Submission date: 11 January 2023.

Abstract:

We consider a definably complete locally o-minimal expansion of an ordered field. The first author proposed the notion of special submanifolds with tubular neighborhoods and demonstrated that a definable set is partitioned into finitely many special submanifolds with tubular neighborhoods in his previous study. This decomposition theorem can be used as a substitute of the definable cell decomposition theorem and the stratification theorem in the o-minimal setting. We demonstrate that (i) a definable closed set is the zero set of a definable definable C^{r-1} map between definable C^r submanifolds in the definable C^r topology. Using (i), we also demonstrate that a regular definable C^r manifold is a definably C^r diffeomorphic to a definable C^r submanifold. It enables us to show that the definable quotient of a definable C^r group by a definable subgroup exists.

Mathematics Subject Classification: Primary 03C64, Secondary 57R40, 54B15

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Full text arXiv 2301.04264: pdf, ps.