LOCALLY DEFINABLE C^{∞}G MANIFOLD STRUCTURES OF LOCALLY DEFINABLE C^{r}G MANIFOLDS

TOMOHIRO KAWAKAMI

Abstract. Let G be a finite abelian group and 1 ≤ r < ∞. We prove that every locally definable C^{r}G manifold admits a unique locally definable C^{∞}G manifold structure up to locally definable C^{∞}G diffeomorphism.

1. Introduction^{1}

Let G be a finite group and 1 ≤ r < ∞. Let M be an o-minimal exponential expansion (ℝ, +, ·, >, e^{x}, …) of the standard structure R = (ℝ, +, ·, <) of the field ℝ of real numbers admits the C^{∞} cell decomposition and has piecewise controlled derivatives.

In this paper we consider existence of locally definable C^{∞}G manifold structures of a locally definable C^{r}G manifold and uniqueness of locally definable C^{∞}G manifold structure up to locally definable C^{∞}G diffeomorphism. If G is a finite abelian group and 1 ≤ s < r < ∞, then unique existence of locally definable C^{r}G manifold structure of a locally definable C^{s}G manifold is studied in [11].

Let 0 ≤ r ≤ ∞. A locally definable C^{r} manifold is a C^{r} manifold admitting a countable system of charts whose gluing maps are of class definable C^{r}. If this system is finite, then it is called a definable C^{r} manifold. Definable C^{r}G manifolds are studied in [5], [6], [7], [8], [9] . A locally definable C^{r} manifold is affine if it can be imbedded into some ℝ^{n} in a locally definable C^{r} way. We can define locally definable C^{r}G manifolds and affine locally definable C^{r}G manifolds in a similar way of equivariant definable cases. Locally definable C^{r}G manifolds are generalizations of definable C^{r}G manifolds and they are studied in [11] when r is a positive integer.

In this paper everything is considered in M, any map is continuous and every manifold does not have boundary unless otherwise stated.

Theorem 1.1. Let G be a finite group and 1 ≤ r < ∞. Then every affine locally definable C^{r}G manifold is locally definably C^{r}G diffeomorphic to some locally definable C^{∞}G manifold.

Theorem 1.2. Let G be a finite group. Then for any two affine locally definable C^{∞} G manifolds, they are C^{1}G diffeomorphic if and only if they are locally definably C^{∞}G diffeomorphic.

If M is polynomially bounded, then Theorem 1.2 is not always true. Even in the non-equivariant Nash category, there exist two affine Nash manifolds such that they are not Nash diffeomorphic but C^{∞} diffeomorphic [14], and that for any two affine Nash manifolds, they are locally Nash diffeomorphic if and only if they are Nash diffeomorphic.

12000 Mathematics Subject Classification. 14P10, 14P20, 57S05, 57S15, 58A05, 58A07, 03C64.

Keywords and P hrases. O-minimal, locally definable C^{∞}G manifolds, locally definable C^{r}G manifolds, definable C^{r}G manifolds, finite groups, exponential.

Existence of C^{ω}G manifold structures of proper C^{∞}G manifolds and uniqueness of them are studied in [3] and [4], respectively, when G is a C^{ω} Lie group. Moreover if G is a compact C^{ω} Lie group, then for any two C^{ω}G manifolds, they are C^{∞}G diffeomorphic if and only if they are C^{ω}G diffeomorphic [13]. Theorem 1.1 and 1.2 are locally definable C^{∞} versions of [2] and [3], respectively, when G is a finite group.

The above theorems are locally definable C^{∞} versions of results of [10].

In the non-equivariant setting, we have the following.

Theorem 1.3. If 1 ≤ r ≤ ∞, then every n-dimensional locally definable C^{r} manifold X is locally definably C^{r} imbeddable into ℝ^{2n+1}.

The above theorem is the locally definable version of Whitney’s imbedding theorem (e.g. 2.14 [2]). The definable C^{r} version of Theorem 1.1 is known in [8] when r is a non-negative integer.

If M = R and r = ∞, then Theorem 1.3 is not true. The assumption that M is exponential is necessary.

As a corollary of Theorem 1.3, we have the following.

Theorem 1.4. Let G be a finite abelian group and 1 ≤ r ≤ ∞. Then every locally definable C^{r}G manifold is affine.

By Theorem 1.1-1.4, we have the following theorem.

Theorem 1.5. Let G be a finite abelian group and 1 ≤ r < ∞. Then every locally definable C^{r}G manifold admits a unique locally definable C^{∞}G manifold structure up to locally definable C^{∞}G diffeomorphism.

2. Locally definable C^{r}G manifolds

Let f : U → ℝ be a definable C^{∞} function on a definable open subset U ⊂ ℝ^{n}. We say that f has controlled derivatives if there exist a definable continuous function u : U → ℝ, real numbers C_{1}, C_{2}, … and positive integers E_{1}, E_{2}, … such that |D^{α}f(x)| ≤ C_{|α|}u(x)^{E_{|α|}} for all x ∈ U and α ∈ (ℕ ∪ \{0\})^{n}, where D^{α} = \frac{∂^{|α|}}{∂x_{1}^{α_{1}}...∂x_{n}^{αn}} and |α| = α_{1} + ⋯ + α_{n}. We say that M has piecewise controlled derivatives if for every definable C^{∞} function f : U → ℝ defined in a definable open subset U of ℝ^{n}, there exist definable open sets U_{1}, …, U_{l} ⊂ U such that dim(U − ∪_{i=1}^{l}U_{i}) < n and each f|U_{i} has controlled derivatives.

A subset X of ℝ^{n} is called locally defin-able if for every x ∈ X there exists a definable open neighborhood U of x in ℝ^{n} such that X ∩ U is definable in ℝ^{n}. Clearly every definable set is locally definable. Remark that any open subset of ℝ^{n} is locally definable and that every compact locally definable set is definable. A more general setting of locally definable sets is studied in [1].

Let U ⊂ ℝ^{n} and V ⊂ ℝ^{m} be locally definable sets. We call a map f : U → V locally definable if for any x ∈ U there exists a definable open neighborhood W_{x} of x in ℝ^{n} such that f|U ∩ W_{x} is definable.

Note that for any locally definable map f between locally definable sets X and Y, if X is compact, then f(X) is a definable set and f : X → f(X) (⊂ Y) is a definable map.

Remark that the maps f_{1}, f_{2} : ℝ → ℝ defined by f_{1}(x) = sin x, f_{2}(x) = cos x, respectively, are analytic but not definable in any o-minimal expansion of R. However they are locally definable in R_{an}. Remark further that the field ℚ (⊂ ℝ) of rational numbers is not a locally definable subset of ℝ.

Proposition 2.1 ([11]). Let X, Y and Z be locally definable sets and let f : X → Y and g : Y → Z be locally definable maps. Then g ∘ f : X → Z is locally definable.

We can define locally definable groups and affine locally definable groups in a similar way of definable cases. But we do not give their definitions here because we restrict our attention to finite groups.

A representation map of G is a group homomorphism from G to some O(n). A representation of G means the representation space of a representation map of G. Recall the definition of locally definable C^{r}G manifolds [11].

Definition 2.2 ([11]). Let 1 ≤ r ≤ ω.

(1) A locally definable C^{r} submanifold of a representation Ω of G is called a locally definable C^{r}G submanifold of Ω if it is G invariant.

(2) A locally definable C^{r}G manifold is a pair (X, θ) consisting of a locally definable C^{r} manifold X and a group action θ of G on X such that θ : G × X → X is a locally definable C^{r} map. For simplicity of notation, we write X instead of (X, θ). Clearly each definable C^{r}G manifold is a locally definable C^{r}G manifold.

(3) Let X and Y be locally definable C^{r}G manifolds. A locally definable C^{r} map is called a locally definable C^{r}G map if it is a G map. We say that X and Y are locally definably C^{r}G diffeomorphic if there exist locally definable C^{r}G maps f : X → Y and h : Y → X such that f ∘ h = id and h ∘ f = id.

(4) A locally definable C^{r}G manifold is said to be affine if it is locally definably C^{r}G diffeomorphic to a locally definable C^{r}G submanifold of some representation of G.

Note that we can define locally definable G manifolds for a locally definable group G, but in this paper we do not use these notions.

Recall existence of definable C^{r}G tubular neighborhoods.

Theorem 2.3 ([9], [6]). Let r be a non-negative integer, ∞ or ω. Then every definable C^{r}G submanifold X of a representation Ω of G has a definable C^{r}G tubular neighborhood (U, θ_{X}) of X in Ω, namely U is a G invariant definable open neighborhood of X in Ω and θ_{X} : U → X is a definable C^{r}G map with θ_{X}|X = id_{X}.

Let G = {g_{1}, …, g_{m}} and let f be a C^{r}G map from a C^{r}G manifold M to a representation Ω of G. Then the averaging map A : M → Ω is

A(f)(x) = \frac{1}{m} ∑_{i=1}^{m} g_{i}^{−1}f(g_{i}x).

By using [7], we have the following lemma.

Proposition 2.4 ([7]). (1) A(f) is equivariant, and A(f) = f if f is equivariant.

(2) If f is a polynomial map, then so is A(f).

(3) If 0 ≤ r ≤ ∞ and f lies in the set C^{r}(M, Ω) of C^{r} maps from M to Ω, then A(f) ∈ C^{r}(M, Ω).

(4) A : C^{r}(M, Ω) → C^{r}(M, Ω), f ↦ A(f) (0 ≤ r ≤ ∞) is continuous in the C^{r} Whitney topology.

(5) If M is a definable C^{r}G manifold, f is a definable C^{r} map and 0 ≤ r ≤ ω, then A(f) is a definable C^{r}G map.

(6) If M is a locally definable C^{r}G manifold, f is a locally definable C^{r} map and 0 ≤ r ≤ ω, then A(f) is a locally definable C^{r}G map.

Let K be a subgroup of G. Suppose that S is an affine definable C^{∞}K manifold. Then we know that the twisted product G ×_{K} S with the standard action G × (G ×_{K} S) → G ×_{K} S, (g, [g′, s]) ↦ [gg′, s] is a definable C^{∞}G manifold [9].

We need the following proposition to prove Theorem 1.1.

Proposition 2.5. Let X be a locally definable C^{∞}G manifold. Suppose that K is a subgroup of G and N is an affine definable C^{∞}K manifold. If f : N → X is a locally definable C^{∞}K map, then

μ(f) : G ×_{K} N → X, μ([g, n]) = gf(n)

is a locally definable C^{∞}G map.

P roof. By the property of quotient manifolds, μ(f) is a C^{∞}G map. Thus it suffices to prove that μ(f) is locally definable. Let π be the orbit map G × N → G ×_{K} N. Then π is a definable C^{∞} map. Take x ∈ G ×_{K} N and y ∈ π^{−1}(x) ⊂ G × N. By the assumption and the definition of the G action on G × N, \overline{μ}(f) : G × N → X, \overline{μ}(f)(g, n) = gf(n) is a locally definable C^{∞}G map. Hence there exist definable open neighborhoods U of y and V of \overline{μ}(f)(y), respectively, such that \overline{μ}(f)(U) ⊂ V and \overline{μ}(f)|U : U → V is a definable C^{∞} map. In particular, \overline{μ}(f)|U : U → V is definable. Hence π(U) is open and definable. Since the graph of μ(f)|π(U) : π(U) → V ⊂ X is the image of that of \overline{μ}(f)|U by π × id_{V}, μ(f)|π(U) is definable. □

Definition 2.6. Let X be a definable C^{∞}G manifold.

(1) We say that a K invariant definable C^{∞} submanifold S of X is a definable K slice if GS is open in X, S is affine as a definable C^{∞}K manifold, and

μ : G ×_{K} S → GS (⊂ X), [g, x] ↦ gx

is a definable C^{∞}G diffeomorphism.

(2) A definable C^{∞}K slice S is called linear if there exist a representation Ω of K and a definable C^{r}K imbedding j : Ω → X such that j(Ω) = S.

(3) We say that a definable C^{∞}K slice (resp. a linear definable C^{∞}K slice) S is a definable C^{∞} slice (resp. a linear definable C^{∞} slice) at x in X if K = G_{x} and x ∈ S (resp. K = G_{x}, x ∈ S and j(0) = x).

Recall existence of definable C^{∞} slices [9] to prove Theorem 1.2 .

Theorem 2.7 ([9]). Let X be an affine definable C^{∞}G manifold, x ∈ X. Then there exists a linear definable C^{∞}G slice at x in X.

3. Proof of Theorem 1.1

The following lemma is obtained by 2.2.8 [2] and Proposition 2.4.

Lemma 3.1. Let K be a finite group. Suppose that f is a definable C^{∞}K map between definable C^{∞}K manifolds M and N. Suppose further that V is an open K invariant subset of M and that P is a K invariant definable C^{∞} submanifold of N with f(V) ⊂ P. Then there exist an open neighborhood 𝔑 of f|V in the set Def_{K}^{∞}(V, P) of definable C^{∞}K maps from V to P such that for any h ∈ 𝔑, the map E(h) : M → N,

E(h)(x) = \{ \begin{matrix}h(x), & x ∈ V \\ f(x), & x ∈ M − V \end{matrix}

is a definable C^{∞}K map and E : 𝔑 → Def_{K}^{∞}(M, N), h ↦ E(h) is continuous in the C^{∞} Whitney topology.

Proposition 3.2. Let X be a locally definable C^{∞}G manifold and Y an affine definable C^{∞}G manifold in a representation Ω of G. Then every C^{∞}G map f : X → Y is approximated by a locally definable C^{∞}G map h : X → Y in the C^{∞} Whitney topology.

In the Nash case, if 1 ≤ r < ∞, then locally C^{r} Nash diffeomorphisms are essentially different from C^{r} Nash diffeomorphisms because there exist two affine Nash manifolds such that they are C^{∞} diffeomorphic but not Nash diffeomorphic [14], and that every C^{r} Nash diffeomorphism between affine Nash manifolds is approximated by a Nash diffeomorphism [15].

Proposition 3.3 ([12]). Every affine definable C^{∞}G manifold is definably C^{∞}G diffeomorphic to a definable C^{∞}G submanifold closed in some representation Ω of G.

For the proof of Proposition 3.3, we need the condition that M is exponential, admits the C^{∞} cell decomposition and has piecewise controlled derivatives.

P roof of P roposition 3.2. By Proposition 3.3, replacing Ω if necessary, we may assume that Y a definable C^{∞}G submanifold closed in Ω. By a way similar to find a C^{∞} partition of unity of C^{∞} manifold, we have a locally definable C^{∞} partition of unity {ϕ_{j}}_{j=1}^{∞} subordinates to some locally finite open definable cover {X_{j}}_{j=1}^{∞} of X such that X = ∪_{j=1}^{∞}supp ϕ_{j} and \overline{X_{j}} is compact. For any j, take an open neighborhood U_{j} of supp ϕ_{j} in X such that \overline{U_{j}} is compact. Applying the polynomial approximation theorem, we have a locally definable C^{∞} map h_{j} : U_{j} → Ω which approximates f|U_{j}. By Theorem 2.3, one can find a definable C^{∞}G tubular neighborhood (U, p) of Y in Ω. If our approximation is sufficiently close, then p ∘ ∑_{j=1}^{∞} ϕ_{j}h_{j} is a (non-equivariant) C^{∞} approximation of f. Since G is a finite group, applying Proposition 2.4, we have the required locally definable C^{∞}G map h as a C^{∞} Whitney approximation of f. □

P roof of T heorem 1.1. Using Lemma 3.1 and Proposition 3.2, a similar proof of 1.1 [11] proves Theorem 1.1. □

4. Proof of Theorem 1.2

In this section we prove the following theorem.

Theorem 4.1. Let G be a finite group and let r be a positive integer. Suppose that Y and Z are affine locally definable C^{∞}G manifolds and there exists a C^{r}G diffeomorphism f : Y → Z. Then there exists a locally C^{∞}G diffeomorphism h : Y → Z which is G homotopic to f.

Theorem 1.2 follows from Theorem 4.1.

Let K be a subgroup of G and let X be an affine definable C^{∞}G manifold. By Theorem 2.7, there exists a linear definable C^{∞}K slice S, namely there exists a definable C^{∞}K diffeomorphism i from some representation Ω of K to S such that GS is open in X, and that μ : G ×_{K} Ω → GS (⊂ X), μ(i)([g, x]) = gi(x) is a definable C^{∞}G diffeomorphism.

For simplicity, we use the following notations. Set B_{s} := {x ∈ Ω|||x|| ≤ s}, B_{s}^{∘} := {x ∈ Ω|||x|| < s}, s > 0, B := B_{1}, and B^{∘} : = B_{1}^{∘}, and let denote D_{s}, D_{s}^{∘}, D and D^{∘} by i(B_{s}), i(B_{s}^{∘}), i(B), and i(B^{∘}), respectively. Let GD (resp. GD^{∘}) denote the closed unit tube (resp. the open unit tube) and let GD_{s} (resp. GD_{s}^{∘}) stand for the closed tube (resp. the open tube) of radius s.

To prove Theorem 4.1 we prepare two preliminary results.

Lemma 4.2. Let Ω and Ξ be representations of G and let M (resp. N) be a definable C^{∞}G submanifold of Ω (resp. Ξ). Suppose that F is a G invariant definable subset of M and that α : M → N is a C^{∞}G map such that α|F : F → N is definable. Let 𝔑 be a neighborhood of α in the set C_{G}^{∞}(M, N) of C^{∞}G maps from M to N and let V_{1} and V_{2} be compact G invariant definable subsets of M such that V_{1} is properly contained in the interior Int V_{2} of V_{2}. Then there exists κ ∈ 𝔑 such that:

(a) κ|F ∪ V_{1} : F ∪ V_{1} → N is definable.

(b) κ = α on M − Int V_{2}

(c) κ is G homotopic to α relative to M − Int V_{2}

P roof. Take a non-negative definable C^{∞} function f : M → ℝ such that f = 0 on V_{1} and f = 1 on M −Int V_{2}. Notice that if M is polynomially bounded, then such an f does not necessarily exist. Since G is a finite group and by Proposition 2.4, we may assume that f is G invariant.

We approximate α by a polynomial G map β on V_{2} using the polynomial approximation theorem and Proposition 2.4. By Theorem 2.3, one can find a definable C^{∞}G tubular neighborhood (U, p) of N in Ξ. If the approximation is sufficiently close, then one can define κ : M → N, κ(x) = p(f(x)α(x) + (1 − f(x))β(x)). Then κ is a C^{∞}G map, and κ satisfies Properties (a) and (b). If this approximation is sufficiently close, then κ ∈ 𝔑 because κ and α coincide with outside of a compact set V_{2}.

The map H : M × [0, 1] → N defined by H(x, t) = p((1 − t)α(x) + tκ(x)) gives a G homotopy relative to M − Int V_{2} from α to κ. □

Proposition 4.3. Let Ω and Ξ be representations of G. Let M ⊂ Ω, N ⊂ Ξ be affine locally definable C^{∞}G manifolds and A a closed G invariant locally definable subset of M. Suppose that f : M → N is a C^{∞}G diffeomorphism such that f|A : A → N is locally definable, and that x ∈ M. Suppose further that j : Ω′ → S is a linear definable C^{∞} slice at x in Ω. If GD_{10} ∩ M is compact, then there exists a C^{∞}G diffeomorphism h : M → N such that:

(1) h|A ∪ (GD ∩ M) : A ∪ (GD ∩ M) → N is locally definable.

(2) h = f on M − GD_{2}^{∘} ∩ M.

(3) h is G homotopic to f relative to M − GD_{2}^{∘} ∩ M.

The condition that GD_{10} ∩ M is compact is not essential. By Theorem 2.7, one can find a linear definable C^{r} slice S at x ∈ M in Ω. Since S is a linear definable C^{∞} slice in Ω, there exists a definable C^{∞}K diffeomorphism j from some representation Ω′ of G_{x} onto S such that j(0) = x, GS is open in Ω, and that

μ(j) : G ×_{G_{x}} Ω′ → GS ( ⊂ Ξ),
μ(j)([g, x]) = gj(x)

is a definable C^{∞}G diffeomorphism. Notice that M is locally compact. Thus replacing smaller S, if necessary, GD_{10} ∩ M is compact because M is locally compact.

P roof of P roposition 4.3. Since GD_{10} ∩M is compact and A is closed in M, A∩GD_{10} (= A ∩ (GD_{10} ∩ M)) is a compact G invariant locally definable subset of GS ∩ M. Thus A ∩ GD_{10} is a G invariant definable subset of Ω. Hence

E := μ(j)^{−1}(A ∩ GD_{10})

is a G invariant definable subset of G ×_{G_{x}} Ω′. Let L = j^{−1}(D_{10}^{∘} ∩ M). The map

α := f ∘ μ(j)|G ×_{G_{x}} L : G ×_{G_{x}} L → Ξ

is a C^{∞}G diffeomorphism onto an open G invariant subset V := f(GD_{10}^{∘} ∩M) of N. Since A∩GD_{10} is compact and f|A is locally definable, f|(A∩GD_{10}) : A∩GD_{10} → f(A∩GD_{10}) ⊂ N ⊂ Ξ is definable. The map α|(E ∩ (G ×_{G_{x}} L)) : E ∩ (G ×_{G_{x}} L) → Ξ is definable because μ(j) and f|(A∩GD_{10}) : A∩GD_{10} → Ξ are definable. Since V is contained in a G invariant compact set f(GD_{10} ∩ M), and since N is a locally definable C^{∞}G submanifold of Ξ, there exists a G invariant definable set W of Ξ such that V ⊂ W ⊂ N and that W is open in N. Notice that W is an affine definable C^{∞}G manifold. Since G ×_{G_{x}} L is contained in a G invariant compact subset of G ×_{G_{x}} j^{−1}(D_{20} ∩ M), G ×_{G_{x}} L is an affine definable C^{∞}G manifold. Applying Lemma 4.2 to α : G ×_{G_{x}} L → W, there exists a C^{∞}G map β : G ×_{G_{x}} L → W as a C^{∞} Whitney approximation of α such that:

(a) β|(G ×_{G_{x}} (j^{−1}(A ∩ D_{10}^{∘}) ∪ (B ∩ L))) : G ×_{G_{x}} (j^{−1}(A ∩ D_{10}^{∘}) ∪ (B ∩ L)) → W (⊂ N) is definable.

(b) β = α on G ×_{G_{x}} (L − B_{2}^{∘} ∩ L).

(c) β is G homotopic to α relative to G ×_{G_{x}} (L − B_{2}^{∘} ∩ L). Then the map h : M → N defined by

h(x) = \{ \begin{matrix}β ∘ μ(j)^{−1}(x), & x ∈ GD_{5} ∩ M \\ f(x), & x ∈ M − M ∩ GD_{5}^{∘} \end{matrix}

is well-defined, and it is a C^{∞}G diffeomorphism if our approximation is sufficiently close. Since h|(A∩GD_{5}) and h|(GD∩M) are definable, and since h|(A∩(M −GD_{5} ∩M)) (= f|(A ∩ (M − GD_{5} ∩ M))) is locally definable, h|A ∪ (GD ∩ M) is locally definable by Proposition 2.1. By the construction of h, h satisfies Properties (2) and (3). □

P roof of T heorem 4.1. Using Proposition 4.3, a similar proof of 4.1 [11] proves Theorem 4.1. □

5. Proof of Theorem 1.3 and 1.4

P roof of T heorem 1.3. By Whitney’s imbedding Theorem (e.g. 2.14 [2]), there exists a C^{∞} imbedding f : X → ℝ^{2n+1}. By Proposition 3.2 and since C^{∞} imbeddings from X to ℝ^{2n+1} are open in the set C^{∞}(X, ℝ^{2n+1}) of C^{∞} maps from X to ℝ^{2n+1}, we have the required a locally definable C^{∞} imbedding h : X → ℝ^{2n+1}. □

P roof of T heorem 1.4. Let G = {g_{1}, …, g_{m}} and X a locally definable C^{∞}G manifold of dimension n. By Theorem 1.3, there exists a locally definable C^{∞} imbedding f : X → ℝ^{2n+1}. Let Ω be the representation of G whose underlying space is ℝ^{(2n+1)m} = ℝ^{2n+1} × ⋯ × ℝ^{2n+1} and its action is defined by the permutation of coordinates (x_{1}, …, x_{m}) ↦ (x_{σ(1)}, …, x_{σ(m)}) induced from (gg_{1}, …, gg_{m}) = (g_{σ(1)}, …, g_{σ(m)}). Then F : X → Ω, F (x) = (f(g_{1}x), …, f(g_{m}x)) is the required locally definable C^{∞}G imbedding. □

References

[1] E Baro and M. Otero, Locally definable homotopy, preprint.

[2] M.W. Hirsch Differential topology, Springer-Verlag (1976).

[3] S. Illman, Every proper smooth actions of a Lie group is equivalent to a real analytic action: a contribution to Hilbert’s fifth problem, Ann. of Math. Stud. 138 (1995), 189-220.

[4] S. Illman, Smoothly equivalent real analytic proper G manifolds are subanalytically equivalent, Math. Ann. 306 (1996), 647-673.

[5] T. Kawakami, Affineness of definable C^{r} manifolds and its applications, Bull. Korean Math. Soc. 40, (2003), 149-157.

[6] T. Kawakami, Equivariant definable C^{r} approximation theorem, definable C^{r}G triviality of definable C^{r}G maps and Nash G compactifications, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. No. 55, (2005), 1–14.

[7] T. Kawakami, Equivariant differential topology in an o-minimal expansion of the field of real numbers, Topology Appl. 123 (2002), 323-349.

[8] T. Kawakami, Every definable C^{r} manifold is affine, Bull. Korean Math. Soc. 42 (2005), 165–167.

[9] T. Kawakami, Imbedding of manifolds defined on an o-minimal structures on (ℝ, +, ·, <), Bull. Korean Math. Soc. 36 (1999), 183-201.

[10] T. Kawakami, Nash G manifold structures of compact or compactifiable C^{∞}G manifolds, J. Math. Soc. Japan 48 (1996), 321-331.

[11] T. Kawakami, Locally definable C^{s}G maifold structures of locally C^{r}G manifolds, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 56 (2006), 1–12.

[12] T. Kawakami, Relative Definable C^{r}G triviality of G invariant proper definable C^{r} functions, preprint.

[13] F. Kutzschebauch On the uniqueness of the real analytic G-structure on a smooth proper G-manifold for G a Lie group, Manuscripta Math. 90 (1996), 17-22.

[14] M. Shiota, Abstract Nash manifolds, Proc. Amer. Math. Soc. 96 (1986), 155-162.

[15] M. Shiota, Approximation theorems for Nash mappings and Nash manifolds, Trans. Amer. Math. Soc. 293 (1986), 319-337.

Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani Wakayama 640-8510, Japan

E-mail address: kawa@center.wakayama-u.ac.jp