LOCALLY DEFINABLE C^sG MANIFOLD STRUCTURES OF LOCALLY DEFINABLE C^rG MANIFOLDS

Let G be a finite group. We define locally definable C^γG (1&le;γ&le;ω) manifolds as generalizations of definable C^γG manifolds (1&le;γ&le;ω). Let 0<γ<s<∞. We prove that every affine locally definable C^γG manifold is locally definably C^γG diffeomorphic to a locally definable C^SG manifold. Moreover we prove that for any two affine locally definable C^γG manifolds, they are C^1G diffeomorphic if and only if they are locally definable C^γG diffeomorphic.