# On Whitney embedding of o-minimal manifolds

## TL;DR

This paper establishes a definable Whitney embedding theorem for o-minimal manifolds, showing they can be embedded into Euclidean space with compatible smooth structures, extending classical results to a definable setting.

## Contribution

It proves a definable Whitney embedding theorem for o-minimal manifolds and demonstrates the existence of compatible higher smoothness atlases.

## Key findings

- Every definable $ ext{C}^p$ manifold can be embedded into some $ ext{R}^N$.
- Such manifolds admit a compatible $ ext{C}^{p+1}$ atlas.
- The result extends classical Whitney embedding to o-minimal structures.

## Abstract

We prove a definable version of the Whitney embedding theorem for abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p<\infty$, namely: every abstract-definable $\mathcal{C}^p$ manifold is abstract-definable $C^p$ embedded into $R^N$, for some positive integer $N$. As a consequence, we show that every abstract-definable $\mathcal{C}^p$ manifold has a compatible $\mathcal{C}^{p+1}$ atlas.