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Allora $\partial M$ è una varietà senza bordo di dimensione $m-1$. \end{proposition} @@ -778,6 +778,19 @@ Brown (Corollario \ref{cor:brown}). \end{remark} + \begin{remark}[$T_x \partial M$ è un iperpiano di $T_x M$] + Sia $M$ una $m$-varietà con bordo. Grazie alla Proposizione \ref{prop:bordo_è_varietà} + sappiamo che $\partial M$ è una $(m-1)$-varietà. \smallskip + + Consideriamo l'inclusione $\iota : \partial M \to M$. Chiaramente $\iota$ è una + mappa liscia tra varietà con differenziale l'inclusione $T_x \partial M \hookrightarrow T_x M$. + In particolare vale: + \[ + \boxed{T_x \partial M \subseteq T_x M} + \] + per ogni punto $x \in \partial M$. + \end{remark} + \subsection{Varietà con bordo da valori regolari} \begin{theorem} \label{thm:varietà_bordata_da_valore_regolare_su_R} @@ -1439,23 +1452,63 @@ $(M, -\Theta)$, dove $-\Theta$ è l'unica altra orientazione possibile. \end{definition} + \begin{remark} + Una varietà è sempre ``localmente orientabile'': è sufficiente + prendere l'orientazione indotta da un'unica parametrizzazione locale. + \end{remark} + \subsection{Orientazione sul bordo della varietà} - \begin{remark} - L'orientazione di una $m$-varietà $M$ con bordo determina la scelta di uno dei semispazi di - $T_x M \setminus T_x \partial M$ per ogni $x$. \smallskip + \begin{remark}[Il semispazio interno è ben definito] + L'orientazione locale di una $m$-varietà $M$ con bordo determina sempre la scelta di uno dei semispazi di + $T_x M \setminus T_x \partial M$ per ogni $x$ sul bordo $\partial M$. \smallskip - Se infatti $g : U \to g(U)$ è una parametrizzazione di un punto $x$ sul bordo $\partial M$ con $g(u) = x$, - il semispazio scelto è proprio $\dif g_u (H^n \setminus \partial H^n)$. \smallskip + Se infatti $g : U \to g(U)$ è una parametrizzazione di un punto $x \in \partial M$ con $g(u) = x$, + il semispazio scelto è proprio: + \[ \dif g_u (H^m \setminus \partial H^m). \] + Questo semispazio non dipende dalla parametrizzazione scelta. Se infatti $h : V \to h(V)$ è + un'altra parametrizzazione con $h(v) = x$, a meno di restringere le mappe possiamo considerare + la funzione di transizione $h\inv \circ g$. Osserviamo che: + \[ + \dif g_u = \dif h_v \circ \dif (h\inv \circ g)_u. + \] + Mostrando allora che $\dif (h\inv \circ g)_u(H^m \setminus \partial H^m) = H^m \setminus \partial H^m$, + otteniamo la tesi. \smallskip - Questo semispazio non dipende dalla parametrizzazione scelta. - ... + Osserviamo che $h\inv \circ g$ è una funzione da un aperto intersecante il bordo di $H^m$ in + un altro aperto dello stesso tipo. Pertanto $\dif (h\inv \circ g) : \RR^{m-1} \times \RR \to \RR^{m-1} \times \RR$ + deve mandare lo spazio tangente $T_u \partial U = \partial H^m$ in $T_v \partial V = \partial H^m$, dal momento che $h\inv \circ g$ + si restringe a una parametrizzazione di $\partial V$ a partire da $\partial U$. \smallskip + + Quindi $J(h\inv \circ g)_u$ deve essere della seguente forma: + \[ + J(h\inv \circ g)_u = \begin{pmatrix} + * & \vline & * \\ + \hline + 0 & \vline & \lambda + \end{pmatrix}. + \] + Se $h\inv \circ g = (\Psi_1, \Psi_2)$, con $\Psi_1$ funzione a valori in $\RR^{m-1}$ e + $\Psi_2$ a valori in $\RR$, allora: + \[ + \lambda = \pd{\Psi_2}{x_m}(u) = \lim_{\eps \to 0^+} \frac{\overbrace{\Psi_2(x + \eps x_m)}^{\mathclap{\geq \, 0}} - \overbrace{\Psi_2(x)}^{\mathclap{= \, 0}}}{\eps} \geq 0. + \] + Quindi $\dif (h\inv \circ g)_u(H^m) = H^m$, da cui segue poi facilmente la tesi. \end{remark} \begin{definition} - Data una varietà orientata $M$, i vettori appartenenti al semispazio - scelto di $T_x M \setminus T_x \partial M$ si dicono \textbf{interni}, - mentre quelli del semispazio complementare sono detti \textbf{esterni}. + Data una $m$-varietà bordata $M$ e un punto $x$ appartenente al bordo + $\partial M$, si definisce il \textbf{semispazio interno} riferito a + $x$ come: + \[ + \boxed{\dif g_u(H^m \setminus \partial H^m),} + \] + dove $g$ è una parametrizzazione locale di $x$ con $g(u) = x$. I suoi + vettori sono detti \textbf{interni}. \smallskip + + Si dice \textbf{semispazio esterno} il semispazio complementare a quello + interno rispetto al taglio dell'iperpiano $T_x \partial M$ in + $T_x M$, e i suoi vettori sono detti \textbf{esterni}. \end{definition} \begin{lemma}[L'orientazione indotta sul bordo è ben definita]