# EVERY DEFINABLE Cr MANIFOLD IS AFFINE

• KAWAKAMI, TOMOHIRO (Department of Mathematics, Faculty of Education, Wakayama University)
• Published : 2005.02.01

#### Abstract

Let M = (R, +, $\cdot$, <, ... ) be an o-minimal expansion of the standard structure R = (R, +, $\cdot$, >) of the field of real numbers. We prove that if 2 $\le$ r < $\infty$, then every n-dimensional definable $C^r$ manifold is definably $C^r$ imbeddable into $R^{2n+l}$. Moreover we prove that if 1 < s < r < $\infty$, then every definable $C^s$ manifold admits a unique definable $C^r$ manifold structure up to definable $C^r$ diffeomorphism.

#### References

1. L. van den Dries, Tame topology and a-minimal structures, London Math. Soc. Lecture Note Ser. 248 (1998)
2. L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540 https://doi.org/10.1215/S0012-7094-96-08416-1
3. M. W. Hirsch, Differential manifolds, Springer, 1976
4. T. Kawakami, Affineness of definable $C^r$ manifolds and its applications, Bull. Korean Math. Soc. 40, 149-157
5. T. Kawakami, Equivariant differential topology in an o-minimal expansion of the field of real numbers, Topology Appl, 123 (2002), 323-349 https://doi.org/10.1016/S0166-8641(01)00200-0
6. T. Kawakami, Imbedding of manifolds defined on an a-minimal structures on (R, +, ., <), Bull. Korean Math. Soc. 36 (1999), 183-201
7. M. Shiota, Abstract Nash manifolds, Proc. Amer. Math. Soc. 96 (1986), 155-162
8. M. Shiota, Geometry of subanalyitc and semialgebraic sets, Progr. Math. 150 (1997)
9. A. G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127-150 https://doi.org/10.1016/0040-9383(69)90005-6

#### Cited by

1. Smooth functions in o-minimal structures vol.218, pp.2, 2008, https://doi.org/10.1016/j.aim.2008.01.002