Imbeddings of manifolds defined on an $O$-minimal structure on $(\mathbb R,+,\cdot,<)$
Bull. Korean Math. Soc. 1999 Vol. 36, No. 1, 183-201
Tomohiro Kawakami
Wakayama University
Abstract : Let $M$ be an $o$-minimal structure on the standard structure
$\frak R:=(\Bbb R, +,$
%\par\noindent
$\cdot, <)$ of the field of real numbers.
We study $C^r \frak S$ manifolds and $C^r \frak S$-$G$ manifolds
$(0 \le r \le \omega)$ which are generalizations of
Nash manifolds and Nash $G$ manifolds.
We prove that if $M$ is polynomially bounded,
then every $C^r \frak S$ $(0 \le r <\infty)$ manifold is $C^r
\frak S$
imbeddable into some $\Bbb R^n$,
and that if $M$ is exponential and $G$ is a compact affine
$C^{\omega} \frak S$ group,
then each compact $C^{\infty} \frak S$-$G$ manifold is
$C^{\infty} \frak S$-$G$ imbeddable into some representation of $G$.
Keywords : Nash $G$ manifolds, $C^{\infty} G$ manifolds, affine, $o$-minimal
MSC numbers : 14P10, 14P15, 14P20, 57S05, 57S15, 58A07, 03C
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