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 Every definable $C^r$ manifold is affine Bull. Korean Math. Soc. 2005 Vol. 42, No. 1, 165-167 Published online March 1, 2005 Tomohiro Kawakami Wakayama University Abstract : Let $\mathcal M=(\bR, +, \cdot, <, \dots)$ be an o-minimal expansion of the standard structure $\mathcal R=(\bR, +, \cdot, >)$ 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 $\bR^{2n+1}$. Moreover we prove that if $1 admits a unique definable$C^r$manifold structure up to definable$C^r$diffeomorphism. Keywords : definable$C^r\$ manifolds, o-minimal, affine MSC numbers : 03C64, 14P10, 14P20, 57R55, 58A05 Downloads: Full-text PDF