Cocurvature
From HandWiki
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 PabPbc = Pac, then the cocurvature [math]\displaystyle{ \bar{R}_P }[/math] is a vector-valued 2-form on M defined by
- [math]\displaystyle{ \bar{R}_P(X,Y) = (\operatorname{Id} - P)[PX,PY] }[/math]
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/.
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