber. |
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In full generality, the horizontal bundle concept is one way to formulate the notion of an [[Ehresmann connection]] on a [[fiber bundle]]. However, the concept is usually applied in more specific contexts. |
In full generality, the horizontal bundle concept is one way to formulate the notion of an [[Ehresmann connection]] on a [[fiber bundle]]. However, the concept is usually applied in more specific contexts. |
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More precisely, if ''e'' |
More precisely, if ''e'' ∈ ''E'' with |
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:''π''(''e'')=''x'' ∈ ''M'', |
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then the '''vertical space''' V<sub>''e''</sub>''E'' at ''e'' is the tangent space T<sub>''e''</sub>(''E''<sub>''x''</sub>) to the fiber ''E''<sub>''x''</sub> through ''e''. A horizontal bundle then determines an '''horizontal space''' H<sub>''e''</sub>''E'' such that T<sub>''e''</sub>''E'' is the [[direct sum]] of V<sub>''e''</sub>''E'' and H<sub>''e''</sub>''E''. |
then the '''vertical space''' V<sub>''e''</sub>''E'' at ''e'' is the tangent space T<sub>''e''</sub>(''E''<sub>''x''</sub>) to the fiber ''E''<sub>''x''</sub> through ''e''. A horizontal bundle then determines an '''horizontal space''' H<sub>''e''</sub>''E'' such that T<sub>''e''</sub>''E'' is the [[direct sum]] of V<sub>''e''</sub>''E'' and H<sub>''e''</sub>''E''. |
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If ''E'' is a [[principal bundle|principal ''G''-bundle]] then the horizontal bundle is usually required to be ''G''-invariant: see [[Connection (principal bundle)]] for further details. In particular, this is the case when ''E'' is the [[frame bundle]], i.e., the set of all [[ordered basis| |
If ''E'' is a [[principal bundle|principal ''G''-bundle]] then the horizontal bundle is usually required to be ''G''-invariant: see [[Connection (principal bundle)]] for further details. In particular, this is the case when ''E'' is the [[frame bundle]], i.e., the set of all [[ordered basis|frame]]s for the tangent spaces of the manifold, and ''G'' = GL<sub>''n''</sub>. |
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[[Category:Connection (mathematics)]] |
[[Category:Connection (mathematics)]] |
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Revision as of 05:57, 16 May 2008
In mathematics, in the field of differential topology, given
- π:E→M,
a smooth fiber bundle over a smooth manifold M, then the vertical bundle VE of E is the subbundle of the tangent bundle TE consisting of the vectors which are tangent to the fibers of E over M. A horizontal bundle is then a particular choice of a subbundle of TE which is complementary to VE, in other words provides a complementary subspace in each fiber.
In full generality, the horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. However, the concept is usually applied in more specific contexts.
More precisely, if e ∈ E with
- π(e)=x ∈ M,
then the vertical space VeE at e is the tangent space Te(Ex) to the fiber Ex through e. A horizontal bundle then determines an horizontal space HeE such that TeE is the direct sum of VeE and HeE.
If E is a principal G-bundle then the horizontal bundle is usually required to be G-invariant: see Connection (principal bundle) for further details. In particular, this is the case when E is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, and G = GLn.
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