EVERY DEFINABLE C^{∞} MANIFOLD IS AFFINE

TOMOHIRO KAWAKAMI

Abstract. Let M = (ℝ, +, ·, <, e^{x}, …) be an exponential o-minimal expansion of the standard structure R = (ℝ, +, ·, <) of the field of real numbers with C^{∞} cell decomposition. We prove that every n-dimensional definable C^{∞} manifold is definably C^{∞} imbeddable into ℝ^{2n+1}.

1. Introduction

M. Shiota proved that if 0 < r < ∞, then every C^{r} Nash manifold is affine [9]. Let M = (ℝ, +, ·, <, …) be an o-minimal expansion of the standard structure R = (ℝ, +, ·, <) of ℝ. Note that if M = R, then a definable C^{r} manifold is a C^{r} Nash manifold. Definable C^{r} categories based on M are generalizations of the C^{r} Nash category.

General references on o-minimal structures are [1], [2], see also [10]. The term “definable” means “definable with parameters in M”.

If r is a non-negative integer, then every definable C^{r} manifold is affine [7]. We have the following theorem as a generalization of this result.

Theorem 1.1. Let M = (ℝ, +, ·, <, e^{x}, …) be an exponential o-minimal expansion of the standard structure R = (ℝ, +, ·, <) of the field of real numbers with C^{∞} cell decomposition. Then every n-dimensional definable C^{∞} manifold X is definably C^{∞} imbeddable into ℝ^{2n+1}.

The above theorem is the definable version of Whitney’s imbedding theorem (e.g. 2.14 [3]). Even in the Nash category (i.e. M = R), we cannot drop the assumption that M is exponential by Theorem 1.1 [9].

Theorem 1.2 ([5]). If 0 ≤ s < ∞ and M is an exponential o-minimal expansion of R = (ℝ, +, ·, <) with C^{∞} cell decomposition, then every definable C^{s} map between definable C^{∞} manifolds is approximated in the definable C^{s} topology by definable C^{∞} maps.

Using Theorem 1.2 and by a way similar to the proof of Theorem 1.1 and 1.3 [4], we have the following theorem.

Theorem 1.3. Let 1 ≦ s < r ≦ ∞, then every definable C^{s} manifold admits a unique definable C^{r} manifold structure up to definable C^{r} diffeomorphism.

2000 Mathematics Subject Classification. 14P10, 14P20, 57R55, 58A05, 03C64.

Keywords and P hrases. Definable C^{∞} manifolds, o-minimal, affine.

2. Proof of our result

Theorem 2.1. Let X be an affine definable C^{∞} manifold and V a definable subset closed in X. Then there exists a non-negative definable C^{∞} function f : X → ℝ such that f^{−1}(0) = V.

P roof. By definition of affineness and 3.2 [8], X is definably C^{∞} diffeomorphic to a definable C^{∞} submanifold of some ℝ^{l} which is closed in ℝ^{l}. We identify X with its image. Thus V is closed in ℝ^{l}. Since M admits C^{∞} cell decomposition, there exists a C^{∞} cell decomposition D partitioning V. For every cell C ∈ D, the closure \overline{C} of C in X lies in V. Thus if V = C_{1} ∪ ⋯ ∪ C_{m}, then V = \overline{C_{1}} ∪ ⋯ ∪ \overline{C_{m}}. If C_{i} is bounded and k-dimensional, then \overline{C_{i}} is definably C^{∞} diffeomorphic to [−1, 1]^{k}. Hence \overline{C_{i}} is the zeros of a definable C^{∞} function. Thus the case where V is compact is proved.

Let \overline{C_{i}} be unbounded. Replacing ℝ^{l} by ℝ^{l+1}, we may assume that 0 ∉ \overline{C_{i}}. Let i : ℝ^{l+1} − \{0\} → ℝ^{l+1} − \{0\}, i(x) = \frac{x}{||x||^{2}} , where ||x|| denotes the norm of x. Then C′_{i} = i(\overline{C_{i}}) ∪ \{0\} is the one point compactification of \overline{C_{i}}. Thus there exists a definable C^{∞} function ψ : ℝ^{l+1} → ℝ with C′_{i} = ψ^{−1}(0). Hence \overline{C_{i}} is definably C^{∞} diffeomorphic to the set C′′_{i} = \{(x, y) ∈ ℝ^{l+1} ×ℝ|ψ(x) = 0, ||x||^{2}y = 1\}. Therefore \overline{C_{i}} is the zeros of a definable C^{∞} function. Since V = \overline{C_{1}} ∪ ⋯ ∪ \overline{C_{m}}, V is the zeros of a definable C^{∞} function ϕ. Thus f := ϕ^{2} : X → ℝ is the required function. □

The following is a definable C^{∞} partition of unity.

Proposition 2.2. Let {U_{i}}_{i=1}^{k} be a definable open covering of a definable C^{∞} manifold X. Then there exist definable C^{∞} functions λ_{i} : X → ℝ (1 ≤ i ≤ k) such that 0 ≤ λ_{i} ≤ 1, supp λ_{i} ⊂ U_{i} and ∑_{i=1}^{k} λ_{i} = 1.

If X is affine, then the definable C^{r} version of Proposition 2.2 is known in 4.8 [6].

P roof. We now prove that there exists a definable open covering {V_{i}}_{i=1}^{k} of X such that \overline{V_{i}} ⊂ U_{i}, (1 ≤ i ≤ k), where \overline{V_{i}} denotes the closure of V_{i} in X.

We proceed by induction on k. If k = 1, then there is nothing to prove. Assume that there exists a definable open covering \{V_{i}\}_{i=1}^{k−1} ∪ \{U_{k}\} of X such that \overline{V_{i}} ⊂ U_{i}, (1 ≤ i ≤ k − 1).

Let X_{k−1} := ∪_{i=1}^{k−1}V_{i}. By the inductive hypothesis, there exists a definable open covering {W_{i}}_{i=1}^{k−1} of X_{k−1} such that cl W_{i} ⊂ V_{i}, where cl W_{i} means the closure of W_{i} in X_{k−1}.

We may assume that U_{k} is affine. Let Z_{k} := U_{k} ∩∪_{i=1}^{k−1}V_{i} and Cl Z_{k} denote the closure of Z_{k} in U_{k}. By Theorem 2.1, there exists a non-negative definable C^{∞} function ϕ_{k} : U_{k} → ℝ such that ϕ_{k}^{−1}(0) = Cl Z_{k}. Since cl W_{1} ⊂ V_{1}, ϕ_{k} is extensible to a non-negative definable C^{∞} function ϕ_{k}^{1} : U_{k} ∪ W_{1} → ℝ such that ϕ_{k}^{1}^{−1}(0) = Cl Z_{k} ∪ W_{1}. Inductively, we have a non-negative definable C^{∞} function ϕ : X → ℝ such that ϕ^{−1}(0) = Cl Z_{k} ∪W_{1} ⋯∪W_{k−1}. Let V_{k} := \{x ∈ U_{k}|ϕ(x) > 0\}. Then V_{k} = \{x ∈ X|ϕ(x) > 0\}, \overline{V_{k}} ⊂ U_{k} and \{V_{i}\}_{i=1}^{k} is the required definable open covering of X.

By Theorem 2.1, we have a non-negative definable C^{∞} function μ_{i} : U_{i} → ℝ such that μ_{i}^{−1}(0) = U_{i} − V_{i}. Thus μ_{i} is extensible to a non-negative definable C^{∞} function μ′_{i} : X → ℝ such that μ′_{i}^{−1}(0) = X − V_{i}. Therefore λ_{i} := μ′_{i}/ ∑_{i=1}^{k} μ′_{i} is the required definable C^{r} partition of unity. □

P roof of T heorem 1.1. Let {ϕ_{i} : U_{i} → ℝ^{n}}_{i=1}^{k} be a definable C^{r} atlas of X. By Proposition 2.2, we have definable C^{∞} functions λ_{i} : X → ℝ, (1 ≤ i ≤ k) such that 0 ≤ λ_{i} ≤ 1, supp λ_{i} ⊂ U and ∑_{i=1}^{k} λ_{i} = 1. Thus the map F : X → ℝ^{nk} × ℝ^{k} defined by F (x) = (λ_{1}(x)ϕ_{1}(x), … λ_{k}(x)ϕ_{k}(x), λ_{1}(x), …, λ_{k}(x)) is a definable C^{∞} imbedding. Hence X is affine. Thus it is either compact or compactifiable by 1.2 [6]. Hence we may assume that X is affine and compact at the beginning. A similar argument of the proof of 1.4 [11], every definable C^{∞} map f : X → ℝ^{2n+1} can be approximated in the C^{r} topology by an injective definable C^{∞} immersion h : X → ℝ^{2n+1}. Since X is compact, h is the required definable C^{∞} imbedding. □

References

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

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