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
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