  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.
