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We consider a definably complete locally o-minimal expansion of an orderedfield. 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 betweendefinable $\mathcal C^r$ submanifolds in the definable $\mathcal C^{r-1}$topology. The second topic is the imbedding theorem for definably compactdefinable $\mathcal C^r$ manifolds. We demonstrate that a definably normaldefinable $\mathcal C^r$ manifold is a definably $\mathcal C^r$ diffeomorphicto a definable $\mathcal C^r$ submanifold. It enables us to show that thedefinable quotient of a definably compact definable $\mathcal C^r$ group by adefinable subgroup exists.

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