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 Affineness of definable $C^r$ manifolds and its applications Bull. Korean Math. Soc. 2003 Vol. 40, No. 1, 149-157 Published online March 1, 2003 Tomohiro Kawakami Wakayama University Abstract : Let $\mathcal M$ be an exponentially bounded o-minimal expansion of the standard structure $\mathcal R=(\bR, +, \cdot, <)$ of the field of real numbers. We prove that if $r$ is a non-negative integer, then every definable $C^r$ manifold is affine. Let $f:X \to Y$ be a definable $C^1$ map between definable $C^1$ manifolds. We show that the set $S$ of critical points of $f$ and $f(S)$ are definable and $\dim f(S)<\dim Y$. Moreover we prove that if $1 unique definable$C^r$manifold structure up to definable$C^r$diffeomorphism. Keywords : definable$C^r$manifolds, definable$C^r\$ maps, o-minimal, Sard's theorem, exponentially bounded MSC numbers : 14P20, 14P10, 57R35, 57R55, 58A05, 03C64 Downloads: Full-text PDF