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zQEPRhKE7kUHm#ZM*0EY6V$-9ja=Lx#l6hM|RjjGx*Cemw_URs6Jz!1fG2Q;l^bl0u z<2%b=nbGpyDx)f{tg!PskC0QNQ~2EIniaIM@$3XY9$la2xFYx6$M}*a28$bjkaF8>5WmdDmnlN$QVwYLZc& zf(+9Y-v}z?>JVNM83-P)aU<0v-Bm%P)k=z#y-wfcjs`z%07} 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 e7b8477..7c7735a 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 @@ -14,6 +14,8 @@ \begin{itemize} \item $\rk$ -- rango di un'applicazione lineare o di una matrice. + \item $M(n)$ -- matrici $n \times n$ a elementi reali. + \item $S(n)$ -- matrici simmetriche $n \times n$ a elementi reali. \end{itemize} \section*{Analisi matematica} @@ -117,6 +119,10 @@ \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. + \item $S^n$ -- sfera $n$-dimensionale reale, ovverosia $\{\vec{x} \in \RR^{n+1} \mid \norm{\vec{x}} = 1\}$. + \item $D^n$ -- disco $n$-dimensionale reale, ovverosia $\{\vec{x} \in \RR^{n} \mid \norm{\vec{x}} \leq 1\}$. + \item $H^n$ -- semispazio $\{ x \in \RR^n \mid x_m \geq 0 \}$. + \item $\partial H^n$ -- bordo del semispazio $\{ x \in \RR^n \mid x_m \geq 0 \}$, ovverosia $\{ x \in \RR^n \mid x_m = 0 \} \cong \RR^{n-1}$. \end{itemize} \section*{Topologia} 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 0dceb3d..85a9901 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 @@ -62,10 +62,10 @@ \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 + Un insieme $M \subseteq \RR^k$ si dice \textbf{varietà (differenziabile liscia) di dimensione $m>0$} (o $m$-varietà) 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 + che ogni $W_x \cap M$ sia un singoletto. \smallskip 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 $M$. @@ -90,7 +90,7 @@ \subsection{Prodotto di varietà} - \begin{proposition}[Prodotto di varietà] + \begin{proposition}[Prodotto di varietà] \label{prop:prodotto_varietà} Siano $M \subseteq \RR^k$ e $N \subseteq \RR^\ell$ varietà di dimensione $m$ e $n$. Allora $M \times N \subseteq \RR^{k+\ell}$ è una varietà di dimensione $m + n$. \smallskip @@ -272,6 +272,14 @@ \end{enumerate} \end{remark} + \begin{proposition}[Differenziale per prodotti di varietà] \label{prop:diff_prodotto} + Sia $f : M \to N \times O$ una mappa liscia, dove $M$, $N$ e $O$ sono varietà. + Se $f(x) = (g(x), p(x))$, allora $g$ e $p$ sono lisce e vale: + \[ + \boxed{df_x(h) = (dg_x(h), dp_x(h)).} + \] + \end{proposition} + \section{Valori regolari e critici} \subsection{Prime definizioni} @@ -378,11 +386,11 @@ come definiti nella dimostrazione della Proposizione \ref{prop:controimmagine_regolare_finita}, allora possiamo definire: \[ - V \defeq \bigcap_{i = 1}^n f(A_i) \setminus f(M \setminus \bigcup_{i = 1}^n A_i). + V \defeq \bigcap_{i = 1}^n f(A_i) \setminus f\left(M \setminus \bigcup_{i = 1}^n A_i\right). \] 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. + è un chiuso in un compatto, e quindi è compatto; allora $f\left(M \setminus \bigcup_{i = 1}^n A_i\right)$ è compatto, + e dunque chiuso essendo $N$ uno spazio T2. Quindi $f\left(M \setminus \bigcup_{i = 1}^n A_i\right)^c$ 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 @@ -447,4 +455,135 @@ cui volume è non nullo. Allora $f(\crit(f))$ non avrebbe misura nulla, che è assurdo per il Teorema \ref{thm:sard}. \end{proof} + + \subsection{Varietà a partire da valori regolari} + + \begin{theorem} + Sia $f : M \to N$ una mappa liscia tra varietà con $\dim M = m \geq n = \dim N$. Se + $y \in N$ è regolare, allora $f\inv(y)$ è una varietà di dimensione $m - n$ + \textnormal{(codimensione $n$)}. + \end{theorem} + + \begin{proof} + Sia $x \in f\inv(y)$. Allora $x$ è per ipotesi un punto regolare, e quindi + $\dif f_x : T_x M \to T_y N$ è una mappa surgettiva, con: + \[ + \dim \ker(\dif f_x) = m - n. + \] + Sia $L : \RR^k \to \RR^{m-n}$ una mappa lineare, dove $T_x M \subseteq \RR^k$ + e $\restr{L}{T_x M}$ è isomorfismo. \smallskip + + Consideriamo la mappa $F : M \to N \times \RR^{m-n}$ tale per cui: + \[ + F(m) = (f(m), L(m)). + \] + Allora, per la Proposizione \ref{prop:diff_prodotto}, vale: + \[ + dF_x(v) = (df_x(v), dL_x(v)) = (df_x(v), L(v)), + \] + dove si è usato che $L$ è una mappa lineare. $dF_x(v)$ si annulla solo + per $v = 0$, essendo $\restr{L}{T_x M}$ un isomorfismo; quindi + $dF_x(v)$ è invertibile, e $x$ è regolare per $F$. \smallskip + + Osserviamo che $F$ è una mappa tra varietà della stessa dimensione (vd. Proposizione + \ref{prop:prodotto_varietà}), e quindi, per la Proposizione + \ref{thm:invertibilità_locale_varietà}, esiste un intorno $U$ di $x$ in $M$ tale per cui + $\restr{F}{U} : U \to V \defeq F(U)$ è un diffeomorfismo. \smallskip + + La restrizione $\restr{F}{U}$ mappa $U \cap f\inv(y)$ su un aperto di $V \cap (\{y\} \times \RR^{m-n})$, + che è diffeomorfo a un aperto di $\RR^{m-n}$ ($\{y\} \times \RR^{m-n} \cong \RR^{m-n}$). In particolare + induce una carta locale per $x$, e quindi $f\inv(y)$ è una varietà di dimensione $m-n$. + \end{proof} + + \begin{proposition} \label{prop:tangente_valore_regolare} + Sia $f : M \to N$ una mappa liscia tra varietà con $\dim M = m \geq n = \dim N$. Se + $y \in N$ è regolare, posto $P \defeq f\inv(y)$, si ha: + \[ + \boxed{T_x P = \ker \dif f_x, \quad \forall x \in P.} + \] + Inoltre $\restr{\dif f_x}{(T_x P)^\perp}$ è un isomorfismo. + \end{proposition} + + \begin{proof} + Poiché $P = f\inv(y)$, l'inclusione $\iota : P \to M$ è tale + per cui $f \circ \iota$ è costante. Tuttavia una mappa costante + ha differenziale nullo, e dunque: + \[ + \dif f_x \circ \iota_{T_x P} = 0 \implies T_x P \subseteq \ker(\dif f_x). + \] + Poiché $\dim T_x P = m - n = \ker(\dif f_x)$, si ottiene + l'uguaglianza $T_x P = \ker(\dif f_x)$. \smallskip + + Osserviamo che $\dim (T_x P)^\perp = n$ e che $(T_x P)^\perp \cap T_x P = \{0\}$. + Allora $\restr{\dif f_x}{(T_x P)^\perp}$ è iniettiva, e per uguaglianza dimensionale + tra $(T_x P)^\perp$ e $T_y N$ si conclude che è un isomorfismo. + \end{proof} + + \begin{proposition} + $S^n \subseteq \RR^{n+1}$ è una varietà di dimensione $n$ e $T_x S^n = x^\perp \subseteq \RR^{n+1}$. + \end{proposition} + + \begin{proof} + Presa $f : \RR^{n+1} \to \RR$ tale per cui: + \[ + f(x) \defeq x_1^2 + \ldots + x_{n+1}^2, + \] + si ha $S^n = f\inv(1)$. Osserviamo che: + \[ + Jf(x) = 2x. + \] + Dunque l'unico valore critico di $f$ è $0$. Pertanto + $S^n = f\inv(1)$ è una varietà di dimensione $(n+1)-1 = n$. \smallskip + + Poiché $\dif f_x(h) = Jf(x) \cdot h = 2x^\top h$, per la + Proposizione \ref{prop:tangente_valore_regolare} vale + anche $T_x S^n = \ker \dif f_x = x^\perp$. + \end{proof} + + \begin{proposition} + $O(n) \subseteq M(n)$ è una varietà di dimensione $\frac{n(n-1)}{2}$. + \end{proposition} + + \begin{proof} + Presa $f : M(n) \cong \RR^{n^2} \to S(n) \cong \RR^{\frac{n(n+1)}{2}}$ tale per cui: + \[ f(A) = AA^\top, \] + si ha $O(n) = f\inv(I)$. Osserviamo che: + \[ df_A : M(n) \to S(n), \quad df_A(B) = AB^\top + B A^\top. \] + Mostriamo che $df_A$ è surgettiva per $A \in f\inv(I) = O(n)$. + Se $AB^\top + BA^\top = C \in S(n)$, $C$ si può spezzare + nella sua parte simmetrica: + \[ + C = \frac{1}{2} C + \frac{1}{2} C^\top, + \] + e quindi, ponendo $\frac{1}{2} C = BA^\top$, si ottiene come + soluzione: + \[ + B = \frac{1}{2} CA. + \] + Dunque $I$ è un valore regolare, e $O(n)$ è una varietà + di dimensione $n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}$. + \end{proof} + + \section{Varietà con bordo} + + \subsection{Semispazio superiore e prime definizioni} + + \begin{definition}[Semispazio superiore] + Si definisce il \textbf{semispazio superiore} $H^n$ in $\RR^n$ come: + \[ + \boxed{H^n \defeq \{x \in \RR^n \mid x_n \geq 0\}.} + \] + \end{definition} + + \begin{definition}[$m$-varietà con bordo] + Si dice che $M \subseteq \RR^k$ è una \textbf{$m$-varietà con bordo} se + ogni punto di $M$ ammette un intorno diffeomorfo ad un aperto del semispazio + superiore $H^n$. Gli intorni e i diffeomorfismi citati formano le + \textbf{carte locali} della varietà, e le inverse di tali diffeomorfismi + sono dette \textbf{parametrizzazioni locali}. \smallskip + + Si dice \textbf{bordo} della varietà $M$ l'insieme dei punti che è immagine + di un punto di $\partial H^n$ tramite qualche parametrizzazione locale, e + si indica con $\partial M$. + \end{definition} \end{multicols*}