How Does Holonomy Relate to Curvature in Higher Dimensional Principal Bundles?

[lavinia]

On an SO(2) bundle over a smooth manifold the holonomy around a closed curve that bounds a disk equals the integral of the curvature 2 form over the interior of the disk.

So holonomy measures curvature and visa vera.

More generally if two closed curves are homologous then the difference in their holonomy is equal to the total curvature of the surface that they mutually bound

What is the relationship of holonomy to curvature for higher dimensional principal budles e.g. SO(3) bundles?

 

[Ben Niehoff]

Try Googling "non-Abelian Stokes theorem". There is a sense in which you can define an area integral of the curvature 2-form such that it gives the holonomy around loops.