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zOSvh}A4fZ_eR#|k?a>+>G4%OGIkMZV+4>v4*)HxkA1~|QqOy1Rq_B+g3>!8-uy-J) zJR^_t=US`vU7io-ATS}e5ROs}TuSscJUO)a=v6i8In>WJ7(2B&>%kB{;CSQNEM7H{ lsSJ8-V?Ui5uD&9s=85m5s!?LU!v^wm^J6hE$g0U<{V(h@Lizvz diff --git a/Corsi/Geometria e topologia differenziale/Scheda/preamble.tex b/Corsi/Geometria e topologia differenziale/Scheda/preamble.tex index f03d744..029d95e 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda/preamble.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda/preamble.tex @@ -83,7 +83,8 @@ \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} -\newcommand{\CC}{\mathcal{C}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\cc}{\mathcal{C}} \newcommand{\TT}{\mathbb{T}} \newcommand{\I}{\mathrm{I}} diff --git a/Corsi/Geometria e topologia differenziale/Scheda/sections/1-curve.tex b/Corsi/Geometria e topologia differenziale/Scheda/sections/1-curve.tex index 7a790c4..6f96536 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda/sections/1-curve.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda/sections/1-curve.tex @@ -531,7 +531,7 @@ \begin{definition}[Cerchio osculatore] Sia $\alpha$ una curva di Frenet. Si definisce il - \textbf{cerchio osculatore} $\CC_\alpha(t)$ al tempo $t$ di $\alpha$ come + \textbf{cerchio osculatore} $\cc_\alpha(t)$ al tempo $t$ di $\alpha$ come il cerchio di raggio $R_\alpha(t)$ e centro $\alpha(t) + R_\alpha(t) N_\alpha(t)$ contenuto nel piano osculatore $\Pi_\alpha(t)$. \end{definition} @@ -541,23 +541,23 @@ \[ f_{P, R}(t) \defeq \norm{\alpha(t) - P}^2 - R^2. \] - Consideriamo i cerchi di raggio $P$ e $R$ nel piano $\Pi_\alpha(t_0)$, denotati con $\CC(P, R)$. + Consideriamo i cerchi di raggio $P$ e $R$ nel piano $\Pi_\alpha(t_0)$, denotati con $\cc(P, R)$. Si pongano le seguenti condizioni: \begin{itemize} - \item $f_{P, R}(t_0) = 0$, ovverosia il cerchio $\CC(P, R)$ passa per $\alpha(t_0)$; + \item $f_{P, R}(t_0) = 0$, ovverosia il cerchio $\cc(P, R)$ passa per $\alpha(t_0)$; \item $f_{P, R}'(t_0) = f_{P, R}''(t_0) = 0$, ovverosia il - cerchio $\CC(P, R)$ approssima $\alpha$ in $t_0$ fino al secondo ordine. + cerchio $\cc(P, R)$ approssima $\alpha$ in $t_0$ fino al secondo ordine. \end{itemize} - Allora l'unico cerchio $\CC(P, R)$ + Allora l'unico cerchio $\cc(P, R)$ soddisfacente le sopracitate condizioni - è il cerchio osculatore $\CC_\alpha(t_0)$ al tempo $t_0$ di $\alpha$. + è il cerchio osculatore $\cc_\alpha(t_0)$ al tempo $t_0$ di $\alpha$. \end{proposition} \begin{proof} Osserviamo che: \[ f_{P, R}'(t) = 2 \alpha'(t) \cdot (\alpha(t) - P), \] e quindi $f_{P, R}'(t_0) = 0$ implica $T_{\alpha}(t_0) \perp \alpha(t_0) - P$. - Dal momento che il cerchio $\CC(P, R)$ deve essere contenuto nel piano osculatore + Dal momento che il cerchio $\cc(P, R)$ deve essere contenuto nel piano osculatore di $\alpha(t_0)$, allora $\alpha(t_0) - P \parallel N_\alpha(t_0)$. \medskip Inoltre: diff --git a/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex b/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex index 7732d36..dc299eb 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex @@ -1961,4 +1961,79 @@ aperto anche un valore regolare di $g$. Per il Lemma \ref{lem:omotopia_intero}, dunque $f$ e $g$ condividono lo stesso grado intero. \end{proof} + + \begin{corollary} + Sia $M$ una varietà chiusa e connessa. Se $f : M \to M$ + è un diffeomorfismo di grado $\deg(f) = -1$, allora + $f$ non è omotopa all'identità $\id_M$, né a una mappa costante $c_x$ per + $x \in M$. + \end{corollary} + + \begin{proof} + La tesi è un'immediata conseguenza del Teorema \ref{thm:fondamentale_grado_intero}, + dal momento che $\deg(\id_M) = 1$ e $\deg(c_x) = 0$ (infatti $c_x$ non è surgettiva). + \end{proof} + + \begin{proposition}[Il grado è moltiplicativo] + Sia $M$ una varietà orientata, chiusa e connessa. Se $f$ e $g$ sono due + mappe lisce da $M$ in sé stessa, allora: + \[ + \boxed{\deg(f \circ g) = \deg(f) \deg(g).} + \] + \end{proposition} + + \begin{proof} + ... + \end{proof} + + \subsection{Applicazioni immediate della teoria del grado intero} + + \begin{lemma} \label{lem:grado_zk} + Sia $f_k : S^1 \to S^1$ tale per cui: + \[ + f_k(z) = z^k \in \CC, + \] + dove si è identificato $S^1 \subseteq \RR^2$ in $\CC$. + Allora $1$ è un valore regolare di $f_k$ e $\deg(f_k) = k$. \smallskip + + Quindi, per $k \neq 0$, $f_k$ \underline{non} può estendersi a una mappa + liscia da $D^2$ a $S^1$. + \end{lemma} + + \begin{proof} + Consideriamo l'elemento $1 \in S^1$. Osserviamo che + $f_k\inv(1)$ è l'insieme delle radici $k$-esime dell'unità, + e quindi contiene esattamente $k$ elementi (a meno del segno). \smallskip + + La funzione $f_k$ si estende a $F_k(z) = z^k$ su tutto $\CC$. Osserviamo + che $T_1 S^1 = \Span(i)$. Per determinare allora il segno di $\dif (f_k)_1$ è + sufficiente considerare la seguente derivata: + \[ + \dif (f_k)_1 (i) = \dertime{e^{i t k}}{t=0} = k i. + \] + Dunque $\dif (f_k)_1$ è un isomorfismo per $k \neq 0$, e preserva l'orientazione + se $k > 0$, mentre non la preserva se $k < 0$. \smallskip + + Sia ora $\xi = e^{i \theta_0} \in f_k\inv(1)$. Consideriamo il diffeomorfismo $h_\xi : S^1 \to S^1$ tale + per cui: + \[ + h_\xi(z) = \xi \cdot z, + \] + ovverosia la rotazione indotta da $\xi$. Si verifica facilmente che: + \[ + H : S^1 \times [0, 1] \to S^1, \quad H(z, t) = e^{i \theta_0 t} z + \] + è un'omotopia liscia da $\id_{S^1}$ a $h_\xi$. Dunque $\deg(h_\xi; 1) = \deg(\id_{S^1}; 1) = 1$. \smallskip + + Osserviamo che $f_k = f_k \circ h_\xi$. Quindi: + \[ + \dif (f_k)_1 = \dif (f_k \circ h_\xi)_1 = \dif (f_k)_\xi \circ \dif (h_\xi)_1 = \dif (f_k)_\xi \circ h_\xi. + \] + Dal momento che $h_\xi$ è invertibile, si deduce che $\dif (f_k)_\xi$ è sempre un isomorfismo, e + dunque che $1$ è un valore regolare. Inoltre, dalla stessa uguaglianza si deduce per la + moltiplicatività del segno dei differenziali che $\sgn(\dif (f_k)_1) = \sgn(\dif (f_k)_\xi)$. \smallskip + + Si conclude facilmente allora che $\deg(f) = \deg(f; 1) = k$. L'ultima affermazione è conseguenza + del Lemma \ref{lem:grado_mappa_estendibile}. + \end{proof} \end{multicols*}