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 ${\displaystyle {\bar {R}}_{P}}$ is a vector-valued 2-form on M defined by

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

where X and Y are vector fields on M.

See also

• Curvature
• Lie bracket
• Frölicher-Nijenhuis bracket