# AFFINENESS OF DEFINABLE Cr MANIFOLDS AND ITS APPLICATIONS

• Kawakami, Tomohiro
• Published : 2003.02.01
• 29 4

#### Abstract

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. 炭᠂ ࠀ豈檧Ā檧Ā̀￿￿֗⨀Ā?ĀĀၧ잖⨀̀ĀĀ逅⨀倅⨀؀肀ქﾖ⨀밟Ԁꀮ֗⨀⃬│胫│惭│郭│塻җ⨀뀯֗⨀찟퀟퀄⨀Ԁ怯֗⨀í│탫│惭│烮│찟퀟퀄⨀퀟ഀĀ顪⨀Ā偫⨀㠈⨀ဩ⨀Ȁᠩ⨀⨀⬀Ā 堪⨀젪⨀阁Ā1֗⨀⡧잖⨀̀⡧잖⨀̀܀屣잖⨀ࠀ褝⎨Ā⎨Ā̀￿￿屣잖⨀檧Ā䁧잖⨀聧잖⨀֗⨀?җ⨀堵֗⨀ഀ Ā

#### Keywords

definable $C^{r}$ manifolds;definable $C^{r}$ maps;o-minimal;Sard's theorem;expotentionally bounded

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