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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
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