Sliding Motions on Non‐Euclidean State Spaces: A Differential‐Geometric Perspective

This paper extends sliding‐mode control theory to nonlinear systems evolving on smooth manifolds. Building on differential geometric methods, we reformulate Filippov's notion of solutions, characterize well‐defined vector fields on quotient spaces, and provide a consistent geometric definition of higher‐order sliding modes. We generalize the regular form to non‐Euclidean settings and design explicit first‐ and second‐order sliding‐mode controllers that respect the manifold structure. Particular attention is given to the role of topological obstructions, which are illustrated through examples on the cylinder, Möbius bundle, SO(3), and 2‐sphere. Our results highlight how geometric and topological properties fundamentally influence sliding dynamics and suggest new directions for robust control on nonlinear spaces.