  Disclinations are ubiquitous lattice defects existing in almost all
crystalline materials. In two-dimensional nanomaterials, disclinations lead to
the warping and deformation of the hosting material, yielding non-Euclidean
geometries. However, such geometries have never been investigated
experimentally in the context of topological phenomena. Here, by creating the
physical realization of disclinations in conical and saddle-shaped acoustic
systems, we demonstrate that disclinations can lead to topologically protected
bound modes in non-Euclidean surfaces. In the designed honeycomb sonic crystal
for p-orbital acoustic waves, non-Euclidean geometry interplay with the
p-orbital physics and the band topology, showing intriguing emergent features
as confirmed by consistent experiments and simulations. Our study opens a
pathway towards topological phenomena in non-Euclidean geometries that may
inspire future studies on, e.g., electrons and phonons in nanomaterials with
curved surfaces.
