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zE@Itx{8~%F?TaPGls#SqCdV)QOr&MWSxR3AW#r=6W=|g`$1_}*%^NcZ)EOR@T6K5> zpCkBb@@pc3<6ac+qi4LY{`455`Ni#|e|;t6Gwq z>!5bD2RqMXpYOZyhXNxUHbl*e_%4jRV^u%QhRyF3z~+IMQIaBB1QPRAHcQ8Hh_91(B#Ok+lL&}|}s>wa1@McM{_4oW1ja9YEFp9x64u5b78~AR& z`FMF|Rr|x1RC{p!$oP(xy|sV8Qzsw(={uT9`*y&5B}LSs5d5LPN&1#IP!okERM=wYOk04XB9md+Kz3nIJ?)&7V2WjCA$rPylTc#;#AICJTY*RQ# z7Gea2PBGE+L_B HVF3RxGnSgS 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 af456b6..2d390cd 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 @@ -134,6 +134,6 @@ \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. + \item T2 -- spazio topologico per cui due punti distinti ammettono l'esistenza di una coppia di intorni disgiunti. \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 fab3655..9ec4cb1 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 @@ -82,6 +82,9 @@ \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. + + \item \textbf{Teorema di approssimazione di Weierstrass} -- Sia $K \subseteq \RR^n$ un compatto, e sia $f : K \to \RR$ una mappa + continua. Allora per ogni $\eps > 0$ esiste una funzione polinomiale $P : K \to \RR$ tale per cui $\norm{f-P}_\infty < \eps$. \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 446506d..4775423 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 @@ -485,7 +485,7 @@ \subsection{Varietà a partire da valori regolari} - \begin{theorem} + \begin{theorem} \label{thm:varietà_da_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, allora $f\inv(y)$ è una varietà di dimensione $m - n$ \textnormal{(codimensione $n$)}. @@ -547,9 +547,9 @@ tra $(T_x P)^\perp$ e $T_y N$ si conclude che è un isomorfismo. \end{proof} - \begin{proposition} \label{prop:sn_è_varietà} + \begin{corollary} \label{cor:sn_è_varietà} $S^n \subseteq \RR^{n+1}$ è una varietà di dimensione $n$ e $T_x S^n = x^\perp \subseteq \RR^{n+1}$. - \end{proposition} + \end{corollary} \begin{proof} Presa $f : \RR^{n+1} \to \RR$ tale per cui: @@ -568,9 +568,9 @@ anche $T_x S^n = \ker \dif f_x = x^\perp$. \end{proof} - \begin{proposition} + \begin{corollary} $O(n) \subseteq M(n)$ è una varietà di dimensione $\frac{n(n-1)}{2}$. - \end{proposition} + \end{corollary} \begin{proof} Presa $f : M(n) \cong \RR^{n^2} \to S(n) \cong \RR^{\frac{n(n+1)}{2}}$ tale per cui: @@ -751,11 +751,11 @@ \subsection{Varietà con bordo da valori regolari} - \begin{lemma} \label{lem:varietà_bordata_da_valore_regolare_su_R} + \begin{theorem} \label{lem:varietà_bordata_da_valore_regolare_su_R} Sia $f : M \to \RR$ una mappa liscia tra varietà, dove $M$ è una $m$-varietà senza bordo. Se - $0 \in N$ è un valore regolare, allora $\{ f \geq 0 \}$ è una + $0$ è un valore regolare per $f$, allora $\{ f \geq 0 \}$ è una $m$-varietà con bordo $f\inv(0)$. - \end{lemma} + \end{theorem} \begin{proof} L'insieme $\{f > 0\}$ è un aperto di $M$ dal momento che $f$ è continua, e quindi @@ -786,37 +786,84 @@ e $a$ valore regolare di $f$. \end{remark} - \begin{proposition} + \begin{corollary} $D^n$ è una varietà $n$-dimensionale con bordo $S^{n-1}$. - \end{proposition} + \end{corollary} \begin{proof} Sia $f : \RR^n \to \RR$ tale per cui: \[ f(x) \defeq x \cdot x = x_1^2 + \ldots + x_{n}^2. \] - Allora, come visto per la Proposizione \ref{prop:sn_è_varietà}, + Allora, come visto per il Corollario \ref{cor:sn_è_varietà}, $1$ è valore regolare di $f$, e quindi $D^n = \{ f \leq 1 \}$ è una $n$-varietà con bordo $S^{n-1}$. \end{proof} - \begin{lemma} - Sia $f : M \to N$ una mappa liscia tra varietà, dove $M$ è una varietà con bordo di dimensione $m$, - e $N$ ha dimensione $n$. Se $y \in N$ è un valore regolare sia per $f$ che per $\restr{f}{\partial M}$, - allora $f\inv(y)$ è una varietà $(m-n)$-dimensionale con bordo $f\inv(y) \cap \partial M$. + \begin{lemma} \label{lem:varietà_bordata_da_valore_regolare_caso_particolare} + Sia $f : H^m \to \RR^n$ con $m > n$ una mappa liscia. Se $y \in \RR^n$ è un valore regolare + sia per $f$ che per $\restr{f}{\partial H^m}$ con $f\inv(y) \neq \emptyset$, allora $f\inv(y)$ è una $(m-n)$-varietà con bordo + $\partial f\inv(y) = f\inv(y) \cap \partial H^m$. \end{lemma} \begin{proof} ... \end{proof} + \begin{theorem} \label{thm:varietà_bordata_da_valore_regolare} + Sia $f : M \to N$ una mappa liscia tra varietà, dove $M$ è una $m$-varietà, + $N$ è una $n$-varietà senza bordo e $m > n$. \smallskip + + Se $y \in N$ è un valore regolare sia per $f$ che per $\restr{f}{\partial M}$ con $f\inv(y) \neq \emptyset$, + allora $f\inv(y)$ è una $(m-n)$-varietà con bordo $\partial f\inv(y) = f\inv(y) \cap \partial M$. + \end{theorem} + + \begin{proof} + Sia $g : U \subseteq H^m \to g(U) \subseteq M$ una parametrizzazione locale di $x \in f\inv(y)$ + con $g(u) = x$. Sia $h : V \subseteq \RR^n \to h(V) \subseteq N$ una parametrizzazione + locale di $y$ con $g(v) = y$. \smallskip + + \[\begin{tikzcd} + M && N \\ + \\ + U && V + \arrow["f"{description}, from=1-1, to=1-3] + \arrow["g", from=3-1, to=1-1] + \arrow["{h\inv \circ f \circ g}"{description}, from=3-1, to=3-3] + \arrow["h", from=3-3, to=1-3] + \end{tikzcd}\] + + A meno di restringere i domini delle due mappe, possiamo considerare $p = h\inv \circ f \circ g$. + Allora $v = h\inv(y)$ è regolare per la mappa $p$. Per possiamo + restringerci a una palla aperta come intorno aperto di $u$ in $p\inv(v)$: se questa è diffeomorfa a $\RR^m$, una carta locale per $u$ + in $p\inv(v)$ è già data dal Teorema \ref{thm:varietà_da_valore_regolare}; se invece è diffeomorfa a + $H^m$, la carta locale è ereditata tramite il Lemma \ref{lem:varietà_bordata_da_valore_regolare_caso_particolare}. \smallskip + + La carta locale trovata si trasferisce tramite $g$ al punto $x$ di $M$, e applicando il Corollario \ref{cor:bordo_param_locale} + il bordo della varietà si trasferisce coerentemente a sua volta. + \end{proof} + \subsection{Mappe dalla varietà al bordo} - \begin{theorem} \label{thm:classificazione_dim_1} - Una varietà compatta di dimensione uno con bordo è necessariamente un'unione - di copie di $S^1$ e di intervalli chiusi di $\RR$. + \begin{theorem} \label{thm:classificazione_dim_1_generale} + Una $1$-varietà con bordo è diffeomorfa a unioni di copie di $S^1$ e + di intervalli di $\RR$. \end{theorem} + \begin{proof} + Si veda l'appendice di [Milnor, J. (1965). \textit{Topology from the Differentiable Viewpoint}. University Press of Virginia]. + \end{proof} + + \begin{corollary} \label{thm:classificazione_dim_1} + Una $1$-varietà compatta con bordo è necessariamente un'unione + di copie di $S^1$ e di intervalli chiusi di $\RR$. + \end{corollary} + + \begin{proof} + Discende immediatamente dal Teorema \ref{thm:classificazione_dim_1_generale} + utilizzando l'ipotesi di compattezza. + \end{proof} + \begin{lemma} Sia $M$ una varietà compatta con bordo $\partial M \neq \emptyset$. Allora \underline{non} esistono mappe lisce $f$ da $M$ in $\partial M$ che fissano il bordo (i.e., con $\restr{f}{\partial M} = \id_{\partial M}$).