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un aperto di $M$ dal momento che $f$ è continua, e quindi + eredita la struttura di varietà di $M$ (vd. Osservazione \ref{rmk:aperti_di_varietà_sono_varietà}). + Dunque, conosciamo già le carte locali di un punto $x \in \{f > 0\}$. \smallskip - ... + Sia $x$ un punto di $\{f = 0\} = f\inv(0)$. Poiché $0$ è un valore regolare, $\dif f_x$ è + surgettiva, e quindi $\dim \ker \dif f_x = m - 1$. Supponiamo che $k$ sia tale per cui $M \subseteq \RR^k$. + Allora possiamo costruire un'applicazione lineare $L : \RR^k \to \RR^{m-1}$ tale per cui + $\restr{L}{\ker \dif f_x}$ è un isomorfismo. \smallskip + + Consideriamo la mappa $F : M \to \RR^{m-1} \times \RR$ tale per cui: + \[ + F(m) = (L(m), f(m)). + \] + Allora $F$ è una mappa liscia tra varietà, il cui differenziale in $x$ è un isomorfismo. Dunque, per il + Teorema \ref{thm:invertibilità_locale_varietà}, esiste un intorno aperto $U$ di $x$ in $M$ per il quale + $\restr{F}{U} : U \to V \defeq F(U)$ è un diffeomorfismo. \smallskip + + Tramite $\restr{F}{U}$ si induce allora un diffeomorfismo tra l'aperto $(\RR^{m-1} \times \RR_{\geq 0}) \cap V = H^m \cap V$ + di $H^m$ e l'aperto $F\inv(H^m \cap V) = \{ f > 0 \} \cap U$, tramite il quale il punto $x$ viene + mappato su $\partial H^m \cap V$. Dunque $\{ f \geq 0 \}$ è una $m$-varietà con bordo $f\inv(0)$. \end{proof} + \begin{remark} + Chiaramente il Lemma \ref{lem:varietà_bordata_da_valore_regolare_su_R} si generalizza + a un valore regolare qualunque. + \end{remark} + \begin{proposition} $D^n$ è una varietà $n$-dimensionale con bordo $S^{n-1}$. \end{proposition}