# zbMATH — the first resource for mathematics

Smooth functions in o-minimal structures. (English) Zbl 1147.03018
Let $${\mathcal M}$$ be an o-minimal expansion of the real exponential field that admits smooth cell decomposition. For every natural number $$m$$, and even for $$m = \infty$$, let $$C^m$$ mean $$m$$ times continuously differentiable. The paper examines definable $$C^m$$ functions from a definable open subset $$U$$ of $$\mathbb R^n$$ to $$\mathbb R$$. In particular, it studies the density of smooth (that is, $$C^\infty$$) functions inside $$C^0$$ functions with respect to the definable version of the Whitney topology. Actually, the author already dealt with the case $$m = 0$$ in his paper [“Smooth approximation of definable continuous functions”, Proc. Am. Math. Soc. 136, No. 7, 2583–2587 (2008; Zbl 1147.03020)]. Here that result is extended to every $$m > 0$$.
As a consequence it is proved that, for all $$m$$, every abstract definable $$C^m$$ manifold of dimension $$n$$ is definably $$C^m$$-diffeomorphic to a definable $$C^m$$ submanifold of $$\mathbb R^{2n+1}$$. Also, two abstract definable smooth manifolds are abstract definably $$C^\infty$$-diffeomorphic if and only if they are abstract definably $$C^1$$-diffeomorphic. Finally, it is shown that, for $$m$$ a positive integer, every definable $$C^m$$ submanifold of $$\mathbb R^n$$ can be approximated in a suitable sense by a definable smooth submanifold.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 14P99 Real algebraic and real-analytic geometry 26E10 $$C^\infty$$-functions, quasi-analytic functions
Full Text:
##### References:
 [1] Blais, F.; Moussu, R.; Rolin, J.-P., Non-oscillating integral curves and o-minimal structures, (), 103-112 · Zbl 1075.03018 [2] Bochnak, J.; Coste, M.; Roy, M.-F., Real algebraic geometry, Ergeb. math. grenzgeb. (3), vol. 36, (1998), Springer-Verlag Berlin · Zbl 0633.14016 [3] Coste, M., An introduction to o-minimal geometry, (2000), Dottorato di Ricerca in Mathematica, Dip. Mat. Univ. Pisa, Instituti Editoriali e Poligrafici Internazionali [4] van den Dries, L., A generalization of the Tarski-seidenberg theorem, and some nondefinability results, Bull. amer. math. soc. (N.S.), 15, 2, 189-193, (1986) · Zbl 0612.03008 [5] van den Dries, L., Tame topology and o-minimal structures, London math. soc. lecture note ser., vol. 248, (1998), Cambridge University Press Cambridge · Zbl 0953.03045 [6] van den Dries, L.; Macintyre, A.; Marker, D., The elementary theory of restricted analytic fields with exponentiation, Ann. of math. (2), 140, 1, 183-205, (1994) · Zbl 0837.12006 [7] van den Dries, L.; Miller, C., Geometric categories and o-minimal structures, Duke math. J., 84, 2, 497-540, (1996) · Zbl 0889.03025 [8] van den Dries, L.; Speissegger, P., The real field with convergent generalized power series, Trans. amer. math. soc., 350, 11, 4377-4421, (1998) · Zbl 0905.03022 [9] van den Dries, L.; Speissegger, P., The field of reals with multisummable series and the exponential function, Proc. London math. soc. (3), 81, 3, 513-565, (2000) · Zbl 1062.03029 [10] Efroymson, G.A., The extension theorem for Nash functions, (), 343-357 [11] Escribano, J., Approximation theorems in o-minimal structures, Illinois J. math., 46, 1, 111-128, (2002) · Zbl 1010.03026 [12] A. Fischer, Peano-differentiable functions in o-minimal structures, Doctoral Thesis, University of Passau, 2006 · Zbl 1364.03001 [13] Fischer, A., O-minimal λm-regular stratification, Ann. pure appl. logic, 147, 1-2, 101-112, (2007) · Zbl 1125.03029 [14] A. Fischer, Smooth approximation of definable continuous functions, Proc. Amer. Math. Soc. (2007), in press [15] G.O. Jones, Local to global methods in o-minimal expansions of fields, Doctoral Thesis, Wolfson College University of Oxford, 2006 [16] T. Kaiser, J.-P. Rolin, P. Speissegger, Transition maps at non-resonant hyperbolic singularities are o-minimal, J. Reine Angew. Math. (2007), in press · Zbl 1203.03051 [17] Kawakami, T., Every definable $$C^r$$ manifold is affine, Bull. Korean math. soc., 42, 1, 165-167, (2005) · Zbl 1063.03023 [18] Kurdyka, K.; Pawlucki, W., Subanalytic version of Whitney’s extension theorem, Studia math., 124, 3, 269-280, (1997) · Zbl 0955.32006 [19] Lion, J.-M.; Speissegger, P., Analytic stratification in the Pfaffian closure of an o-minimal structure, Duke math. J., 103, 2, 215-231, (2000) · Zbl 0970.32009 [20] Malgrange, B., Ideals of differentiable functions, Tata inst. fund. res. stud. math., vol. 3, (1967), Tata Institute of Fundamental Research Bombay, Oxford University Press, London [21] Miller, C., Infinite differentiability in polynomially bounded o-minimal structures, Proc. amer. math. soc., 123, 8, 2551-2555, (1995) · Zbl 0823.03019 [22] Miller, C., Exponentiation is hard to avoid, Proc. amer. math. soc., 122, 1, 257-259, (1994) · Zbl 0808.03022 [23] Milnor, J., On manifolds homeomorphic to the 7-sphere, Ann. of math. (2), 64, 399-405, (1956) · Zbl 0072.18402 [24] Parusinski, A., Subanalytic functions, Trans. amer. math. soc., 344, 2, 583-595, (1994) · Zbl 0819.32006 [25] Pecker, D., On Efroymson’s extension theorem for Nash functions, J. pure appl. algebra, 37, 2, 193-203, (1985) · Zbl 0581.14016 [26] Rolin, J.-P.; Speissegger, P.; Wilkie, A.J., Quasianalytic Denjoy-Carleman classes and o-minimality, J. amer. math. soc., 16, 4, 751-777, (2003) · Zbl 1095.26018 [27] Shiota, M., Approximation theorems for Nash mappings and Nash manifolds, Trans. amer. math. soc., 293, 1, 319-337, (1986) · Zbl 0601.58005 [28] Shiota, M., Abstract Nash manifolds, Proc. amer. math. soc., 96, 1, 155-162, (1986) · Zbl 0594.58006 [29] Shiota, M., Nash manifolds, Lecture notes in math., vol. 1269, (1987), Springer-Verlag Berlin · Zbl 0629.58002 [30] Speissegger, P., The Pfaffian closure of an o-minimal structure, J. reine angew. math., 508, 189-211, (1999) · Zbl 1067.14519 [31] Whitney, H., Differentiable manifolds, Ann. of math. (2), 37, 3, 645-680, (1936) · JFM 62.1454.01 [32] Wilkie, A.J., On defining $$C^\infty$$, J. symbolic logic, 59, 1, 344, (1994) · Zbl 0808.03023 [33] Wilkie, A.J., A theorem of the complement and some new o-minimal structures, Selecta math. (N.S.), 5, 4, 397-421, (1999) · Zbl 0948.03037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.