Cocurvature
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In mathematics in the branch of differential geometry, the cocurvature of a connection on a manifold is the
obstruction Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway o ...
to the integrability of the
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.


Definition

If ''M'' is a manifold and ''P'' is a connection on ''M'', that is a vector-valued 1-form on ''M'' which is a projection on T''M'' such that ''PabPbc'' = ''Pac'', then the cocurvature \bar_P is a vector-valued 2-form on ''M'' defined by :\bar_P(X,Y) = (\operatorname - P) X,PY/math> where ''X'' and ''Y'' are vector fields on ''M''.


See also

* Curvature *
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
* Frölicher-Nijenhuis bracket Differential geometry Curvature (mathematics) {{differential-geometry-stub