  We propose a Wigner quasiprobability distribution function for Hamiltonian
systems in spaces of constant curvature --in this paper on hyperboloids--,
which returns the correct marginals and has the covariance of the Shapiro
functions under SO(D,1) transformations. To the free systems obeying the
Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator
potential in the hyperbolic coordinate. As an example, we analyze the
1-dimensional case on a hyperbola branch, where this conic-oscillator is the
Poschl-Teller potential. We present the analytical solutions and plot the
computed results. The standard theory of quantum oscillators is regained in the
contraction limit to the space of zero curvature.
