• Title, Summary, Keyword: definable C$^r$ manifolds

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AFFINENESS OF DEFINABLE Cr MANIFOLDS AND ITS APPLICATIONS

  • Kawakami, Tomohiro
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.1
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    • pp.149-157
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    • 2003
  • Let M be an exponentially bounded o-minimal expansion of the standard structure R = (R ,+,.,<) 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 ${\longrightarrow}$ 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 < s < ${\gamma}$ < $\infty$, then every definable $C^{s}$ manifold admits a unique definable $C^{r}$ manifold structure up to definable $C^{r}$ diffeomorphism.

EVERY DEFINABLE Cr MANIFOLD IS AFFINE

  • KAWAKAMI, TOMOHIRO
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.1
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    • pp.165-167
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    • 2005
  • Let M = (R, +, $\cdot$, <, ... ) be an o-minimal expansion of the standard structure R = (R, +, $\cdot$, >) 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 $R^{2n+l}$. Moreover we prove that if 1 < s < r < $\infty$, then every definable $C^s$ manifold admits a unique definable $C^r$ manifold structure up to definable $C^r$ diffeomorphism.