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We propose a Wigner quasiprobability distribution function for Hamiltoniansystems in spaces of constant curvature --in this paper on hyperboloids--,which returns the correct marginals and has the covariance of the Shapirofunctions under SO(D,1) transformations. To the free systems obeying theLaplace-Beltrami equation on the hyperboloid, we add a conic-oscillatorpotential in the hyperbolic coordinate. As an example, we analyze the1-dimensional case on a hyperbola branch, where this conic-oscillator is thePoschl-Teller potential. We present the analytical solutions and plot thecomputed results. The standard theory of quantum oscillators is regained in thecontraction limit to the space of zero curvature.Comment: 25 pages (included one figure), Latex fil

Wigner functions for curved spaces. I. On hyperboloids