Relative Definable Cg Triviality of G Invariant Proper Definable C Functions

Let G be a compact definable C group and 1 ≤ r < ∞. Let X be an affine definable CG manifold and X1, . . . , Xk definable CG submanifolds of X such that X1, . . . , Xk are in general position in X. Suppose that f : X → R is a G invariant proper surjective submersive definable C function such that for every 1 ≤ i1 < · · · < is ≤ k, f |Xi1 ∩ · · · ∩ Xis : Xi1 ∩ · · · ∩ Xis → R is a proper surjective submersion. We prove that there exists a definable CG diffeomorphism h = (h′, f) : (X;X1, . . . , Xk)→ (X∗;X∗ 1 , . . . , X ∗ k)× R, where Z∗denotes Z ∩ f−1(0) for a subset Z of X Moreover we prove an equivariant definable C∞ version under some conditions and its application.