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differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex new file mode 100644 index 0000000..3010eab --- /dev/null +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex @@ -0,0 +1,120 @@ +%-------------------------------------------------------------------- +\chapter{Varietà e teoria del grado} +\setlength{\parindent}{2pt} + +\begin{multicols*}{2} + \section{Varietà differenziabili e prime definizioni} + + \subsection{Mappe \texorpdfstring{$C^\infty$}{C∞} e diffeomorfismi} + + \begin{definition}[Mappe lisce tra due sottinsiemi] Siano $X \subseteq \RR^k$, + $Y \subseteq \RR^\ell$ sottinsiemi qualsiasi. Allora una funzione + $f : X \to Y$ si dice di classe $C^\infty$ (o \textit{liscia}) se per ogni + $x \in X$ esistono un aperto $W_x$ e una funzione $F : W_x \to \RR^\ell$, + chiamata \textit{estensione}, di classe $C^\infty$ per cui: + \[ + \restr{F}{W_x \cap X} = \restr{f}{W_x \cap X}. + \] + \end{definition} + + \begin{remark} + Osserviamo che i sottinsiemi di $X$ della forma $W \cap X$ con $W$ aperto sono esattamente gli + aperti per la topologia di sottospazio di $X$. + \end{remark} + + \begin{definition}[Diffeomorfismo] + Siano $X \subseteq \RR^k$, + $Y \subseteq \RR^\ell$ sottinsiemi qualsiasi. + Allora una funzione $f : X \to Y$ si dice \textbf{diffeomorfismo} se + è un omeomorfismo, è liscia e ammette inversa liscia. + \end{definition} + + \begin{remark} + Le definizioni di mappa liscia e diffeomorfismo date sono chiaramente compatibili + con le usuali definizioni date su aperti di $\RR^n$. + \end{remark} + + \begin{proposition} + La composizione di mappe lisce è liscia. La restrizione di una mappa liscia + è liscia. + \end{proposition} + + \begin{proof} + Siano $f : X \to Y$ e $g : Y \to Z$ due mappe lisce con $X \subseteq \RR^k$, + $Y \subseteq \RR^\ell$, $Z \subseteq \RR^p$. Sia $x \in X$. Allora, + poiché $f$ è liscia, esistono $W_x \subseteq \RR^k$ aperto e $F : W_x \to \RR^\ell$ liscia tale per cui: + \[ + \restr{F}{W_x \cap X} = \restr{f}{W_x \cap X}. + \] + Analogamente, per $f(x) \in Y$ esistono $U_{f(x)} \subseteq \RR^\ell$ aperto e + $G : U_{f(x)} \to \RR^p$ liscia tale per cui: + \[ + \restr{G}{U_{f(x)} \cap Y} = \restr{g}{U_{f(x)} \cap Y}. + \] + Pertanto, a meno di restringere $W_x$ per ottenere $F(W_x \cap X) \subseteq U_{f(x)} \cap Y$, + si ha: + \[ + \restr{G \circ F}{W_x \cap X} = \restr{g \circ f}{W_x \cap X}, + \] + dove $g \circ f$ è liscia. + \end{proof} + + \subsection{Varietà differenziabili, carte, atlanti e parametrizzazioni locali} + + \begin{definition}[Varietà differenziabile liscia in $\RR^k$] + Un insieme $M \subseteq \RR^k$ si dice \textbf{varietà (differenziabile liscia) di dimensione $m>0$} se per ogni + suo punto $x$ esistono un intorno aperto $W_x$ in $\RR^k$ e un diffeomorfismo + $f_x : W_x \cap M \to U$ verso un aperto $U$ in $\RR^m$. Per $m = 0$, si richiede invece + che ogni $W_x \cap M$ sia un singoletto. \medskip + + Gli insiemi della forma $W_x \cap M$ si dicono \textbf{carte locali}, e formano un \textbf{atlante} della varietà. L'inversa di + $f_x$ si dice invece \textbf{parametrizzazione locale} di $x$ in $M$. + \end{definition} + + \begin{remark} + Le varietà di dimensione zero sono esattamente le unioni di punti isolati. + \end{remark} + + \begin{remark} + Le carte locali inducono un ricoprimento aperto di $M$, e quindi, qualora $M$ fosse compatta, si potrebbe + sempre prendere un atlante finito. + \end{remark} + + \begin{remark} + Ogni aperto di $\RR^k$ è una varietà di dimensione $k$. Le superfici sono invece varietà + di dimensione $2$, le cui parametrizzazioni locali sono date dalle parametrizzazioni + regolari. + \end{remark} + + \section{Spazio tangente} + + \subsection{Differenziale e spazio tangente} + + Ricordiamo la definizione di differenziale per mappe su aperti di spazi reali: + + \begin{definition}[Differenziale per $f : U \subseteq \RR^k \to \RR^\ell$] + Sia $U$ un aperto di $\RR^k$ e sia $f : U \to \RR^\ell$ una funzione liscia. + Allora il \textbf{differenziale $df_x : \RR^k \to \RR^\ell$} nel + punto $x \in U$ è la funzione tale per cui: + \[ + df_x(h) \defeq \lim_{t \to 0} \frac{f(x+th) - f(x)}{t}. + \] + Equivalentemente $df_x$ è l'unica funzione lineare tale per cui: + \[ + f(x+h) = f(x) + df_x(h) + o(\norm{h}). + \] + \end{definition} + + \begin{remark} + Il differenziale $df_x$ rispetta alcune proprietà fondamentali: + \begin{enumerate}[(i.)] + \item La matrice di $df_x$ è data dallo jacobiano $Jf(x) = (\partial_{x_j} f_i(x))_{i, j}$. + \item Il differenziale rispetta la \textit{regola della catena} (chain rule): + \[ d(f \circ g)_x = df_{g(x)} \circ dg_x. \] + \item Data $\id_U$ su un aperto $U \subseteq \RR^k$, allora $d(\id_U)_x = \id_{\RR^k}$ per ogni $x \in U$. + \item Dati $U' \subseteq U \subseteq \RR^k$ con $U'$ aperto, l'inclusione $\iota : U' \to U$ è tale per cui + $d(\iota)_x = \id_{\RR^k}$ per ogni $x \in U'$. + \item Se $L : U \to \RR^\ell$ è lineare, allora $d L_x = L$ per ogni $x \in U$. + \end{enumerate} + \end{remark} +\end{multicols*}