Relativistic Heavy Ion Collider: Difference between revisions
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In [[mathematics]], the '''exterior covariant derivative''', sometimes also '''covariant exterior derivative''', is a very useful notion for [[calculus on manifolds]], which makes it possible to simplify formulas which use a [[Connection (principal bundle)|principal connection]]. | |||
==Definition== | |||
Let ''P'' → ''M'' be a [[principal bundle|principal ''G''-bundle]] on a [[smooth manifold]] ''M''. If ϕ is a [[tensorial form|tensorial ''k''-form]] on ''P'', then its exterior covariant derivative is defined by | |||
:<math>D\phi(X_0,X_1,\dots,X_k)=\mathrm{d}\phi(h(X_0),h(X_1),\dots,h(X_k))</math> | |||
where ''h'' denotes the projection to the [[Horizontal space|horizontal subspace]], ''H<sub>x</sub>'' defined by the connection, with kernel ''V<sub>x</sub>'' (the [[vertical subspace]]) of the tangent bundle of the [[total space]] of the [[fiber bundle]]. Here ''X<sub>i</sub>'' are any vector fields on ''P''. ''D''ϕ is a tensorial (''k'' + 1)-form on ''P''. | |||
==Properties== | |||
Unlike the usual [[exterior derivative]], which squares to 0 (that is d<sup>2</sup> = 0), we have | |||
:<math>D^2\phi=\Omega\wedge\phi</math> | |||
where Ω denotes the [[curvature form]]. In particular ''D''<sup>2</sup> vanishes for a [[flat connection]]. | |||
==See also== | |||
*[[Connection_form#Exterior_connections|Exterior connections]] | |||
==References== | |||
*{{cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = [[Foundations of Differential Geometry]], Vol. 1 | publisher=Wiley-Interscience | year=1996 (New edition) |isbn = 0-471-15733-3}} | |||
{{tensor}} | |||
[[Category:Connection (mathematics)]] | |||
[[Category:Differential geometry]] | |||
[[Category:Fiber bundles]] | |||
{{differential-geometry-stub}} | |||
Revision as of 16:16, 29 November 2013
In mathematics, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a principal connection.
Definition
Let P → M be a principal G-bundle on a smooth manifold M. If ϕ is a tensorial k-form on P, then its exterior covariant derivative is defined by
where h denotes the projection to the horizontal subspace, Hx defined by the connection, with kernel Vx (the vertical subspace) of the tangent bundle of the total space of the fiber bundle. Here Xi are any vector fields on P. Dϕ is a tensorial (k + 1)-form on P.
Properties
Unlike the usual exterior derivative, which squares to 0 (that is d2 = 0), we have
where Ω denotes the curvature form. In particular D2 vanishes for a flat connection.
See also
References
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