Editing Curvature form (section)

==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}}