The nonlinear propagation of cylindrical and spherical positron-acoustic solitary waves (PASWs) in an unmagnetized, collisionless, relativistic, degenerate plasma system that consists of nonrelativistic inertial cold positrons, relativistic degenerate electron and hot positron fluids, and positively-charged immobile ions has been investigated theoretically. By using a reductive perturbation technique, we derive the Korteweg-de Vries (K-dV) equation. The solitary wave solution has been numerically analyzed to comprehend the localized electrostatic disturbances. We observed that the effects of the degenerate pressure, relativity, and the number densities of inertial cold positrons, hot positrons, electrons, and positively-charged static ions notably modify the fundamental features of the cylindrical and the spherical PASWs. The implications of our results for dense plasmas in astrophysical compact objects (e.g., non-rotating white dwarfs, neutron stars, etc.) are briefly discussed.