Let
M
M
be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential
2
2
-form
ξ
∈
Ω
2
(
M
)
\xi \in \Omega ^2(M)
defines a bounded cocycle
c
ξ
∈
C
b
2
(
M
)
c_\xi \in C_b^2(M)
by integrating
ξ
\xi
over straightened
2
2
-simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when
M
M
is a closed hyperbolic surface,
Ω
2
(
M
)
\Omega ^2(M)
injects this way in
H
b
2
(
M
)
H_b^2(M)
as an infinite dimensional subspace. We show that the cup product of any class of the form
[
c
ξ
]
[c_\xi ]
, where
ξ
\xi
is an exact differential 2-form, and any other bounded cohomology class is trivial in
H
b
∙
(
M
)
H_b^{\bullet }(M)
.