Cocurvature

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In mathematics in the branch of differential geometry, the cocurvature of a connection on a manifold is the obstruction to the integrability of the vertical bundle.

Definition[edit]

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 TM such that PabPbc = Pac, then the cocurvature \bar{R}_P is a vector-valued 2-form on M defined by

\bar{R}_P(X,Y) = (\operatorname{Id} - P)[PX,PY]

where X and Y are vector fields on M.

See also[edit]