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diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex index 750a31d..862be60 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex @@ -108,6 +108,9 @@ \newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} +\let\oldemptyset\emptyset +\let\emptyset\varnothing + \newcommand{\dA}{\dif{A}} \newcommand{\dx}{\dif{x}} \newcommand{\dy}{\dif{y}} 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 3010eab..0759d94 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 @@ -34,7 +34,7 @@ con le usuali definizioni date su aperti di $\RR^n$. \end{remark} - \begin{proposition} + \begin{proposition} \label{prop:comp_liscia} La composizione di mappe lisce è liscia. La restrizione di una mappa liscia è liscia. \end{proposition} @@ -86,35 +86,182 @@ regolari. \end{remark} - \section{Spazio tangente} + \section{Spazio tangente e differenziale su mappe tra varietà} - \subsection{Differenziale e spazio tangente} + \subsection{Differenziale su aperti di \texorpdfstring{$\RR^n$}{ℝⁿ}} 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 + Allora il \textbf{differenziale $\dif f_x : \RR^k \to \RR^\ell$} nel punto $x \in U$ è la funzione tale per cui: \[ - df_x(h) \defeq \lim_{t \to 0} \frac{f(x+th) - f(x)}{t}. + \boxed{\dif f_x(h) \defeq \lim_{t \to 0} \frac{f(x+th) - f(x)}{t}.} \] - Equivalentemente $df_x$ è l'unica funzione lineare tale per cui: + Equivalentemente $\dif f_x$ è l'unica funzione lineare tale per cui: \[ - f(x+h) = f(x) + df_x(h) + o(\norm{h}). + f(x+h) = f(x) + \dif f_x(h) + o(\norm{h}). \] \end{definition} \begin{remark} - Il differenziale $df_x$ rispetta alcune proprietà fondamentali: + Il differenziale $\dif f_x$ 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 La matrice di $\dif f_x$ 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 $x \in U$. + \[ \dif(f \circ g)_x = \dif f_{g(x)} \circ \dif g_x. \] + \item Data $\id_U$ su un aperto $U \subseteq \RR^k$, allora $\dif (\id_U)_x = \id_{\RR^k}$ 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 $x \in U$. + $\dif \iota_x = \id_{\RR^k}$ per ogni $x \in U'$. + \item Se $L : U \to \RR^\ell$ è lineare, allora $\dif L_x = L$ per ogni $x \in U$. + \end{enumerate} + \end{remark} + + \begin{proposition} \label{prop:diffeomorfismo_iso_diff} + Siano $U \subseteq \RR^k$, $V \subseteq \RR^\ell$ aperti. Sia $f : U \to V$ un diffeomorfismo. + Allora $k = \ell$ e $\dif f_x$ è un isomorfismo di $\RR^k$ per ogni $x \in U$. + \end{proposition} + + \begin{proof} + Sia $g : V \to U$ l'inversa di $f$. Poiché $f$ è un diffeomorfismo, $g$ è liscia. Sia $x \in U$. Allora: + \begin{enumerate}[(i.)] + \item $\id_{\RR^k} = \dif (\id_U)_x = \dif (g \circ f)_x = \dif g_{f(x)} \circ \dif f_x$, + \item $\id_{\RR^\ell} = \dif (\id_V)_{f(x)} = \dif (f \circ g)_{f(x)} = \dif f_x \circ \dif g_{f(x)}$. \end{enumerate} + Da (i.) si deduce che $\dif f_x$ è iniettiva, mentre da (ii.) si deduce che è surgettiva. Dunque + $\dif f_x$ è un isomorfismo (e quindi vale anche $k = \ell$). + \end{proof} + + \subsection{Spazio tangente in un punto di una varietà} + + \begin{remark}[Lo spazio tangente è ben definito] + Sia $M$ una varietà di dimensione $m$. Siano $g : U \to W \cap M$ e + $h : U' \to W' \cap M$ due parametrizzazioni locali di $x \in M$ + con $g(u) = h(u') = x$. \smallskip + + Supponiamo senza perdita di generalità che $W' = W$ (è sufficiente restringere + le immagini). La funzione + $g \circ h\inv$ è un diffeomorfismo in quanto composizione di diffeomorfismi + (vd. Proposizione \ref{prop:comp_liscia}). Allora per la Proposizione \ref{prop:diffeomorfismo_iso_diff} + $\dif (h\inv \circ g)_u$ è un isomorfismo. \smallskip + + Osserviamo che: + \[ + \dif g_u = \dif (h \circ (h\inv \circ g))_u = \dif h_{u'} \circ \dif (h\inv \circ g)_u. + \] + Dal momento che $\dif (h\inv \circ g)_u$ è in particolare surgettiva, si ha: + \[ + \dif g_u(\RR^m) = \dif h_{u'}(\RR^m). + \] \end{remark} + + \begin{definition}[Spazio tangente] + Sia $x$ un punto di una varietà $M$ di dimensione $m$. Presa una + parametrizzazione locale $g : U \subseteq \RR^m \to W \cap M$ di + $x$ con $g(u) = x$, si definisce lo \textbf{spazio tangente di $M$ in $x$} come: + \[ + \boxed{T_x M \defeq \dif g_u(\RR^m).} + \] + \end{definition} + + \begin{proposition} + Sia $x$ un punto di una varietà $M$ di dimensione $m$. Allora: + \[ + \boxed{\dim T_x M = m.} + \] + \end{proposition} + + \begin{proof} + Si prenda una parametrizzazione locale + $g : U \subseteq \RR^m \to W \cap M$ di + $x$ con $g(u) = x$. È sufficiente dimostrare + che $\dif g_u$ è una mappa iniettiva. \smallskip + + La mappa $g$ è indotta dalla carta locale tramite + un'estensione $F : W \to \RR^m$ con: + \[ + \restr{F}{W \cap M} = g\inv. + \] + Osserviamo allora che: + \[ + \id_{\RR^m} = \dif (F \circ g)_u = \dif F_{x} \circ \dif g_u, + \] + da cui si deduce che $\dif g_u$ ammette un'inversa sinistra, ed è + dunque iniettiva. + \end{proof} + + \subsection{Differenziale per mappe lisce tra varietà} + + \begin{remark}[Il differenziale per mappe lisce è ben definito] + Siano $M \subseteq \RR^k$ una varietà di dimensione $m$, $N \subseteq \RR^\ell$ + un'altra varietà, e sia + $f : M \to N$ una mappa liscia tra varietà. \smallskip + + Sia $F : W \to \RR^\ell$ un'estensione di $f$ per un intorno aperto di + $x \in M$. Siano $g : U \subseteq \RR^m \to W \cap M$ una parametrizzazione + locale di $x$ e $h : V \to N$ una parametrizzazione locale di $f(x)$ con + $g(u) = x$ e $h(v) = f(x)$. + + \[\begin{tikzcd} + M && W && N \\ + \\ + U &&&& V + \arrow["\iota", from=1-1, to=1-3] + \arrow["F", from=1-3, to=1-5] + \arrow["g", from=3-1, to=1-1] + \arrow["{h\inv \circ F \circ g}"', dashed, from=3-1, to=3-5] + \arrow["h"', from=3-5, to=1-5] + \end{tikzcd}\] + + Dal diagramma commutativo si deduce che: + \[ + \dif F_x \circ \dif g_u = \dif h_v \circ \dif (h\inv \circ F \circ g)_u. + \] + Pertanto $\dif F_x(T_x M)$ \underline{non} dipende dalla scelta dell'estensione $F$ e vale: + \[ \dif F_x(T_x M) \subseteq T_{f(x)} N. \] + \end{remark} + + \begin{definition}[Differenziale su mappe tra varietà] + Sia $f : M \to N$ una mappa tra varietà. Se $F$ è un'estensione di $f$ + in $x$, si definisce il \textbf{differenziale di $f$ in $x$} + $\dif f_x : T_x M \to T_{f(x)} N$ come segue: + \[ + \boxed{\dif f_x \defeq \restr{\dif F_x}{T_x M}.} + \] + \end{definition} + + \begin{remark} + Le proprietà del differenziale su aperti di $\RR^n$ si trasferiscono + facilmente al differenziale su mappe tra varietà: + \begin{enumerate}[(i.)] + \item Il differenziale rispetta la \textit{regola della catena} (chain rule): + \[ \dif(f \circ g)_x = \dif f_{g(x)} \circ \dif g_x. \] + \item Data $\id_M$ per una varietà $M$, allora $\dif (\id_M)_x = \id_{T_x M}$ per ogni $x \in M$. + \item Dati $M'$ e $M$ sono varietà con $M' \subseteq M$, l'inclusione $\iota : M' \to M$ è liscia, + $\dif \iota_x : T_x M' \to T_x M$ è iniettiva e $T_x M'$ è un sottospazio di $T_x M$. + \item Se $f : M \to N$ è un diffeomorfismo, allora $\dif f_x$ è un isomorfismo per ogni $x \in M$. + \end{enumerate} + \end{remark} + + \subsection{Punti e valori regolari o critici} + + \begin{definition}[Punti regolari o critici] + Sia $f : M \to N$ una mappa liscia tra varietà, con + $\dim M = m$ e $\dim N = n$. \smallskip + + 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}. + \end{definition} + + \begin{definition}[Valori regolari o critici] + Sia $f : M \to N$ una mappa liscia tra varietà, con + $\dim M = m$ e $\dim N = n$. \smallskip + + Sia $y \in N$. Si dice che $y$ è un \textbf{valore critico} + se è immagine di almeno un punto critico, e altrimenti si dice che + è un \textbf{valore regolare} (in particolare lo è se + $f\inv(y) = \emptyset$). + \end{definition} \end{multicols*}