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[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 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[edit]

• Curvature
• Lie bracket
• Frölicher-Nijenhuis bracket

References[edit]

• Kolář, Ivan; Michor, Peter W.; Slovák, Jan (1993). Natural operations in differential geometry. Berlin: Springer-Verlag. ISBN 3-540-56235-4. MR 1202431. Zbl 0782.53013.

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