  KLT relations almost factorize closed string amplitudes on $S_2$ by two open
string tree amplitudes which correspond to the left- and the right- moving
sectors. In this paper, we investigate string amplitudes on $D_2$ and $RP_2$.
We find that KLT factorization relations do not hold in these two cases. The
relations between closed and open string amplitudes have new forms. On $D_2$
and $RP_2$, the left- and the right- moving sectors are connected into a single
sector. Then an amplitude with closed strings on $D_2$ or $RP_2$ can be given
by one open string tree amplitude except for a phase factor. The relations
depends on the topologies of the world-sheets.Under T-duality, the relations on
$D_2$ and $RP_2$ give the amplitudes between closed strings scattering from
D-brane and O-plane respectively by open string partial amplitudes.In the low
energy limits of these two cases, the factorization relations for graviton
amplitudes do not hold. The amplitudes for gravitons must be given by the new
relations instead.
