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On negatively curved compact manifolds, it is possible to associate to everyclosed form a bounded cocycle - hence a bounded cohomology class - viaintegration over straight simplices. The kernel of this map is contained in thespace of exact forms. We show that in degree 2 this kernel is trivial, incontrast with higher degree. In other words, exact non-zero $2$-forms definenon-trivial bounded cohomology classes. This result is the higher dimensionalversion of a classical theorem by Barge and Ghys for surfaces. As aconsequence, one gets that the second bounded cohomology of negatively curvedmanifolds contains an infinite dimensional space, whose classes are explicitlydescribed by integration of forms. This also showcases that some recent resultsby Marasco (arXiv:2202.04419, arXiv:2209.00560) can be applied in higherdimension to obtain new non-trivial results on the vanishing of certain cupproducts and Massey products. Some other applications are discussed.

Bounded Cohomology Classes of Exact Forms