Editing Curvature form (section)

===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.