Obviously, if one pulls back an exact (with respect to the de Rham d) differential form by any map, then one obtains an exact form on the submanifold.
But if one starts out with a form that isn't exact, how can we guarantee that we don't pull it back to an exact form on the embedded submanifold?
Since I'm working with closed embedded surfaces, Stokes's theorem provides an easy criterion to determine if the pullback is exact or not. But this doesn't provide me much information about the original form on the ambient manifold.
I would like to be able to explicitly construct a 2-form on the ambient submanifold that I know - even without testing it with Stokes's theorem - will not pull back to exact form. Is there a special characteristic class of the normal bundle of the submanifold that I should use? Will some theory about Poincare duals of submanifolds help me here?