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z1+?RX>da|k-8=O?UsOGhDJmaa95NgR5y@Qk2ug>bj?2U>PXW>bzwGfl3Ao>HaP{Z2 z$crd4b`#g$z3)BX0OUK&E7Bvu75A1?o-L#-2F9EXC+C zWCN)*?L3yR10+MHUfg2lFvbDeg*GdMQc<*zr!zVNWpv=MMN6efg9S-sf3w6Y^SKP$*A ztQE^w+zGbqDfd;#0nI+P?vnJMY+I!rV7qoBJ`y7~8Y9>0BR=X;v$UlO$Kbt*awB+h zncx4I5Igs#xF#fd202cd$vFv@bO_tZeJC}Ze5Q8`5Umtazdt}}I^o`L{?~-hvcH{c z!)ZN|`|NxmU^${(WyfhgGRbYjX+Pp*Fn!WW`(YzfiKhN`1PnIveX9UolC%m-N^Z0C zvN<^8JM-sh)o;&HF1H%XtBvMu$L5r}%jWUaUz+|k;&V5B_-Iwat1tGH%@>uwD70v) zmt1%O^R_EGMwMSGypD&IjM17DMwB{T_{~-~7-UHgs2$K@gWlj92f4&_XotddByzK`Y!} zI9_W-b}zqhd`O-LBk$@dyj=)m=!wby)(4*`;wb z6lGA2GuT+lvdd@c&Wt5l{vXjF20Y7Q7WwdH;@<1zeS+VvN9Md1^^hj0EGD{dQ2CpL zcN8IknmpiA;25d%wFu96qCEu8M>d0fP+Mm!t~w5Q`Bi~ZQi;Z3!Bx6HY zu|>;IN8KE{G 1$ o $\partial M = \emptyset$] + Una \textbf{varietà orientata di dimensione $m$} (con $\underline{m>1}$ o $\underline{\partial M = \emptyset}$) + è una coppia $(M, \Theta)$, dove $M$ è una $m$-varietà, eventualmente con bordo, e $\Theta = \{\Theta_x\}_{x \in M}$ + è una famiglia di orientazioni degli spazi tangenti dei punti di $M$ tale per cui: + + \begin{quote} + Per ogni $x \in M$ esiste una parametrizzazione locale $g : U \subseteq \RR^m \to g(U) \subseteq M$ con + $\dif g_u (\Theta_0) = \Theta_{g(U)}$ per ogni $u \in U$ (\textbf{condizione di compatibilità di $g$ con $\Theta$}). + \end{quote} + + Una varietà $M$ per cui esiste una famiglia di orientazioni tali per cui $(M, \Theta)$ è orientata si + dice \textbf{orientabile}. + \end{definition} + + \begin{definition}[$1$-varietà compatta orientata bordata] + Un'orientazione su una $1$-varietà connessa compatta con bordo è per definizione una famiglia + $\Theta = \{\dif \varphi_t(\Theta_0)\}_{t \in [0, 1]}$ dove + $\varphi : [0, 1] \to M$ è un diffeomorfismo. \smallskip + Se $M$ è sconnessa, un'orientazione è un'orientazione su ciascuna componente connessa. + \end{definition} + + \begin{proposition} + Una varietà connessa, eventualmente con bordo, ammette al più due orientazioni. + \end{proposition} + + \begin{proof} Si fissi un'orientazione $\Theta$ per la varietà $M$. Sia $\Theta'$ un'altra orientazione. + Dividiamo la dimostrazione in due casi: + \begin{itemize} + \item $\boxed{\dim M > 1}$ Si definiscano i seguenti due insiemi: + \begin{equation*} + \begin{aligned} + A & \defeq \{ x \in M \mid \Theta_x = \Theta'_x \}, \\[1ex] + B & \defeq \{ x \in M \mid \Theta_x = -\Theta'_x \}. + \end{aligned} + \end{equation*} + Osserviamo che $A$ e $B$ sono disgiunti, e che la loro unione è la varietà $M$. Poiché + $M$ è connessa, mostrando che $A$ e $B$ sono aperti, uno dei due sarebbe vuoto, da cui la tesi. \smallskip + + Dimostriamo dunque la tesi. Senza perdita di generalità, mostriamo solo che $A$ è aperto. Sia + $g : U \to g(U)$ una parametrizzazione locale di $x$ compatibile con $\Theta$ e $g(u) = x$, e sia $h : V \to h(V)$ + una parametrizzazione compatibile con $\Theta'$ e $h(v) = x$. Assumiamo senza perdita di generalità che $g(U) = h(V)$. \smallskip + + Poiché $g$ è compatibile con $\Theta$, si ha $\dif g(\Theta_0) = \Theta_x$, e così + per $h$ si ha $\dif h_x(\Theta_0) = \Theta'_x$. Quindi: + \[ + \dif g_u(\Theta_0) = \dif h_v(\Theta_0) \implies \dif (g\inv \circ h)_v (\Theta_0) = \Theta_0. + \] + Pertanto $\det(J(g\inv \circ h)_v) > 0$. Per continuità esiste allora un intorno $J$ di $v$ in cui: + \[ + \det(J(g\inv \circ h)_{v'}) > 0, \quad \forall v' \in J. + \] + Questo si traduce nell'avere $\Theta_{h(v')} = \Theta'_{h(v')}$ su tutto $J$, e quindi + $h(J)$ è un aperto di $M$ contenente $x$ e contenuto in $A$; dunque $A$ è aperto. + + \item $\boxed{\dim M = 1}$ Se $\varphi$ e $\psi$ sono due diffeomorfismi da $[0, 1]$ in $M$ che inducono + $\Theta$ e $\Theta'$, allora $\varphi\inv \circ \psi$ è un diffeomorfismo da $[0, 1]$ in sé. In quanto + tale, la sua derivata è ovunque non nulla, e il suo segno determina se $\Theta'$ è $\Theta$ o $-\Theta$. + \end{itemize} + \end{proof} + + \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. + \end{definition} + + \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 + + 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 + + Questo semispazio non dipende dalla parametrizzazione scelta. + ... + \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}. + \end{definition} + + \begin{lemma}[L'orientazione indotta sul bordo è ben definita] + ... + \end{lemma} + + \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 + + Per $M$ orientata tramite un diffeomorfismo $\varphi : [0, 1] \to M$, si associa + $-1$ a $\varphi(0)$ e $+1$ a $\varphi(1)$. + \end{definition} \end{multicols*}