Cup product in bounded cohomology of negatively curved manifolds

Let M M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential alttext="2"> 2 encoding="application/x-tex">2 -form alttext="xi element-of normal upper Omega squared left-parenthesis M right-parenthesis"> ξ ∈ Ω<!-- Ω </mml:msup> ( stretchy="false">) encoding="application/x-tex">\xi \in \Omega ^2(M) defines bounded cocycle alttext="c Subscript xi Baseline C b Superscript 2 c C b encoding="application/x-tex">c_\xi C_b^2(M) by integrating alttext="xi"> encoding="application/x-tex">\xi over straightened -simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when is hyperbolic surface, alttext="normal encoding="application/x-tex">\Omega injects this way in H H encoding="application/x-tex">H_b^2(M) as an infinite dimensional subspace. We show that the cup product of any class form alttext="left-bracket c right-bracket"> stretchy="false">[ stretchy="false">] encoding="application/x-tex">[c_\xi ] , where exact 2-form, other cohomology trivial bullet ∙<!-- ∙ encoding="application/x-tex">H_b^{\bullet }(M) .