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
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