  The effect of many-body interaction in curved space is studied based on the
extended Bose--Hubbard model on hyperbolic lattices. Using the mean-field
approximation and quantum Monte Carlo simulation, the phase diagram is
explicitly mapped out, which contains the superfluid, supersolid and insulator
phases at various fillings. Particularly, it is revealed that the sizes of the
Mott lobes shrink and the supersolid is stabilized at smaller nearest-neighbor
interaction as $q$ in the Schl\"afli symbol increases. The underlying physical
mechanism is attributed to the increase of the coordination number, and hence
the kinetic energy and the nearest-neighbor interaction. The results suggest
that the hyperbolic lattices may be a unique platform to study the effect of
the coordination number on quantum phase transitions, which may be relevant to
the experiments of ultracold atoms in optical lattices.
