Editing Curvature form

{{Short description|Term in differential geometry}}
In [[differential geometry]], the '''curvature form''' describes [[curvature]] of a [[connection form|connection]] on a [[principal bundle]]. The [[Riemann curvature tensor]] in [[Riemannian geometry]] can be considered as a special case.

==Definition==

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 [[Lie algebra-valued form|<math>\mathfrak g</math>-valued]] [[Differential form|one-form]] on ''P''). 

Then the '''curvature form''' is the <math>\mathfrak g</math>-valued 2-form on ''P'' defined by

:<math>\Omega=d\omega + {1 \over 2}[\omega \wedge \omega] = D \omega.</math>

(In another convention, 1/2 does not appear.) Here <math>d</math> stands for [[exterior derivative]], <math>[\cdot \wedge \cdot]</math> is defined in the article "[[Lie algebra-valued form]]" and ''D'' denotes the [[exterior covariant derivative]]. In other terms,<ref>since <math>[\omega \wedge \omega](X, Y) = \frac{1}{2}([\omega(X), \omega(Y)] - [\omega(Y), \omega(X)])</math>. Here we use also the <math>\sigma=2 </math> Kobayashi convention  for the exterior derivative of a one form which is then <math>d\omega(X, Y) = \frac12(X\omega(Y) - Y \omega(X) - \omega([X, Y])) </math></ref>
:<math>\,\Omega(X, Y)= d\omega(X,Y) + {1 \over 2}[\omega(X),\omega(Y)]</math>
where ''X'', ''Y'' are tangent vectors to ''P''.<!-- do not remove 1/2 -->

There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', then<ref>Proof: <math>\sigma\Omega(X, Y) = \sigma d\omega(X, Y) = X\omega(Y) - Y \omega(X) - \omega([X, Y]) = -\omega([X, Y]).</math></ref>
:<math>\sigma\Omega(X, Y) = -\omega([X, Y]) = -[X, Y] + h[X, Y]</math>
where ''hZ'' means the horizontal component of ''Z'', on the right we identified a vertical vector field and a Lie algebra element generating it ([[fundamental vector field]]), and <math>\sigma\in \{1, 2\}</math> is the inverse of the normalization factor used by convention in the formula for the [[exterior derivative#In terms of invariant formula|exterior derivative]].

A connection is said to be '''[[flat vector bundle|flat]]''' if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology.

===Curvature form in a vector bundle===
If ''E'' → ''B'' is a vector bundle, then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation of E.&nbsp;Cartan:

:<math>\,\Omega = d\omega + \omega \wedge \omega, </math>

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

:<math>\Omega^i_j = d{\omega^i}_j + \sum_k {\omega^i}_k \wedge {\omega^k}_j.</math>

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 the Lie algebra of O(''n''), i.e. 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.

:<math>\,R(X, Y) = \Omega(X, Y),</math>

using the standard notation for the Riemannian curvature tensor.

==Bianchi identities==
{{see also|Contracted Bianchi identities}}
{{see also|Riemann curvature tensor#Symmetries and identities}}
If <math>\theta</math> is the canonical vector-valued 1-form on the [[frame bundle]], the [[Connection form#Torsion|torsion]] <math>\Theta</math> of the [[connection form]] <math>\omega</math> is the vector-valued 2-form defined by the structure equation

:<math>\Theta = d\theta + \omega\wedge\theta = D\theta,</math>

where as above ''D'' denotes the [[exterior covariant derivative]].

The first Bianchi identity takes the form

:<math>D\Theta = \Omega\wedge\theta.</math>

The second Bianchi identity takes the form

:<math>\, D \Omega = 0 </math>

and is valid more generally for any [[Connection form|connection]] in a [[principal bundle]].

The Bianchi identities can be written in tensor notation as:
<math> R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m} = 0.</math>

The [[contracted Bianchi identities]] are used to derive the [[Einstein tensor]] in the [[Einstein field equations]], a key component in the [[general theory of relativity]].{{clarify|reason=what is 'bulk of general relativity'?|date=October 2022}}

==Notes==
{{reflist}}

==References==
* [[Shoshichi Kobayashi]] and [[Katsumi Nomizu]] (1963) [[Foundations of Differential Geometry]], Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, [[Wiley Interscience]].

==See also==

* [[Connection (principal bundle)]]
* [[Basic introduction to the mathematics of curved spacetime]]
* [[Contracted Bianchi identities]]
* [[Einstein tensor]]
* [[Einstein field equations]]
* [[General theory of relativity]]
* [[Chern-Simons form]]
* [[Curvature of Riemannian manifolds]]
* [[Gauge theory]]

{{curvature}}

[[Category:Curvature tensors]]
[[Category:Differential geometry]]