  We study low-temperature properties of the $XY$ spin model on a negatively
curved surface. Geometric curvature of the surface gives rise to frustration in
local spin configuration, which results in the formation of high-energy spin
clusters scattered over the system. Asymptotic behavior of the spin-glass
susceptibility suggests a zero-temperature glass transition, which is
attributed to multiple optimal configurations of spin clusters due to nonzero
surface curvature of the system. It implies that a constant ferromagnetic spin
interaction on a regular lattice can exhibit glasslike behavior without
possessing any disorder if the lattice is put on top of a negatively curved
space such as a hyperbolic surface.
