  Fractional Chern insulators (FCIs) have attracted intensive attention for the
realization of fractional quantum Hall states in the absence of an external
magnetic field. Most of FCIs have been proposed on two-dimensional (2D)
Euclidean lattice models with various boundary conditions. In this work, we
investigate hyperbolic FCIs which are constructed in hyperbolic geometry with
constant negative curvature. Through the studies on hyperbolic analogs of
kagome lattices with hard-core bosons loaded into topological flat bands, we
find convincing numerical evidences of two types of $\nu=1/2$ FCI states,
{\emph {i.e.}}, the conventional and unconventional FCIs. Multiple branches of
edge excitations and geometry-dependent wave functions for both conventional
and unconventional $\nu=1/2$ FCI states are revealed, however, the geometric
degree of freedom in these FCIs plays various roles. Additionally, a
center-localized orbital plays a crucial role in the unconventional FCI state.
