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differenziale/Scheda riassuntiva/preamble.tex @@ -89,6 +89,9 @@ \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\crit}{crit} + \DeclareMathOperator{\Span}{span} \newcommand{\defeq}{\overset{\mathrm{def}}{=}} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex index 44c2c5d..10f084a 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-notazioni.tex @@ -9,10 +9,18 @@ vettori o campi vettoriali; caratteri greci minuscoli ($\alpha$, $\beta$, $\gamma$) per indicare curve; e caratteri greci maiuscoli ($\Sigma$) per indicare superfici. + \section*{Algebra lineare} + \addcontentsline{toc}{section}{Algebra lineare} + + \begin{itemize} + \item $\rk$ -- rango di un'applicazione lineare o di una matrice. + \end{itemize} + \section*{Analisi matematica} \addcontentsline{toc}{section}{Analisi matematica} \begin{itemize} + \item $\restr{f}{A}$ -- restrizione di una funzione $f$ al sottinsieme $A$ del dominio. \item $f_i$ -- nel caso di una funzione $f : \RR^n \to \RR^m$, la proiezione di $f$ sulla $i$-esima coordinata, ovverosia $\pi_i \circ f : \RR^n \to \RR$. \item $\der{f}{t}(x)$, $f'(x)$, $\dot{f}$ -- derivata di una funzione $f : I \subseteq \RR \to \RR^n$. Nel caso $n > 1$, coincide con @@ -33,8 +41,8 @@ \item diffeomorfismo di classe $C^k$ -- funzione di classe $C^k$ con inversa di classe $C^k$. \end{itemize} - \section*{Geometria differenziale} - \addcontentsline{toc}{section}{Geometria differenziale} + \section*{Geometria differenziale delle curve e delle superfici} + \addcontentsline{toc}{section}{Geometria differenziale delle curve e delle superfici} \begin{itemize} \item $B_R(P, \RR^n)$, $B_R(P)$ -- la palla $n$-dimensionale di raggio $R$ e centro $P$. Ometteremo $\RR^n$ quando la dimensione si deduce dal contesto. @@ -88,4 +96,33 @@ \item $\chi(\Sigma)$ -- caratteristica di Eulero di una superficie con bordo. \end{itemize} + \section*{Teoria della misura} + \addcontentsline{toc}{section}{Teoria della misura} + + \begin{itemize} + \item $\vol$ -- applicazione per calcolare il volume di un rettangolo in $\RR^n$; tale per cui + $\vol([a_1, b_1] \times \cdots \times [a_n, b_n]) = \prod_i {(b_i - a_i)}$. + \item $m$ -- misura di Lebesgue su $\RR^n$. + \end{itemize} + + \section*{Teoria delle varietà} + \addcontentsline{toc}{section}{Teoria delle varietà} + + \begin{itemize} + \item $\dim M$ -- dimensione di una varietà $M$. + \item $(f, W \cap M)$ -- carta locale di una varietà $M$; si sottintende che $W$ sia un aperto dello spazio ambiente in cui è contenuto $M$ e + che $f$ sia un diffeomorfismo con dominio $W \cap M$ verso un aperto di $\RR^{\dim M}$. + \item $T_x M$ -- spazio tangente di un punto $x$ di una varietà $M$. + \item $\dif f_x$ -- differenziale di una mappa liscia $f : M \to N$ tra varietà nel punto $x$. + \item $\crit(f)$ -- insieme dei punti critici di una mappa liscia $f : M \to N$ tra varietà. + \end{itemize} + + \section*{Topologia} + \addcontentsline{toc}{section}{Topologia} + + \begin{itemize} + \item II-numerabile -- spazio topologico che ammette una base numerabile. + \item T1 -- spazio topologico i cui singoletti sono insiemi chiusi. + \item T2 -- spazio topologico in cui per due punti distinti esiste una coppia di intorni disgiunti dei due punti. + \end{itemize} \end{multicols*} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex index 9119f79..d0bce26 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/0-prerequisiti.tex @@ -5,8 +5,8 @@ \begin{multicols*}{2} - \section*{Analisi matematica} - \addcontentsline{toc}{section}{Analisi matematica} + \section*{Analisi matematica e teoria della misura} + \addcontentsline{toc}{section}{Analisi matematica e teoria della misura} \begin{itemize} \item \textbf{Teorema di Schwarz} -- Sia $f : \RR^n \to \RR$ una funzione che ammette derivate seconde @@ -71,6 +71,17 @@ \right\} \] è un aperto di $\RR \times \Omega$ e l'applicazione $\Phi : \mathcal{D} \to \RR^n$ è di classe $C^k$. + + \item \textbf{Caratterizzazione dell'annullamento della misura di Lebesgue} -- Sia $A$ un sottinsieme di $\RR^n$. Allora + $A$ ha misura nulla se e solo se, per ogni scelta di $\eps > 0$, esiste una famiglia numerabile $\{B_i\}_{i \geq 0}$ + di rettangoli $B_i \subseteq \RR^n$ tali per cui: + \[ A \subseteq \bigcup_{i \geq 0} B_i, \quad \sum_{i \geq 0} \vol(B_i) < \eps. \] + + \item \textbf{Lemma per la nullità della misura su un unione numerabile di insiemi di misura nulla} -- Se $\{A_k\}_{k \geq 0}$ + è una famiglia di sottinsiemi di misura nulla di $\RR^n$, allora anche $\bigcup_{k \geq 0} A_k$ ha misura nulla. + + \item \textbf{Teorema di Sard sugli aperti di $\RR^n$} -- Sia $U \subseteq \RR^n$ aperto, e sia $f : U \to \RR^n$ una mappa + liscia. Allora l'insieme $\crit(f)$ dei valori critici di $f$ ha misura nulla. \end{itemize} \end{multicols*} \ No newline at end of file diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex index 0759d94..b29e736 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/4-varieta-e-teoria-del-grado.tex @@ -67,7 +67,7 @@ $f_x : W_x \cap M \to U$ 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 + Le coppie della forma $(f_x, W_x \cap M)$ 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} @@ -77,7 +77,9 @@ \begin{remark} Le carte locali inducono un ricoprimento aperto di $M$, e quindi, qualora $M$ fosse compatta, si potrebbe - sempre prendere un atlante finito. + sempre prendere un atlante finito. \smallskip + + Inoltre, poiché $\RR^k$ è II-numerabile, si può sempre prendere un atlante numerabile. \end{remark} \begin{remark} @@ -244,7 +246,9 @@ \end{enumerate} \end{remark} - \subsection{Punti e valori regolari o critici} + \section{Valori regolari e critici} + + \subsection{Prime definizioni} \begin{definition}[Punti regolari o critici] Sia $f : M \to N$ una mappa liscia tra varietà, con @@ -252,7 +256,10 @@ Sia $x \in M$. Si dice che $x$ è un \textbf{punto critico} se $\rk(\dif f_x) < n$, e altrimenti si dice che è un - \textbf{punto regolare}. + \textbf{punto regolare}. \smallskip + + Indichiamo con $\crit(f)$ l'insieme dei punti critici di + $f$. \end{definition} \begin{definition}[Valori regolari o critici] @@ -264,4 +271,115 @@ è un \textbf{valore regolare} (in particolare lo è se $f\inv(y) = \emptyset$). \end{definition} + + \begin{remark} + È immediato osservare che l'insieme dei valori critici di + $f$ è esattamente $f(\crit(f))$. + \end{remark} + + \subsection{Teorema di invertibilità locale per varietà e lemma della pila di dischi} + + \begin{theorem}[di invertibilità locale per varietà] \label{thm:invertibilità_locale_varietà} + Siano $M$ e $N$ due varietà di stessa dimensione. Sia + $f : M \to N$ una mappa liscia. Se $x \in M$ è regolare, + allora esiste un intorno $A$ di $x$ in $M$ tale per cui + $\restr{f}{A} : A \to f(A)$ è un diffeomorfismo. + \end{theorem} + + \begin{proof} + Sia $g : U \to g(U)$ una parametrizzazione locale di $x$ in $M$ + con $g(u) = x$. + Sia $h : V \to h(V)$ una parametrizzazione locale di $f(x)$ in $N$ + con $h(v) = f(x)$. + A meno di restringere il dominio di $g$, possiamo supporre che + $f(g(U)) \subseteq h(V)$. + + \[\begin{tikzcd} + M && N \\ + \\ + U && V + \arrow["f"{description}, from=1-1, to=1-3] + \arrow["g", from=3-1, to=1-1] + \arrow["{h\inv \circ f \circ g}"{description}, dashed, from=3-1, to=3-3] + \arrow["h", from=3-3, to=1-3] + \end{tikzcd}\] + + Osserviamo che: + \[ \dif f_x \circ \dif g_u = \dif h_{f(x)} \circ \dif (h\inv \circ f \circ g)_u. \] + Dal momento che $x$ è regolare, $\dif f_x$ è un isomorfismo. Poiché anche $\dif g_u$ + e $\dif h_{f(x)}$ sono isomorfismi ($g$ e $h$ sono diffeomorfismi, vd. Proposizione + \ref{prop:diffeomorfismo_iso_diff}), allora anche $\dif (h\inv \circ f \circ g)_u$ è + un isomorfismo. \smallskip + + Per il Teorema di invertibilità locale sugli aperti di $\RR^n$, allora esiste un + aperto $A'$ di $U$ su cui $\restr{h\inv \circ f \circ g}{A'}$ è un diffeomorfismo. + Osserviamo che: + \[ + \restr{f}{g(A')} = \restr{h}{h\inv(f(g(A')))} \circ \restr{h\inv \circ f \circ g}{A'} \circ \restr{g\inv}{g(A')}. + \] + Quindi $\restr{f}{g(A')}$ è un diffeomorfismo in quanto composizione di restrizioni di + diffeomorfismi (vd. Proposizione \ref{prop:comp_liscia}), e $A \defeq g(A')$ è l'intorno cercato. + \end{proof} + + \begin{proposition} \label{prop:controimmagine_regolare_finita} + Sia $M$ una varietà compatta. Sia $f : M \to N$ una mappa liscia tra varietà della + \underline{stessa dimensione}. Se $y \in N$ è un valore regolare, allora $f\inv(y)$ è un insieme finito. + \end{proposition} + + \begin{proof} + Poiché $N$ è T1, $f\inv(y)$ è un chiuso di $M$; pertanto, essendo $M$ compatta, $f\inv(y)$ è + un compatto. \smallskip + + Mostriamo ora che $f\inv(y)$ è discreto. Sia $x \in f\inv(y)$. Dal momento che $y$ è un valore regolare, + $x$ è un punto regolare. Quindi esiste per il Teorema \ref{thm:invertibilità_locale_varietà} un + intorno aperto $A \subseteq M$ di $x$ per cui $\restr{f}{A}$ è un diffeomorfismo. In questo intorno $x$ + è l'unica controimmagine di $y$ mediante $f$, e quindi $f\inv(y)$ è discreto. \smallskip + + Dal momento che $f\inv(y)$ è sia compatto che discreto, $f\inv(y)$ è finito. + \end{proof} + + \begin{lemma}[della pila dei dischi] Siano $M$ e $N$ varietà della \underline{stessa dimensione}. + Sia $M$ compatta. Sia $f : M \to N$ una mappa liscia con $y \in N$ valore regolare. + Allora esiste un intorno $V$ di $y$ tale per cui: + \[ + \abs{f\inv(y')} = \abs{f\inv(y)}, \quad \forall y' \in V. + \] + \end{lemma} + + \begin{proof} + Sappiamo dalla Proposizione \ref{prop:controimmagine_regolare_finita} che + $f\inv(y)$ è finito. Se $f\inv(y) = \{x_1, \ldots, x_n\}$, riprendendo gli intorni aperti $A_i$ + come definiti nella dimostrazione della Proposizione \ref{prop:controimmagine_regolare_finita}, + allora possiamo definire: + \[ + V \defeq \bigcap_{i = 1}^n f(A_i) \setminus f(M \setminus \bigcup_{i = 1}^n A_i). + \] + Gli $f(A_i)$ sono aperti dal momento che $f$ è un diffeomorfismo. L'insieme $M \setminus \bigcup_{i = 1}^n A_i$ + è un chiuso in un compatto, e quindi è compatto; allora $f(M \setminus \bigcup_{i = 1}^n A_i)$ è compatto, + e dunque chiuso essendo $N$ uno spazio T2. Quindi $f(M \setminus \bigcup_{i = 1}^n A_i)^c$ è aperto. + Si conclude dunque che $V$ è aperto. \smallskip + + Su $V$, $\abs{f\inv(\cdot)}$ è necessariamente costante: la prima intersezione assicura che esistano + almeno $\abs{f\inv(y)}$ controimmagini, e la sottrazione insiemistica assicura che non possano esisterne di più. + \end{proof} + + \subsection{Misura nulla e teoremi di Sard e Brown} + + \begin{definition}[Sottinsiemi di varietà di misura nulla] + Sia $A$ un sottinsieme di una varietà $M$ di dimensione $m$. + Si dice che + $A$ ha \textbf{misura nulla (rispetto a $M$)} se per ogni + carta locale $(f, W \cap M)$, $f(A \cap W)$ ha misura nulla + in $\RR^m$. + \end{definition} + + \begin{theorem}[di Sard, per le varietà] Sia $f : M \to N$ una + mappa liscia tra due varietà. Allora l'insieme dei valori + critici $f(\crit(f))$ ha misura nulla in $N$. + \end{theorem} + + \begin{corollary}[di Brown] Sia $f : M \to N$ una + mappa liscia tra due varietà. Allora l'insieme dei valori + regolari di $f$ è denso in $N$. + \end{corollary} \end{multicols*}