Hyperbolic lattices constitute a unique platform to study various novel quantum phenomena on a negatively curved space and identify the imprints of spatial curvatures therein. In this talk, first I will show a simple classification scheme on the family of two-dimensional bipartite hyperbolic lattices, resulting from simple but canonical spin-independent tight-binding model for free electrons, yielding three classes of ballistic quantum fluids on negatively curved space, namely Dirac liquids, Fermi liquids, and flat bands, that are, respectively, characterized by a vanishing, a finite, and a diverging density of states near the band center or zero-energy.