This is a general question that asks whether there is geometric significance to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle.

For an SO(2) bundle this is certainly true since the exterior derivative of such a form is the pull back of the Euler class of the bundle. In this case, the form itself is just a connection 1-form and is also a "global angular form".

The connection 1-form on an oriented circle bundle is a special case of a differential form that restricts to a G-invariant volume element on the fibers of a principal G-bundle. So do higher dimensional examples also contain topological significance? Do they generalize the idea of a global angular form?

Strangely for the tangent bundle, dimension 2 is the only dimension where the exterior derivative of this form on the tangent SO(n) bundle has the same dimension as the manifold. In higher dimensions, the dimension of the form is bigger than the dimension of the manifold. So for the tangent bundle one can not pull something back from the base except in dimension 2.

Still one might ask if such higher dimensional forms ever have geometric significance, perhaps even on the principal bundle itself. One might look at principal SO(3)or SU(2) bundles over 4 manifolds for instance or principal SO(4) bundles over 8 manifolds.

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    $\begingroup$ I don't quite understand your question, but indeed, various characteristic classes can be represented in terms of integrals of forms related to a connection on the bundle (e.g., its curvature). $\endgroup$ – Alex Degtyarev Mar 20 '15 at 11:42
  • $\begingroup$ Thanks for your response Alex. The Euler class, Chern classes and Pontryagin classes are invariant polynomials in the curvature 2 form that are "horizontal" and thus pull back to forms on the base manifold. They are not G-invariant volume elements to my knowledge except in the case of oriented circle bundles. One can generalize the Euler class to an oriented sphere bundle in which case one has a global angular form on the sphere bundle but this is not quite the same as what I am asking. $\endgroup$ – Joe S Mar 20 '15 at 12:04
  • $\begingroup$ Also a G-invariant volume element is not horizontal so it does not naturally descend to the base manifold. If the bundle has a section, e.g. a parallelizable manifold, then one can pull the form back but since the form is not horizontal this would depend on the section. $\endgroup$ – Joe S Mar 20 '15 at 12:09
  • $\begingroup$ I think some of these questions will be answered for you if you look at the original paper by Chern and Simons, "Characteristic forms and geometric invariants". $\endgroup$ – Robert Bryant Mar 20 '15 at 12:14
  • $\begingroup$ @JoeS You might want to clarify what you are asking. At the moment it is even difficult to find the question in the text, which means that many people will just not have a look at it! Some maths: In the context of symplectic topology, when you have a Hamiltonian fibration, i.e. a fibre bundle $M \to E \to B$ whose fibre is a symplectic manifold $(M, \omega)$ and the structure group is $Ham(M, \omega)$, you can construct a two-form $\Omega$ on the total space $E$ which restricts to $\omega$ in each fibre. This form is called a coupling form... $\endgroup$ – Oldřich Spáčil Mar 20 '15 at 17:10

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