Approximation and zero set of definable functions in a definably complete locally o-minimal structure

TL;DR

This paper studies approximation and embedding properties of definable functions and manifolds within a specific mathematical structure, advancing understanding of their geometric and topological features.

Contribution

It introduces a definable $ ext{C}^r$ approximation method and proves an embedding theorem for definably compact $ ext{C}^r$ manifolds in a locally o-minimal structure.

Findings

Definable $ ext{C}^r$ approximation of maps is established.

Definably normal $ ext{C}^r$ manifolds are diffeomorphic to submanifolds.

Existence of quotients of definably compact $ ext{C}^r$ groups by subgroups is shown.

Abstract

We consider a definably complete locally o-minimal expansion of an ordered field. We treat two topics in this paper. The first topic is a definable $\mathcal C^r$ approximation of a definable $\mathcal C^{r-1}$ map between definable $\mathcal C^r$ submanifolds in the definable $\mathcal C^{r-1}$ topology. The second topic is the imbedding theorem for definably compact definable $\mathcal C^r$ manifolds. We demonstrate that a definably normal definable $\mathcal C^r$ manifold is a definably $\mathcal C^r$ diffeomorphic to a definable $\mathcal C^r$ submanifold. It enables us to show that the definable quotient of a definably compact definable $\mathcal C^r$ group by a definable subgroup exists.