Cocurvature Explained

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

References

• Book: Kolář. Ivan. Michor. Peter W.. Slovák. Jan. Natural operations in differential geometry. Springer-Verlag. Berlin. 1993. 3-540-56235-4. 1202431. 0782.53013.