# Smooth functions in o-minimal structures

@article{Fischer2008SmoothFI,
title={Smooth functions in o-minimal structures},
author={Andreas Fischer},
year={2008},
volume={218},
pages={496-514}
}
• A. Fischer
• Published 2008
• Mathematics
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.
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 smoothExpand
O-minimal Analytic Separation of Sets in Dimension 2
• A. Fischer
• 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
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 deflnabilityExpand
THE RIEMANN MAPPING THEOREM FOR o-MINIMAL FUNCTIONS
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 diffeomorphismsExpand
Tempered distributions and Schwartz functions on definable manifolds
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 classicalExpand
DEFINABILITY OF ALGEBRAIC MODELS
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 isExpand
ALGEBRAIC MODELS FOR O-MINIMAL MANIFOLDS
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-minimalExpand
Volumes of definable sets in o-minimal expansions and affine GAGA theorems
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 theExpand
DEFINABLE C∞G MANIFOLD STRUCTURES OF DEFINABLE CG MANIFOLDS
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 whichExpand

#### References

SHOWING 1-10 OF 27 REFERENCES
Smooth approximation of definable continuous functions
Let M be an o-minimal expansion of the real exponential field which possesses smooth cell decomposition. We prove that for every definable open set, the definable indefinitely continuouslyExpand
EVERY DEFINABLE C r MANIFOLD IS AFFINE
Let M = (R, +, , , >) of the field of real numbers. We prove that if 2 r , then every n-dimensional definable manifold is definably imbeddable into . Moreover we prove that if 1 , then everyExpand
Transition maps at non-resonant hyperbolic singularities are o-minimal
• Mathematics
• 2006
Abstract We construct a model complete and o-minimal expansion of the field of real numbers such that, for any planar analytic vector field ξ and any isolated, non-resonant hyperbolic singularity pExpand
Tame Topology and O-minimal Structures
• L. Dries
• Mathematics, Computer Science
• 1998
1. Some elementary results 2. Semialgebraic sets 3. Cell decomposition 4. Definable invariants: Dimension and Euler characteristic 5. The Vapnik-Chernovenkis property in o-minimal structures 6.Expand
Infinite differentiability in polynomially bounded o-minimal structures
Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion 9l of the ordered field of real numbers are shown to have some of the nice properties of real analyticExpand
The Pfaffian closure of an o-minimal structure
Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C1 coefficients are definable.
THE REAL FIELD WITH CONVERGENT GENERALIZED POWER SERIES
• Mathematics
• 1998
We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn 1 is definable. ThisExpand
The Elementary Theory of Restricted Analytic Fields with Exponentiation
• Mathematics
• 1994
numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of theExpand
Quasianalytic Denjoy-Carleman classes and o-minimality
• Mathematics
• 2003
We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closureExpand
Analytic stratification in the Pfaffian closure of an o-minimal structure
• Mathematics
• 2000
Introduction. Let U ⊆ R be open andω = a1dx1+·· ·+andxn a nonsingular, integrable 1-form onU of classC1, and let be the foliation onU associated toω. A leafL ⊆ U of is aRolle leaf if anyC1 curveγ :Expand