  Nucleon spin structure functions have been investigated mainly by
longitudinally-polarized ones for finding the origin of the nucleon spin. Other
types of spin structure functions are transversely-polarized ones. In
particular, quark transversity distributions in the nucleons have very
different properties from the longitudinally-polarized quark distribution
functions, especially in scaling violation, because they are decoupled from the
gluon transversity, due to the fact that they are helicity-flip (chiral-odd)
distributions. Such studies are valuable for finding not only the origin of the
nucleon spin but also a signature on physics beyond the standard model, because
the electric dipole moment of the neutron is proportional to the transversity
distributions. Now, there is experimental progress on the quark transversity
distributions; however, there is no experimental information on gluon
transversity. In fact, the gluon transversity does not exist for the spin-1/2
nucleon due to the helicity-conservation constraint. One needs a hadron with
spin more than or equal to one, so that the helicity flip of two units is
allowed. A stable spin-1 target is, for example, the deuteron for studying the
gluon transversity. In this work, we propose a possibility for finding the
gluon transversity at hadron-accelerator facilities, especially in the
proton-deuteron Drell-Yan process with the linearly-polarized deuteron, by
showing theoretical formalism and numerical results. We show the dependencies
of the Drell-Yan cross section on the dimuon-mass squared $M_{\mu\mu}^{\,2}$,
the dimuon transverse-momentum $q_T$, the dimuon rapidity $y$ in the
center-of-momentum frame, and the magnitude of the gluon transversity $\Delta_T
g$.
