Abstract We consider a new approach to the description of dense nuclear matter in the framework of chirally symmetric, quark-based hadron models. As previously in the Skyrme model, the dense environment is described in terms of hyperspherical cells of unit baryon number. The intrinsic curvature of these cells generates a new gauge interaction for the quark fields which mediates interactions with the ambient matter. We apply this approach to the Nambu-Jona-Lasinio (NJL) model, construct its curved-space quark propagator and solve the ladder Bethe-Salpeter equation for the pion. We find a high-density phase transition to chiral restoration, discuss the density dependence of the chiral order parameter and of the pion properties, and compare with results of the conventional chemical-potential approach. The new approach can additionally describe baryon-density-free cavities in the dense medium.