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We know given a Connection 1-form $\omega$ on a Principle bundle $P(M,G)$ we can define a curvature 2-form $\Omega$ of $\omega$. We also know that given a Connection 1-form $\omega$ we can define a Connection $\lambda$ (a smooth distribution on $P$) .

My question is the following:

Is there any equivalent notion of Curvature 2- form which arise from the notion ( when Connection is given as a smooth distribution on $P$) and independent of the notion(Connection 1-form)? Since we already know that when a Connection is given as a distribution then we can define the notion of Parallel transport and Holonomy groups independent of the notion of Connection 1-form. Can we some way define the notion of Curvature in terms of Parallel transport and Holonomy groups independent of considering the corresponding connection 1-form of the given Connection so that "this" notion of curvature obtained in this way will be equivalent to the our standard notion of Curvature 2-form of a Connection 1 -form?

I think Ambrose-Singer Holonomy Theorem relates partially the Curvature 2-form of a Connection form and the Holonomy of the corresponding Connection. Can it be used in some way to define alternative independent equivalent notion of Curvature of a Connection?

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  • $\begingroup$ @FrancoisZiegler Thanks .. I saw the reference. But its unanswered. Is there such an equivalent notion of the Curvature? $\endgroup$ – Wandereradi Feb 22 at 16:56
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    $\begingroup$ Here’s an answer for Riemann curvature but I’m pretty sure the same proof works for the curvature of a connection on a principal bundle. deaneyang.com/papers/holonomy.pdf $\endgroup$ – Deane Yang Feb 22 at 18:27
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    $\begingroup$ @Wandereradi I'd say that no, besides the 2-form there isn’t another notion or mathematical object one calls curvature, at least not in the same way that “connection” can be the horizontal distribution, 1-form, or covariant derivative $\nabla$ depending on who’s talking. (Spivak’s comprehensive vol. II seems to confirm this, which is all for the better if you ask me.) $\endgroup$ – Francois Ziegler Feb 22 at 18:54
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    $\begingroup$ The projection of the bracket of two horizontal vector fields to TP/TH defines a tensor. $\endgroup$ – Martin de Borbon Feb 22 at 23:26
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    $\begingroup$ If you refresh the link deaneyang.com/papers/holonomy.pdf, you'll get a new version that does the same calculation but for a connection on a vector bundle. And indeed the idea is to take a limit of loops whose lengths converge to zero, as well as the areas of the disks spanned by them. $\endgroup$ – Deane Yang Feb 23 at 6:56

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