Curved Odd Elasticity

Living materials such as membranes, cytoskeletal assemblies, cell collectives, and tissues can often be described as active solids-materials that are energized from within, with elastic response about a well-defined reference configuration. These materials often live in complex and curved manifolds, yet most descriptions of active solids are flat. Here, we explore the interplay between curvature and nonreciprocal elasticity via a covariant effective theory on curved manifolds in combination with numerical simulations. We find that curvature spatially patterns activity, gaps the spectrum, modifies exceptional points, and introduces non-Hermitian defect modes. Together, these results establish a foundation for hydrodynamic and rheological models on curved manifolds, with direct implications for living matter and active metamaterials.