EQUIVARIANT DEFINABLE C^r APPROXIMATION THEOREM, DEFINABLE C^rG TRIVIALITY OF G INVARIANT DEFINABLE C^r FUNCTIONS AND COMPACTIFICATIONS

Let G be a compact subgroup of GLn(R) and 0 &le; s < r < ∞. We prove that every definable C^SG map between affine definable C^rG manifolds is approximated in the definable C^s topology by definable C^rG maps. We show that each G invariant proper submersive surjective definable C^r function defined on an affine definable C^rG manifold is definably C^rG trivial. Moreover we prove that every noncompact affine definable C^rG manifold admits a unique affine definable CrG compactification up to definable CTG diffeomorphism when r &ge; 2.