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 ${\overline{R}}_P$ is a vector-valued 2-form on M defined by

${\overline{R}}_P(X,Y)=(\mathrm{Id}-P)[PX,PY]$

where X and Y are vector fields on M.

See also

• Curvature
• Lie bracket
• Frölicher-Nijenhuis bracket

References

• Kolář, Ivan; Michor, Peter W.; Slovák, Jan (1993). Natural operations in differential geometry. Berlin: Springer-Verlag. ISBN 3-540-56235-4. https://www.emis.de///monographs/KSM/.