Augmented Dickey–Fuller test: Difference between revisions
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In [[mathematics]] in the branch of [[differential geometry]], the '''cocurvature''' of a [[Connection (mathematics)|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 T''M'' such that ''P<sub>a</sub><sup>b</sup>P<sub>b</sub><sup>c</sup>'' = ''P<sub>a</sub><sup>c</sup>'', then the cocurvature <math>\bar{R}_P</math> is a vector-valued 2-form on ''M'' defined by | |||
:<math>\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]] | |||
{{curvature}} | |||
[[Category:Differential geometry]] | |||
[[Category:Curvature (mathematics)]] | |||
{{differential-geometry-stub}} | |||
Revision as of 03:32, 9 January 2014
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.