  Topological insulators (TIs) with robust boundary states against
perturbations and disorders provide a unique approach for manipulating waves,
whereas curved space can effectively control the wave propagation on curved
surfaces by the geometric potential effect as well. In general, two-dimensional
(2D) TIs are designed on a flat surface; however, in most practical cases,
curved topological structures are required. In this study, we design a 2D
curved acoustic TI by perforation on a curved rigid plate. We experimentally
demonstrate that a topological localized state stands erect in the bulk gap,
and the corresponding pressure distributions are confined at the position with
the maximal curvature. Moreover, we experimentally verify the robustness of the
topological localized state by introducing defects near the localized position.
To understand the underlying mechanism of the topological localized state, a
tight-binding model considering the geometric potential effect is proposed. The
interaction between the geometrical curvature and topology in the system
provides a novel scheme for manipulating and trapping wave propagation along
the boundary of curved TIs, thereby offering potential applications in flexible
devices.
