From 816571de50720d0dfeab6b91e426e49857672fb1 Mon Sep 17 00:00:00 2001 From: hearot Date: Mon, 29 Dec 2025 12:45:44 +0100 Subject: [PATCH] =?UTF-8?q?gtd(scheda):=20migliora=20l'esposizione=20del?= =?UTF-8?q?=20prodotto=20di=20variet=C3=A0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../Scheda riassuntiva/README.md | 3 +- .../Scheda riassuntiva/main.pdf | Bin 645733 -> 645893 bytes .../Scheda riassuntiva/main.tex | 2 +- .../sections/4-varieta-e-teoria-del-grado.tex | 74 ++++++++++++------ 4 files changed, 50 insertions(+), 29 deletions(-) diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/README.md b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/README.md index 0b2bc19..31021d8 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/README.md +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/README.md @@ -3,7 +3,6 @@ - [Programma del corso 📘](https://unipi.coursecatalogue.cineca.it/insegnamenti/2025/50256_686307_66441/2008/50256/10299?coorte=2023&schemaid=8700) - [Registro del corso 📑](https://unimap.unipi.it/registri/dettregistriNEW.php?re=11083221::::&ri=8527) -Questa cartella contiene una *Scheda riassuntiva*, che, come suggerisce il nome, vorrebbe essere un recap completo di tutta la -teoria del corso. +Questa cartella contiene un recap completo di tutta la teoria del corso di *Geometria e topologia differenziale*. Il progetto si basa su un layout di [Luca Lombardo](https://lukefleed.xyz/), utilizzato in particolare nelle [Schede riassuntive di Geometria 2](https://github.com/lukefleed/G2-cheat-sheet), basate sulle [dispense-capolavoro](https://www.overleaf.com/read/vsdktbwrgpth) di [Francesco Sorce](mailto:f.sorce@studenti.unipi.it). \ No newline at end of file diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.pdf b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/main.pdf index 5a09a82038d5fd147249e4dabb2536c9338385df..648a4352bd534a4c58d1fa0371436be79d7217e4 100644 GIT binary patch delta 91659 zcmZshQ+S|Vux4Z1NyoNr+jhsc{l&IzTOD?6yJOqzIO+Lkp1C{cZr9DOtJ?di^{(Sq zs<1Yy#sg?jmW-f6SQ;QsZNg!b8Kvig&Wq0dE)+kR)V&6r9V!@qWAvJ0LL@0OFp!~K zSGD!$ZV`2fa}FamBG09-BlQ4bmeFR~YlOgV-7I6PQRM(H-TF;X(Q;v&nItcDnQ?de z_ML$-!Aldedy^)Zx1RLP%iGp)Kx7ObhG)0j1@%u4eJz;5CJsPzFLw&Dy_Y$7EQH8a zsq58l^p!p0nI&t?v=y;kly;Y@;m6Lsnx*zf*9M5PXr%I`A 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zXe#@=+U~ofd0YQfTa7pSxTkI5n{(>aJ7<`AuoR+ayjr#gjp;b#lVW+Edl&*Lc?5Zp za#vu`JACpAOP7rcS!CliPixzTFbYpW-bXd$TWTTR%eTl$JwKu401SHha2)k4c)kTE zo~CgrcwZf6Of7Pr-cIg+VTo+@?25d;#a&R@1l{NSbp`l$ZBD8~ze93X__skiNaRcWYarJ@{@B*0KYzlE22`{t zvW;5AD%)ugaM+h~agH3-Lc9}%DsZQec-@7ZayyXsQ92MD$%MEH9kuifX7^nuq^`|F z3Blu}#g3{b-rAL7jPqAXIjLIX8?}xzM&7?F4+w-_4Ckn+j2~25qE5YX!K)Q2E)3-2 kwGJ$DrM|^KBYsA=?5CK?ge_yzU0$} (o $m$-varietà) se per ogni + \begin{definition}[Varietà differenziabile liscia senza bordo] + Un insieme $M \subseteq \RR^k$ si dice \textbf{varietà (differenziabile liscia senza bordo) 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. \smallskip @@ -79,7 +84,7 @@ Le carte locali inducono un ricoprimento aperto di $M$, e quindi, qualora $M$ fosse compatta, si potrebbe sempre prendere un atlante finito. \smallskip - Inoltre, poiché $\RR^k$ è II-numerabile, si può sempre prendere un atlante numerabile. + Inoltre, poiché $\RR^k$ è II-numerabile, si può sempre prendere un \underline{atlante numerabile}. \end{remark} \begin{remark} @@ -95,7 +100,8 @@ $m$ e $n$. Allora $M \times N \subseteq \RR^{k+\ell}$ è una varietà di dimensione $m + n$. \smallskip - Un atlante per $M \times N$ è $\{(f_i \times g_j, (W_i \times Q_j) \cap (M \times N))\}_{i, j}$, + Un atlante per $M \times N$ è dato da: + \[ \{(f_i \times g_j, (W_i \times Q_j) \cap (M \times N))\}_{i, j}, \] dove $\{(f_i, W_i \cap M)\}_i$ è un atlante di $M$ e $\{(g_i, Q_j \cap N)\}_j$ è un atlante di $N$. \end{proposition} @@ -210,14 +216,20 @@ dunque iniettiva. \end{proof} - \begin{proposition}[Spazio tangente in un prodotto di varietà] - Siano $M$ e $N$ due varietà di dimensione $m$ e $n$. Allora vale: + \begin{remark}[Spazio tangente in un prodotto di varietà] \label{rmk:tangente_prodotto} + Siano $M$ e $N$ due varietà di dimensione $m$ e $n$. + Se $f_i \times g_j$ è una carta locale di $M \times N$, + come ottenuto nella Proposizione \ref{prop:prodotto_varietà}, allora: + \begin{align*} + T_{(m, n)}(M \times N) & = d(f_i^{-1} \times g_j^{-1})_{(m, n)}(\mathbb{R}^{m+n}) \\ + & \cong d(f_i^{-1})(\mathbb{R}^m) \times d(g_j^{-1})(\mathbb{R}^n) \\ + & = T_m M \times T_n N + \end{align*} + Quindi vale il seguente isomorfismo canonico, ottenuto proiettando sulle componenti: \[ - \boxed{T_{(m, n)} (M \times N) \cong \dif \iota^M_{m}(T_m M) \oplus \dif \iota^N_{n}(T_{n} N),} + \boxed{T_{(m, n)} (M \times N) \cong T_m M \times T_n N.} \] - dove $\iota^M$ è l'immersione di $M$ in $M \times N$ che fissa $n$ nella seconda componente, - e $\iota^N$ è l'immersione di $N$ che fissa $m$ nella prima. - \end{proposition} + \end{remark} \subsection{Differenziale per mappe lisce tra varietà} @@ -466,7 +478,8 @@ \end{theorem} \begin{proof} - Sia $x \in f\inv(y)$. Allora $x$ è per ipotesi un punto regolare, e quindi + Sia $k$ tale per cui $M \subseteq \RR^k$, e 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. @@ -474,7 +487,7 @@ 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: + Consideriamo la mappa $F : M \subseteq \RR^k \to N \times \RR^{m-n}$ tale per cui: \[ F(m) = (f(m), L(m)). \] @@ -510,7 +523,7 @@ per cui $f \circ \iota$ è costante. Tuttavia una mappa costante ha differenziale nullo, e dunque: \[ - \dif f_x \circ \iota_{T_x P} = 0 \implies T_x P \subseteq \ker(\dif f_x). + \dif f_x \circ \underbrace{\iota_{T_x P}}_{= \dif \iota^P_x} = 0 \implies T_x P \subseteq \ker(\dif f_x). \] Poiché $\dim T_x P = m - n = \ker(\dif f_x)$, si ottiene l'uguaglianza $T_x P = \ker(\dif f_x)$. \smallskip @@ -527,7 +540,7 @@ \begin{proof} Presa $f : \RR^{n+1} \to \RR$ tale per cui: \[ - f(x) \defeq x_1^2 + \ldots + x_{n+1}^2, + f(x) \defeq x \cdot x = x_1^2 + \ldots + x_{n+1}^2, \] si ha $S^n = f\inv(1)$. Osserviamo che: \[ @@ -549,17 +562,18 @@ 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: + \[ \dif f_A : M(n) \to S(n), \quad \dif f_A(B) = AB^\top + B A^\top. \] + Mostriamo che $\dif f_A$ è surgettiva per $A \in f\inv(I) = O(n)$. \smallskip + + Sia $C \in S(n)$ simmetrica. Allora $C$ è uguale + alla 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: + e quindi, ponendo $\frac{1}{2} C = AB^\top$, si ottiene la seguente + soluzione a $AB^\top + BA^\top = C$: \[ - B = \frac{1}{2} CA. + B = \frac{1}{2} C^TA. \] Dunque $I$ è un valore regolare, e $O(n)$ è una varietà di dimensione $n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2}$. @@ -588,6 +602,14 @@ si indica con $\partial M$. \end{definition} + \begin{remark} + La definizione data è coerente con la definizione di varietà senza bordo: una + varietà senza bordo $M$ è esattamente una varietà con bordo $M$ tale per cui + $\partial M = \emptyset$. \smallskip + + Utilizzeremo dunque indistintamente le due caratterizzazioni. + \end{remark} + \subsection{Differenziale e spazio tangente su varietà con bordo} \begin{remark}[Il differenziale sul bordo di $H^n$ è ben definito] @@ -843,7 +865,7 @@ ... \end{proof} - \subsection{Grado modulo $2$ e buona definizione} + \subsection{Grado modulo \texorpdfstring{$2$}{2} e buona definizione} \begin{theorem} Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ compatta