局所デファイナブルC^rG多様体の局所デファイナブルC^sG多様体構造について

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 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.