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una varietà orientata. Allora una base per $T_x M$ si dice + \textbf{positiva} se è della stessa orientazione di $\Theta_x$; altrimenti + si dice \textbf{negativa}. + \end{definition} + \begin{definition}[Varietà di orientazione opposta] Data $(M, \Theta)$ una varietà orientata, indichiamo con $-M$ la varietà $(M, -\Theta)$, dove $-\Theta$ è l'unica altra orientazione possibile. @@ -1457,7 +1463,33 @@ prendere l'orientazione indotta da un'unica parametrizzazione locale. \end{remark} - \subsection{Orientazione sul bordo della varietà} + \begin{proposition}[$m$-varietà immerse in $\RR^m$ sono orientabili] \label{prop:orientazione_immersa_Rm} + Sia $M$ una $m$-varietà immersa in $\RR^m$. Allora + $M$ è orientabile secondo l'orientazione canonica di $\RR^m$. + \end{proposition} + + \begin{proof} + Sia $x \in M$. Osserviamo + che deve valere necessariamente $T_x M = \RR^m$, dal momento che + $T_x M$ è uno spazio vettoriale $m$-dimensionale immerso in $\RR^m$. + Definiamo allora $\Theta_x \defeq \Theta_0$, + dove $\Theta_0$ è l'orientazione canonica di $\RR^m$. \smallskip + + Se $g : U \subseteq H^m \to g(U)$ è una parametrizzazione locale di $x$, possiamo restringere e + riparametrizzare $g$ + in modo tale che il suo dominio sia una semipalla di $\RR^m$. Se $\dif g_{-}$ preserva + l'orientazione canonica, scegliamo $g$ come parametrizzazione locale compatibile + per $\Theta \defeq \{\Theta_x\}_{x \in M}$. Altrimenti possiamo precomporre $g$ + con una riflessione rispetto all'asse della semipalla e ottenere una nuova parametrizzazione, + stavolta compatibile. \smallskip + + Infatti, $g$ si estende a un diffeomorfismo tra aperti di $\RR^m$ e, + per il Teorema della permanenza del segno, lo jacobiano deve essere localmente o positivo + o negativo, e quindi $g$ preserva localmente l'orientazione. + Dunque $M$ è orientabile secondo l'orientazione canonica di $\RR^m$. + \end{proof} + + \subsection{Orientazione sul bordo della varietà, semispazio interno o esterno} \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 @@ -1511,16 +1543,110 @@ $T_x M$, e i suoi vettori sono detti \textbf{esterni}. \end{definition} - \begin{lemma}[L'orientazione indotta sul bordo è ben definita] - ... - \end{lemma} + \begin{remark}[L'orientazione indotta sul bordo è ben definita -- \textit{esistenza}] + Sia $x \in \partial M$ un punto della $m$-varietà orientata $(M, \Theta)$, con $m > 1$. + Sia $g : U \subseteq H^m \to g(U)$ una parametrizzazione locale di $x$ con $g(u) = x$ che + sia compatibile con l'orientazione $\Theta$. Allora: - \begin{definition} - Data una varietà orientata $M$ con bordo e $\dim M > 1$, la famiglia $\{ \Theta_x^{\partial M} \}_{x \in \partial m}$ - di orientazioni di $T_x \partial M$ indotte dall'orientazione di $M$ è un'orientazione su $\partial M$ - detta \textbf{orientazione indotta sul bordo} (o \textit{orientazione di bordo}). \smallskip + \begin{itemize} + \item per definizione, $\dif g_u(-e_m)$ è + un vettore \textit{esterno} per $x$; + \item $\{\dif g_u(e_i)\}_{i=1 -- m-1}$ è una base + di $T_x \partial M$, dacché $\restr{g}{\partial H^m}$ + si identifica come una parametrizzazione locale di $x$ + in $\partial M$; + \item $\{\dif g_u(e_i)\}_{i=1 -- m}$ è una base positiva + di $T_x M$, dacché $g$ è compatibile e $\{e_i\}_i$ ha + l'orientazione canonica in $\RR^m$. + \end{itemize} + \end{remark} + + \begin{remark}[L'orientazione indotta sul bordo è ben definita -- \textit{unicità}] + Sia $x \in \partial M$ un punto della $m$-varietà orientata $(M, \Theta)$, con $m > 1$. + Sia $\{v_1, \ldots, v_n\}$ una base di $T_x M$ tale per cui + $v_1$ un vettore \underline{esterno} per $x \in \partial M$ e + $\{v_2, \ldots, v_n\}$ è base di $T_x \partial M$. \smallskip + + Sia $\{v_1', \ldots, v_n'\}$ un'altra tale base di $T_x M$. + Sia $g : U \to g(U)$ una parametrizzazione locale di $x$ con + $g(u) = x$. Allora $\dif g_u$ è un isomorfismo, e in quanto tale + lascia invariate le relazioni di orientazioni delle basi di $T_u U = \RR^m$ + quando portate in $T_x M$. \smallskip + + Sia $w_i = \dif g_u\inv(v_i)$ e sia $w_i' = \dif g_u\inv(v_i')$. Dal momento + che $v_1$ e $v_1'$ sono vettori esterni, si deve avere necessariamente + $(w_1)_m$, $(w_1')_m < 0$. Dal momento che $\restr{g}{\partial H^m}$ si identifica + naturalmente come una parametrizzazione locale di $x$ in $\partial M$, e che i $v_i$ e i $v_i'$ per $i > 1$ formano + una base di $T_x \partial M$, si ha $(w_i)_m = (w_i')_m = 0$ per ogni $i > 1$. \smallskip + + Dunque la matrice di cambio di base da $\{v_1, \ldots, v_n\}$ a + $\{v_1', \ldots, v_n'\}$ è della seguente forma: + \[ + M = \begin{pmatrix} + \lambda & \vline & 0 \\ + \hline + * & \vline & A + \end{pmatrix}, + \] + dove $\lambda > 0$ affinché $w_1$ e $w_1'$ abbiano ancora lo stesso segno + sull'ultima coordinata. Dal momento che $\{v_1, \ldots, v_n\}$ e + $\{v_1', \ldots, v_n'\}$ sono basi positive di $T_x M$, allora hanno + stesso orientazione, e quindi $\det(M) > 0$. Ne segue che $\det(A) > 0$. \smallskip + + Osserviamo che $A$ è proprio la matrice di cambio di base da $\{v_2, \ldots, v_n\}$ a + $\{v_2', \ldots, v_n'\}$. Dunque queste due basi hanno stessa orientazione. + \end{remark} + + \begin{remark}[L'orientazione indotta sul bordo è ben definita -- \textit{è effettivamente un'orientazione}] + Sia $x \in \partial M$ un punto della $m$-varietà orientata $(M, \Theta)$, con $m > 1$. + Denotiamo con $\Theta_x^{\partial M}$ l'orientazione indotta da $\{v_2, \ldots, v_n\}$ + su $T_x \partial M$ da una base positiva $\{v_1, v_2, \ldots, v_n\}$ di $T_x M$ + con $v_1$ \underline{esterno} e $\{v_2, \ldots, v_n\}$ base di $T_x \partial M$. \smallskip + + Definiamo: + \[ \Theta^{\partial M} \defeq \{ \Theta_x^{\partial M} \}_{x \in \partial M}. \] + Data $g : U \subseteq H^m \to g(U) \subseteq M$ parametrizzazione locale + compatibile di + $x \in \partial M$ in $M$, $\restr{g}{\partial U}$, identificata come parametrizzazione da $\RR^{m-1}$, + è compatibile rispetto a $\Theta^{\partial M}$, a meno di restringimento del dominio a una palla + con conseguente riflessione rispetto a un asse. \smallskip + + Si può infatti estendere in tal caso la base canonica di $\RR^{m-1} \cong \partial H^m$ a una base di $\RR^m$ con l'aggiunta di + un vettore la cui immagine tramite $\dif g_{-}$ risulta essere sempre esterna. Quindi + $\Theta^{\partial M}$ è un'orientazione per $\partial M$. + \end{remark} + + \begin{definition}[Orientazione indotta sul bordo] + Sia $(M, \Theta)$ una $m$-varietà bordata e orientata con $m > 1$. Per $x \in \partial M$ denotiamo con + $\Theta_x^{\partial M}$ l'orientazione indotta da $\{v_2, \ldots, v_n\}$ su $T_x \partial M$ da una base positiva $\{v_1, v_2, \ldots, v_n\}$ di $T_x M$ + con $v_1$ \underline{esterno} e $\{v_2, \ldots, v_n\}$ base di $T_x \partial M$. \smallskip + + Si definisce allora $\Theta^{\partial M} = \{ \Theta_x^{\partial M} \}_{x \in \partial M}$ come + l'\textbf{orientazione indotta sul bordo} (o \textit{orientazione di bordo}) per $\partial M$. - Per $M$ orientata tramite un diffeomorfismo $\varphi : [0, 1] \to M$, si associa + Per $M \cong [0, 1]$ orientata tramite un diffeomorfismo $\varphi : [0, 1] \to M$, si associa $-1$ a $\varphi(0)$ e $+1$ a $\varphi(1)$. \end{definition} + + \begin{corollary} \label{cor:bordo_orientabile_immersa_Rm} + Sia $M$ una $m$-varietà orientabile immersa in $\RR^m$. Allora + $\partial M$ è orientabile con l'orientazione indotta sul bordo dall'orientazione + canonica di $\RR^m$. + \end{corollary} + + \begin{proof} + Segue immediatamente dalla Proposizione \ref{prop:orientazione_immersa_Rm}. + \end{proof} + + \begin{corollary} + $S^n$ è orientabile per ogni $n$. + \end{corollary} + + \begin{proof} + Segue immediatamente dal Corollario \ref{cor:bordo_orientabile_immersa_Rm}, + dacché $S^n = \partial D^{n+1}$ e $D^{n+1}$ è una $(n+1)$-varietà immersa + in $\RR^{n+1}$. + \end{proof} + + %\section{Teoria del grado su \texorpdfstring{$\ZZ$}{ℤ}} \end{multicols*}