Let G be a finite group. We define locally definable C^γG (1≤γ≤ω) manifolds as generalizations of definable C^γG manifolds (1≤γ≤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 CSG manifold. Moreover we prove that for any two affine locally definable C^γG manifolds, they are C¹G diffeomorphic if and only if they are locally definable C^γG diffeomorphic.
Tomohiro KAWAKAMI (Tue,) studied this question.