Non-Hermitian exceptional physics in RP² hyperbolic media

Conventional momentum space provides an orientable base space of a torus for topological classifications based on band theory. Here, we introduce a non-orientable momentum space isomorphic to the real projective plane RP^2 within the low-symmetry media. We show that the local band fluidity can be characterized by an expanded dihedral group with non-Abelian properties, while the global band fluidity offers a versatile platform to explore the evolution of non-Hermitian exceptional manifolds, including order-1, higher-order, hybrid exceptional manifolds, diabolic points and even bound states in the continuum. Furthermore, the non-orientable momentum space can pave the way for exploring the emergence of phenomena for exceptional manifolds.