Topological mechanics of hyperbolic and spherical Maxwell lattices

Maxwell lattices, which are at the verge of mechanical instability, are analogs of topological insulators. In fact, they can host boundary-localized excitations that are topologically protected via a bulk-boundary correspondence. In periodic Euclidean lattices, it is known that the boundary modes' location along the edge is determined by the unit cell's geometry in the bulk. Straight lines of bonds (fibers) can be polarized, i.e. floppy modes can accumulate at one fiber's endpoint while depleting at the other. By considering homogeneous Maxwell lattices embedded in manifolds of constant positive and negative Gaussian curvature, we show that the lattice's intrinsic curvature provides an additional way to manipulate the localization of the boundary modes. We test this idea on hyperbolic and spherical versions of the Euclidean-space kagome lattice. Thanks to spatial curvature, rings of corner-sharing triangles can have an arbitrary number of edges. Euclidean straight fibers of bonds are replaced by diverging geodesics which can have both endpoints on the same boundary component. By polarizing the fibers, soft modes can be moved to selected regions along the boundary. We show that curvature can interplay with topological polarization leading to rich new phenomena.