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| In [[differential geometry]], the '''curvature form''' describes [[curvature]] of a [[connection form|connection]] on a [[principal bundle]]. It can be considered as an alternative to or generalization of [[Riemann curvature tensor|curvature tensor]] in [[Riemannian geometry]].
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| ==Definition==
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| Let ''G'' be a [[Lie group]] with [[Lie algebra]] <math>\mathfrak g</math>, and ''P'' → ''B'' be a [[principal bundle|principal ''G''-bundle]]. Let ω be an [[Ehresmann connection]] on ''P'' (which is a <math>\mathfrak g</math>-valued [[Differential form|one-form]] on ''P'').
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| Then the '''curvature form''' is the <math>\mathfrak g</math>-valued 2-form on ''P'' defined by
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| :<math>\Omega=d\omega +{1\over 2}[\omega,\omega]=D\omega.</math> | |
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| Here <math>d</math> stands for [[exterior derivative]], <math>[\cdot,\cdot]</math> is defined by <math>[\alpha \otimes X, \beta \otimes Y] := \alpha \wedge \beta \otimes [X, Y]_\mathfrak{g}</math> and ''D'' denotes the [[exterior covariant derivative]]. In other terms,
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| :<math>\,\Omega(X,Y)=d\omega(X,Y) + [\omega(X),\omega(Y)]. </math>
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| ===Curvature form in a vector bundle===
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| If ''E'' → ''B'' is a vector bundle. then one can also think of ω as
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| a matrix of 1-forms and the above formula becomes the structure equation:
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| :<math>\,\Omega=d\omega +\omega\wedge \omega, </math>
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| where <math>\wedge</math> is the [[Exterior power|wedge product]]. More precisely, if <math>\omega^i_{\ j}</math> and <math>\Omega^i_{\ j}</math> denote components of ω and Ω correspondingly, (so each <math>\omega^i_{\ j}</math> is a usual 1-form and each <math>\Omega^i_{\ j}</math> is a usual 2-form) then
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| :<math>\Omega^i_{\ j}=d\omega^i_{\ j} +\sum_k \omega^i_{\ k}\wedge\omega^k_{\ j}.</math> | |
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| For example, for the [[tangent bundle]] of a [[Riemannian manifold]], the structure group is O(''n'') and Ω is a 2-form with values in O(''n''), the [[skew-symmetric matrix|antisymmetric matrices]]. In this case the form Ω is an alternative description of the [[Riemann curvature tensor|curvature tensor]], i.e.
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| :<math>\,R(X,Y)=\Omega(X,Y),</math> | |
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| using the standard notation for the Riemannian curvature tensor.
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| ==Bianchi identities==
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| If <math>\theta</math> is the canonical vector-valued 1-form on the frame bundle,
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| the [[Connection form#Torsion|torsion]] <math>\Theta</math> of the [[connection form]]
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| <math>\omega</math>
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| is the vector-valued 2-form defined by the structure equation
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| :<math>\Theta=d\theta + \omega\wedge\theta = D\theta,</math>
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| where as above ''D'' denotes the [[Connection form#Exterior covariant derivative|exterior covariant derivative]].
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| The first Bianchi identity takes the form
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| :<math>D\Theta=\Omega\wedge\theta.</math> | |
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| The second Bianchi identity takes the form
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| :<math>\, D \Omega = 0 </math>
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| and is valid more generally for any [[Connection form#Connection|connection]] in a [[principal bundle]].
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| ==References==
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| * [[Shoshichi Kobayashi]] and [[Katsumi Nomizu]] (1963) [[Foundations of Differential Geometry]], Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, [[Wiley Interscience]].
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| ==See also==
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| *[[Connection (principal bundle)]]
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| *[[Basic introduction to the mathematics of curved spacetime]]
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| *[[Chern-Simons form]]
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| *[[Curvature of Riemannian manifolds]]
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| *[[Gauge theory]]
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| {{curvature}}
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| [[Category:Differential geometry]]
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| [[Category:Curvature (mathematics)]]
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