Let G be a compact subgroup of GLn (R) and 0 ≤ s < r < ∞. We prove that every definable CSG map between affine definable CʳG manifolds is approximated in the definable Cˢ topology by definable CʳG maps. We show that each G invariant proper submersive surjective definable Cʳ function defined on an affine definable CʳG manifold is definably CʳG trivial. Moreover we prove that every noncompact affine definable CʳG manifold admits a unique affine definable CrG compactification up to definable CTG diffeomorphism when r ≥ 2.
Tomohiro KAWAKAMI (Mon,) studied this question.