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zN~cL*y3iKNU)1!5AVEhEE>=!icQeSIXDnfJJGAwHz@fuI^@N6^Yw|q;z-e<7-2r{Z z$q`xGRn5s+{|ELP-k{9H?{c@Kp(T4pC|WFIa+~7GQdPsAI{u^FV=F7y;h=gK-aWv{ zHOFRT^6tTh57Q^y94R^g9a;P9BOPCF-;TI|dxB29J0{<%e?Ju=K1~b>EAM%cyuq`DFeH-=>{Jn<zG2AL=>=L@b$xJYVsrZ}-TiOXQB(@N^#J!!OS#YN%|+CQZU{~yCvECx4$kX|W> zZg@+o62VY=brEBQMk d#3Q~N;ekRLj~pf)j)0Ju5H>TjlD0DT{{q{IxPt%y diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex index 7ebde4d..9f372c2 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/preamble.tex @@ -105,7 +105,7 @@ \newcommand{\der}[2]{\frac{\mathop{}\!\textnormal{\slshape d} #1}{\mathop{}\!\textnormal{\slshape d} #2}} \newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} -\newcommand{\dertime}[1]{\frac{\mathop{}\!\textnormal{\slshape d}}{\mathop{}\!\textnormal{\slshape d} t} \bigg|_{#1}} +\newcommand{\dertime}[2]{\left. \der{}{t} #1 \right|_{#2}} \newcommand*\dif{\mathop{}\!\textnormal{\slshape d}} diff --git a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex index a8438b8..3f3c03f 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda riassuntiva/sections/3-curve_su_superfici.tex @@ -64,7 +64,7 @@ Pertanto: \[ \begin{aligned} - (f \circ \alpha)'(0) & = \dertime{t=0} (f \circ x)(u(t), v(t)) \\[1em] + (f \circ \alpha)'(0) & = \dertime{(f \circ x)(u(t), v(t))}{t=0} \\[1em] & = (f \circ x)_u u'(0) + (f \circ x)_v v'(0). \end{aligned} \] @@ -616,7 +616,7 @@ Un piano e la sfera \underline{non} sono localmente isometrici. \end{corollary} - \section{Trasporto parallelo e geodetiche} + \section{Trasporto parallelo e campi vettoriali} \subsection{Campi vettoriali e derivata covariante} @@ -729,7 +729,9 @@ $\varphi < 0$ su tutto $[0, 1]$, da cui la tesi. \end{proof} - \subsection{Geodetiche} + \section{Geodetiche} + + \subsection{Relazione tra geodetiche e trasporto parallelo} \begin{definition} Una curva $\alpha$ su una superficie $\Sigma$ si dice @@ -752,6 +754,19 @@ \end{equation*} \end{remark} + \begin{proposition} + Sia $\alpha$ una geodetica su $\Sigma$. Allora + $\norm{\alpha'}$ è costante. + \end{proposition} + + \begin{proof} + Derivando $\alpha' \cdot \alpha'$ si ottiene + $2 (\alpha'' \cdot \alpha')$, che però è nullo dal momento + che $\alpha'' \perp T_{\alpha} \Sigma$, da cui la tesi. + \end{proof} + + \subsection{Mappa esponenziale, coordinate normali e intorno normale} + \begin{proposition} Sia $\Sigma$ una superficie e $q$ un suo punto. Allora per ogni $v \in T_q \Sigma$ esiste $\eps > 0$ e un'unica geodetica $\gamma_v : (-\eps, \eps) \to \Sigma$ tale per cui $\gamma_v(0) = q$ e @@ -776,27 +791,27 @@ $\gamma_v : [0, 1] \to \Sigma$. \smallskip Dal Teorema di dipendenza liscia dai dati iniziali è ben definita e liscia allora l'applicazione - $v \mapsto \gamma_v(1)$, dove $v \in U_q$. + $v \mapsto \gamma_v(1)$, dove $\norm{v} < \eps_{\textnormal{min}}$. \end{remark} \begin{definition}[Mappa esponenziale] Sia $P$ un punto su una superficie $\Sigma$ e sia $\vec{x}$ una sua parametrizzazione. Sia: \[ - U_q = \{ v \in T_P \Sigma \mid \norm{v} < \eps_{\textnormal{min}} \}, + \boxed{U_P = \{ v \in T_P \Sigma \mid \norm{v} < \eps_{\textnormal{min}} \},} \] dove $\eps_{\textnormal{min}}$ è definito secondo l'Osservazione \ref{rmk:geodetiche}. \smallskip - Si definisce allora la \textbf{mappa esponenziale} $\exp_P : U_q \to \vec{x}(U_q)$ come l'applicazione + Si definisce allora la \textbf{mappa esponenziale} $\exp_P : U_P \to \vec{x}(U_P)$ come l'applicazione con la seguente proprietà: \[ - \exp_P(v) = \gamma_v(1). + \boxed{\exp_P(v) = \gamma_v(1).} \] \end{definition} \begin{definition}[Intorno normale] Sia $P$ un punto su una superficie $\Sigma$ e sia $\vec{x}$ una sua parametrizzazione. - Si dice che l'immagine $\exp_P(U_q)$ è un \textbf{intorno normale} di $P$. + Si dice che l'immagine $N_P \defeq \exp_P(U_P)$ è un \textbf{intorno normale} di $P$. \end{definition} \begin{remark} @@ -805,6 +820,82 @@ \[ \vec{y}(u, v) = \exp_P(u \vec{e_1} + v \vec{e_2}), \] - dove $\{\vec{e_1}, \vec{e_2}\}$ è una base ortonormale di $T_P \Sigma$. + dove $\{\vec{e_1}, \vec{e_2}\}$ è una base ortonormale di $T_P \Sigma$. \smallskip + + Questa mappa, detta indotta dalle \textbf{coordinate normali}, soddisfa alcune + importanti proprietà: + \begin{itemize} + \item $\vec{y}(0, 0) = P$, + \item $\vec{y}_u(0, 0) = \dertime{\exp_P(t \vec{e_1})}{t=0} = + \dertime{\gamma_{t \vec{e_1}}(1)}{t=0} = + \dertime{\gamma_{\vec{e_1}} (u)}{t=0} = \vec{e_1}$, + \item $\vec{y}_v(0, 0) = \cdots = \vec{e_2}$. + \end{itemize} \end{remark} + + \subsection{Lemma di Gauss e minimizzazione locale delle distante} + + \begin{lemma}[Gauss, le geodetiche sono ortogonali ai cerchi indotti dalla mappa esponenziale] \label{lem:gauss} + Sia $\{\vec{e_1}, \vec{e_2}\}$ una base ortonormale di $T_P \Sigma$. + Sia $\eps$ lo stesso valore che definisce $U_P$. + Sia + $v_k : [0, 2\pi] \to U_P$ la curva definita come: + \[ + v_k(t) = k(\cos(t) \vec{e_1} + \sin(t) \vec{e_2}), \quad k < \eps. + \] + Allora ogni geodetica + $\gamma_w$ è ortogonale a $\exp_P \circ \, v_k$ per $k < \eps$ e $w \in U_P$. + \end{lemma} + + \begin{proof} + Sia $\vec{x} : [0, 1] \cdot \RR \to \Sigma$ definita come: + \[ \vec{x}(u, t) = \exp_P(u \cdot v_k(t)) = \gamma_{v_k(t)}(u). \] + Dimostriamo la tesi mostrando che $\vec{x}_u$ (che parametrizza le geodetiche) e $\vec{x}_t$ (che parametrizza + i cerchi) sono ortogonali. \smallskip + + Osserviamo che: + \[ (\vec{x}_u \cdot \vec{x}_t)_u = \vec{x}_{uu} \cdot \vec{x}_t + \vec{x}_u \cdot \vec{x}_{tu}. \] + Inoltre $\vec{x}_{uu}(u, t_0) = (\gamma_{v_k(t_0)}(u))''$ è normale a $T_P \Sigma$ dacché + $\gamma_{v_k(t_0)}$ è una geodetica, e quindi $\vec{x}_{uu} \cdot \vec{x}_t = 0$. \smallskip + + Ancora $\vec{x}_u \cdot \vec{x}_u = \norm{v_k(t)}^2 = k^2$ è costante, e quindi, derivando + $\vec{x}_u \cdot \vec{x}_u$ per $t$, si ottiene anche $\vec{x}_u \cdot \vec{x}_{ut} = 0$. \smallskip + + Pertanto $\vec{x}_u \cdot \vec{x}_t$ è costante lungo $u$. È sufficiente allora mostrare la tesi per + $u = 0$: + \[ + \vec{x}(0, t) = P \implies \vec{x}_t(0, t) = 0, + \] + e quindi $\vec{x}_u \cdot \vec{x}_t = 0$ lungo $u = 0$. + \end{proof} + + \begin{corollary}[Le geodetiche minimizzano localmente le distanze] + Sia $P' = \gamma_v(1)$ con $v \in U_P$ un punto distinto da $P$. Se $\alpha : [0, 1] \to N_P$ + è una curva liscia con $\alpha(0) = P$ e $\alpha(1) = P'$, allora: + \[ + \boxed{\ell(\alpha) \geq \ell(\gamma_{v}).} + \] + \end{corollary} + + \begin{proof} + Parametrizziamo $N_P$ usando $\vec{x}$ come definita nella dimostrazione del + Lemma di Gauss (\ref{lem:gauss}) per un $k = \norm{v}$. \smallskip + + Per la Proposizione \ref{prop:coordinate_curva_parametrizzazione}, + $\alpha$ si scrive come: + \[ + \alpha(t) = \vec{x}(u(t), \tau(t)), + \] + con $u$ e $\tau$ lisce. Allora, derivando $\alpha$ e applicando il Lemma di Gauss + (\ref{lem:gauss}) si ottiene: + \[ + \norm{\alpha'}^2 \geq (u')^2 \norm{v}^2. + \] + Allora: + \[ + \ell(\alpha) = \int_0^1 \norm{\alpha'} \dt \geq \norm{v} (u(1) - u(0)) = \norm{v} = \ell(\gamma_v), + \] + dove si è usato che $u(0) = 0$ ($P$ nelle coordinate normali corrisponde a $(0, 0)$) e che + $u(1) = 1$ ($P'$ ha già raggio $k$). + \end{proof} \end{multicols*}