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 PabPbc = Pac, then the cocurvature is a vector-valued 2-form on M defined by
![{\displaystyle {\bar {R}}_{P}(X,Y)=(\operatorname {Id} -P)[PX,PY]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14509678a5db234a8fd50ab81f1ce2820e349514)
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. MR 1202431. Zbl 0782.53013.
- v
- t
- e
Various notions of curvature defined in differential geometry
of curves
- Curvature
- Torsion of a curve
- Frenet–Serret formulas
- Radius of curvature (applications)
- Affine curvature
- Total curvature
- Total absolute curvature
of surfaces
- Principal curvatures
- Gaussian curvature
- Mean curvature
- Darboux frame
- Gauss–Codazzi equations
- First fundamental form
- Second fundamental form
- Third fundamental form
- Curvature form
- Torsion tensor
- Cocurvature
- Holonomy
 | This differential geometry-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e
