# Smooth functions in o-minimal structures

@article{Fischer2008SmoothFI, title={Smooth functions in o-minimal structures}, author={Andreas Fischer}, journal={Advances in Mathematics}, year={2008}, volume={218}, pages={496-514} }

Abstract Fix an o -minimal expansion of the real exponential field that admits smooth cell decomposition. We study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the Whitney topology. This implies that abstract definable smooth manifolds are affine. Moreover, abstract definable smooth manifolds are definably C ∞ -diffeomorphic if and only if they are definably C 1 -diffeomorphic.

#### 22 Citations

Transversality of smooth definable maps in O-minimal structures

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2019

Abstract We present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth… Expand

O-minimal Analytic Separation of Sets in Dimension 2

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 2009

If the set of analytic germs is dense in the Hardy field, then it can definably analytically separate sets in R 2, and the same statement holds for definable smooth functions. Expand

On smooth locally o-minimal functions

- Mathematics
- 2014

We study the smooth functions which are locally deflnable in an o-minimal expansion of the real exponential fleld with some additional smooth- ness conditions. Here, the local deflnability… Expand

THE RIEMANN MAPPING THEOREM FOR o-MINIMAL FUNCTIONS

- 2008

The proof of the Riemann mapping theorem is not constructive. We study versions of it for sets and functions which are definable in an ominimal expansion of the real field. The diffeomorphisms… Expand

Tempered distributions and Schwartz functions on definable manifolds

- Mathematics
- Journal of Functional Analysis
- 2020

Abstract We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical… Expand

DEFINABILITY OF ALGEBRAIC MODELS

- Mathematics
- 2008

A dierentiable manifold admits an algebraic model if M is dif- feomorphic to some nonsingular real algebraic set. We study algebraic models for dierentiable manifolds whose underlying set is… Expand

ALGEBRAIC MODELS FOR O-MINIMAL MANIFOLDS

- Mathematics
- 2008

A difierentiable manifold admits an algebraic model if it is difieo- morphic to some non-singular real algebraic set. We prove that every manifold whose underlying set is deflnable in some o-minimal… Expand

Volumes of definable sets in o-minimal expansions and affine GAGA theorems

- Mathematics
- 2021

I show that a d-dimensional definable set S ⊆ R in an o-minimal expansion of the ordered field of real numbers satisfies the volume estimate H({x ∈ S : ‖x‖ < r}) ≤ Cr, where H denotes the… Expand

DEFINABLE C∞G MANIFOLD STRUCTURES OF DEFINABLE CG MANIFOLDS

- 2009

Let G be a compact definable C∞ group and 2 ≤ r < ∞. Let X be a noncompact affine definable CG manifold and X1, . . . , Xk noncompact codimension one definable CG submanifolds of X such that X1, . .… Expand

O-minimal De Rham cohomology

- Mathematics
- 2017

In the present paper we elaborate an o-minimal de Rham cohomology theory for abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p\leq \infty$ in an o-minimal expansion of the real field which… Expand

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