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+ \subsection{Geodetiche} + + \begin{definition} + Una curva $\alpha$ su una superficie $\Sigma$ si dice + \textbf{geodetica} se il campo $\alpha'$ è parallelo lungo + $\alpha$, ovverosia se: + \[ + \boxed{\nabla_{\alpha'} \alpha' = (\alpha'')^\top = 0.} + \] + \end{definition} + + \begin{remark} + Per le geodetiche è necessario specializzare correttamente + le equazioni del trasporto parallelo. Ponendo + $\alpha(t) = \vec{x}(u(t), v(t))$, otteniamo: + \begin{equation*} \label{eq:Geo} + \boxed{\textnormal{(Geo): } \begin{cases} + u'' + \Gamma_{uu}^u \, (u')^2 + 2 \Gamma_{uv}^u \, u'v' + \Gamma_{vv}^u \, (v')^2 = 0, \\ + v'' + \Gamma_{uu}^v \, (u')^2 + 2 \Gamma_{uv}^v \, u'v' + \Gamma_{vv}^v \, (v')^2 = 0, + \end{cases}} + \end{equation*} + \end{remark} + + \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 + $\gamma_v'(0) = v$. + \end{proposition} + + \begin{proof} + Segue dall'applicazione del Teorema di Cauchy-Lipschitz per l'esistenza e l'unicità locale di una + soluzione per una sistema di equazioni differenziali al sistema dell'Osservazione \ref{eq:Geo}. + \end{proof} + + \begin{remark} + Possiamo in realtà assumere che $\gamma_v(t)$ sia definita su $[0, 1]$ senza alcuna perdita di generalità. + Infatti, se $\gamma_v(t)$ è definita su $(-\eps, \eps)$, la curva + $\gamma_v(st)$ è ben definita per $t \in [0, 1]$, dove $s \in (0, \eps)$. Per unicità di $\gamma_{sv}(t)$, dal momento che + $\gamma_v(st)$ ne rispetta le condizioni iniziali, si ha $\gamma_v(st) = \gamma_{sv}(t)$. \smallskip + \end{remark} + + \begin{remark} \label{rmk:geodetiche} + Per compattezza di $\{v \in T_P \Sigma \mid \norm{v} = 1\}$, esiste un $\eps_{\textnormal{min}}$ per il + quale per ogni vettore $v$ di norma $\norm{v} < \eps_{\textnormal{min}}$ possiamo considerare la geodetica + $\gamma_v : [0, 1] \to \Sigma$. \smallskip + + Dal Teorema di dipendenza liscia dai dati iniziali è ben definita e liscia allora l'applicazione + $v \mapsto \gamma_v(1)$, dove $v \in U_q$. + \end{remark} + + \begin{definition}[Mappa esponenziale] + Sia $P$ un punto su una superficie $\Sigma$ e sia $\vec{x}$ una sua parametrizzazione. + Sia: + \[ + U_q = \{ v \in T_P \Sigma \mid \norm{v} < \eps_{\textnormal{min}} \}, + \] + 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 + con la seguente proprietà: + \[ + \exp_P(v) = \gamma_v(1). + \] + \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 $P$. + \end{definition} + + \begin{remark} + Una volta ben definita la mappa esponenziale $\exp_P$, possiamo riparametrizzare + $\Sigma$ utilizzando $\exp_P$, definendo sul suo dominio naturale l'applicazione: + \[ + \vec{y}(u, v) = \exp_P(u \vec{e_1} + v \vec{e_2}), + \] + dove $\{\vec{e_1}, \vec{e_2}\}$ è una base ortonormale di $T_P \Sigma$. + \end{remark} \end{multicols*}