This is a general question that asks whether there is geometric significant 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 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. 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.

  • $\begingroup$ I think this is something like a Thom class. $\endgroup$ – Qiaochu Yuan Mar 20 '15 at 0:17
  • $\begingroup$ Euler classes are related to Thom classes but these forms are not related to Euler classes except in the case of oriented circle bundles. On a sphere bundle one has a similar class - what is called a global angular form - that is the volume element on each fiber sphere and whose exterior derivative is the pull back of the Euler class. The Thom class is a form with compact supports on a vector bundle whose integral along each fiber equals one. $\endgroup$ – Joe S Mar 21 '15 at 1:17

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