Strict $C^1$-triangulations in o-minimal structures

Abstract

Inspired by the recent articles of T. Ohmoto and M. Shiota \cite{OS1}, \cite{OS2} on $\mathcal C^1$-triangulations of semialgebraic sets, we prove here by using different methods the following theorem: {\it Let $R$ be a real closed field and let an expansion of $ R$ to an o-minimal structure be given. Then for any closed bounded definable subset $A$ of $ R^n$ and a finite family $B_1,\dots,B_r$ of definable subsets of $A$ there exists a definable triangulation \linebreak $h\colon |\mathcal K|\rightarrow A$ of $A$ compatible with $B_1,\dots,B_r$ such that $\mathcal K$ is a simplicial complex in $R^n$, $h$ is a $C^1$-embedding of each \rom{(}open\rom{)} simplex $\Delta\in \mathcal K$ and $h$ extends to a definable $C^1$-mapping defined on a definable open neighborhood of $|\mathcal K|$ in $R^n$.} This improves Ohmoto-Shiota's theorem in three ways; firstly, $h$ is a $\mathcal C^1$-embedding on each simplex; secondly, the simplicial complex $\mathcal K$ is in the same space as $A$ and thirdly, our proof is performed for any o-minimal structure. The possibility to have $h$ with the first of these properties was stated by Ohmoto and Shiota as an open problem (see \cite{OS1}).