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

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.