Cocurvature

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

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 P_a^bP_b^c = P_a^c, 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

• curvature
• Lie bracket
• Frölicher-Nijenhuis bracket

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