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| -rw-r--r-- | ggs/n.lyx | 11 | ||||
| -rw-r--r-- | ggs/n2.lyx | 2142 |
2 files changed, 2143 insertions, 10 deletions
@@ -161,15 +161,11 @@ filename "n1.lyx" \end_layout -\begin_layout Standard -\begin_inset Note Note -status open - \begin_layout Chapter Geodésicas \end_layout -\begin_layout Plain Layout +\begin_layout Standard \begin_inset CommandInset include LatexCommand input filename "n2.lyx" @@ -179,10 +175,5 @@ filename "n2.lyx" \end_layout -\end_inset - - -\end_layout - \end_body \end_document diff --git a/ggs/n2.lyx b/ggs/n2.lyx new file mode 100644 index 0000000..a0f5b5e --- /dev/null +++ b/ggs/n2.lyx @@ -0,0 +1,2142 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +Una curva +\begin_inset Formula $\gamma:I\to S$ +\end_inset + + es una +\series bold +geodésica +\series default + de la superficie regular +\begin_inset Formula $S$ +\end_inset + + si +\begin_inset Formula $\gamma'$ +\end_inset + + es paralelo. + Propiedades: Sea +\begin_inset Formula $\gamma:I\to S$ +\end_inset + + una geodésica: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\Vert\gamma'(t)\Vert$ +\end_inset + + es constante. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\gamma$ +\end_inset + + es constante si y sólo si existe +\begin_inset Formula $t_{0}\in I$ +\end_inset + + con +\begin_inset Formula $\gamma'(t_{0})=0$ +\end_inset + +, por lo que toda geodésica no constante es una curva regular. +\end_layout + +\begin_deeper +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Obvio. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Para +\begin_inset Formula $t\in I$ +\end_inset + +, +\begin_inset Formula $\Vert\gamma'(t)\Vert=\Vert\gamma'(t_{0})\Vert=0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La condición de geodésica se conserva por isometrías locales. +\end_layout + +\begin_deeper +\begin_layout Standard +La derivada covariante se conserva por ser un concepto intrínseco. +\end_layout + +\end_deeper +\begin_layout Enumerate +Si +\begin_inset Formula $\gamma$ +\end_inset + + no es constante, una reparametrización suya es una geodésica si y sólo + si el cambio de parámetro es afín. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $h:J\to I$ +\end_inset + + un cambio de parámetro y +\begin_inset Formula $\alpha:=\gamma\circ h$ +\end_inset + +, entonces +\begin_inset Formula $\alpha'(s)=h'(s)\gamma'(h(s))$ +\end_inset + + y +\begin_inset Formula +\begin{align*} +\frac{D\alpha'}{ds}(s) & =(h''(s)\gamma'(h(s))+h'(s)^{2}\gamma''(h(s)))^{\top}=h''(s)\gamma'(h(s))+h'(s)^{2}\frac{D\gamma'}{dt}(h(s))=\\ + & =h''(s)\gamma'(h(s)), +\end{align*} + +\end_inset + +pues +\begin_inset Formula $\frac{D\gamma'}{dt}(h(s))=0$ +\end_inset + + por ser +\begin_inset Formula $\gamma$ +\end_inset + + una geodésica. + Como +\begin_inset Formula $\gamma$ +\end_inset + + no es constante, +\begin_inset Formula $\gamma'(h(s))\neq0$ +\end_inset + + en todo +\begin_inset Formula $s$ +\end_inset + +, luego +\begin_inset Formula $\frac{D\alpha'}{ds}(s)=h''(s)\gamma'(h(s))=0\iff h''(s)=0\iff\exists a,b\in\mathbb{R}:h(s)=as+b$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular, +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva regular y +\begin_inset Formula $N:I\to\mathbb{R}^{3}$ +\end_inset + + un campo normal unitario a lo largo de +\begin_inset Formula $\alpha$ +\end_inset + +, entonces +\begin_inset Formula $\alpha$ +\end_inset + + es una geodésica si y sólo si +\begin_inset Formula +\[ +\alpha''(t)+\langle\alpha'(t),N'(t)\rangle N(t)=0, +\] + +\end_inset + +sustituyendo en la e.d.o. + extrínseca de los campos paralelos. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $(U,X)$ +\end_inset + + es una parametrización de +\begin_inset Formula $S$ +\end_inset + +, +\begin_inset Formula $\alpha:I\to X(U)$ +\end_inset + + es una curva y +\begin_inset Formula $(u,v):=X^{-1}\circ\alpha:I\to U$ +\end_inset + +, +\begin_inset Formula $\alpha$ +\end_inset + + es una geodésica de +\begin_inset Formula $S$ +\end_inset + + si y sólo si +\begin_inset Formula +\[ +\left\{ \begin{aligned}u''+(u')^{2}\Gamma_{11}^{1}(u,v)+2u'v'\Gamma_{12}^{1}(u,v)+(v')^{2}\Gamma_{22}^{1}(u,v) & =0,\\ +v''+(u')^{2}\Gamma_{11}^{2}(u,v)+2u'v'\Gamma_{12}^{2}(u,v)+(v')^{2}\Gamma_{22}^{2}(u,v) & =0. +\end{aligned} +\right. +\] + +\end_inset + +En efecto, como +\begin_inset Formula $\alpha=X(u,v)$ +\end_inset + +, +\begin_inset Formula $\alpha'=dX_{(u,v)}(u',v')=u'X_{u}(u,v)+v'X_{v}(u,v)$ +\end_inset + +, y solo hay que sustituir en la e.d.o. + intrínseca de los campos paralelos. +\end_layout + +\begin_layout Section +Geodésicas maximales +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{EDO} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Picard en un abierto: +\series default + Sean +\begin_inset Formula $\Omega\subseteq\mathbb{R}\times\mathbb{R}^{n}$ +\end_inset + + abierto y +\begin_inset Formula $f:\Omega\to\mathbb{R}^{n}$ +\end_inset + + continua y localmente lipschitziana respecto a la segunda variable, para + +\begin_inset Formula $(t_{0},x_{0})\in\Omega$ +\end_inset + + existe +\begin_inset Formula $K:=[t_{0}-\alpha,t_{0}+\alpha]\times\overline{B}(x_{0},b)\subseteq\Omega$ +\end_inset + + tal que +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =f(t,x)\\ +x(t_{0}) & =x_{0} +\end{aligned} +\right. +\] + +\end_inset + +tiene solución única definida en +\begin_inset Formula $[t_{0}-\alpha,t_{0}+\alpha]$ +\end_inset + + con gráfica contenida en +\begin_inset Formula $K$ +\end_inset + +. +\end_layout + +\begin_layout Standard +[...] Sea +\begin_inset Formula $\Omega\subseteq\mathbb{R}\times\mathbb{R}^{n}$ +\end_inset + + abierto, si para cada +\begin_inset Formula $(t_{0},x_{0})\in\Omega$ +\end_inset + + existe un intervalo en que el problema de Cauchy +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =f(t,x)\\ +x(t_{0}) & =x_{0} +\end{aligned} +\right. +\] + +\end_inset + +tiene solución única, entonces para cualesquiera soluciones +\begin_inset Formula $x$ +\end_inset + + e +\begin_inset Formula $y$ +\end_inset + + de +\begin_inset Formula $\dot{x}=f(t,x)$ +\end_inset + + definidas respectivamente en +\begin_inset Formula $I_{x}$ +\end_inset + + e +\begin_inset Formula $I_{y}$ +\end_inset + +, si ambas coinciden en un +\begin_inset Formula $\xi\in I_{x}\cap I_{y}$ +\end_inset + +, coinciden en toda la intersección. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dados un abierto +\begin_inset Formula $\Omega\subseteq\mathbb{R}^{m}$ +\end_inset + + y +\begin_inset Formula $f:\Omega\to\mathbb{R}^{n}$ +\end_inset + + diferenciable, +\begin_inset Formula $f$ +\end_inset + + es localmente lipschitziana. + En efecto, para +\begin_inset Formula $x\in\Omega$ +\end_inset + + existe +\begin_inset Formula $\varepsilon$ +\end_inset + + tal que +\begin_inset Formula $\overline{B}(x,\varepsilon)\subseteq\Omega$ +\end_inset + +, y al ser +\begin_inset Formula $f'$ +\end_inset + + continua, +\begin_inset Formula $(f')(\overline{B}(x,\varepsilon))$ +\end_inset + + está acotada por un cierto +\begin_inset Formula $M$ +\end_inset + + y, para +\begin_inset Formula $a,b\in B(x,\varepsilon)$ +\end_inset + +, +\begin_inset Formula $\Vert f(a)-f(b)\Vert\leq M\Vert a-b\Vert$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, sean +\begin_inset Formula $S$ +\end_inset + + es una superficie regular, +\begin_inset Formula $p\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p}S$ +\end_inset + +, existe una única geodésica +\begin_inset Formula $\gamma_{v}:I_{v}\to S$ +\end_inset + + tal que +\begin_inset Formula $0\in I_{v}$ +\end_inset + +, +\begin_inset Formula $\gamma_{v}(0)=p$ +\end_inset + +, +\begin_inset Formula $\gamma'_{v}(0)=v$ +\end_inset + + y cualquier otra geodésica que cumpla estas condiciones es una restricción + de esta a un subintervalo, y llamamos +\series bold +geodésica maximal +\series default + con +\series bold +condiciones iniciales +\series default + +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $v$ +\end_inset + + a +\begin_inset Formula $\gamma_{v}$ +\end_inset + + e +\series bold +intervalo maximal de existencia +\series default + a +\begin_inset Formula $I_{v}$ +\end_inset + +. + +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula ${\cal J}_{p,v}:=\{(I,\alpha):\alpha:I\to S\text{ geodésica},0\in I,\alpha(0)=p,\alpha'(0)=v\}$ +\end_inset + +. + Sean +\begin_inset Formula $(X,U)$ +\end_inset + + una carta local de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +, +\begin_inset Formula $(u_{0},v_{0}):=X^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $a,b\in\mathbb{R}$ +\end_inset + + con +\begin_inset Formula $v=aX_{u}(u_{0},v_{0})+bX_{u}(u_{0},v_{0})$ +\end_inset + +, por el teorema de Picard, existe una solución +\begin_inset Formula $(u,v):(-\varepsilon,\varepsilon)\to U$ +\end_inset + + de la e.d.o. + intrínseca de los campos paralelos con +\begin_inset Formula $u(0)=u_{0}$ +\end_inset + +, +\begin_inset Formula $v(0)=v_{0}$ +\end_inset + +, +\begin_inset Formula $u'(0)=a$ +\end_inset + + y +\begin_inset Formula $v'(0)=b$ +\end_inset + +, y entonces +\begin_inset Formula $\alpha(t):=X(u(t),v(t))$ +\end_inset + + es una geodésica con +\begin_inset Formula $\alpha(0)=X(u_{0},v_{0})=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=dX_{(u_{0},v_{0})}(a,b)=aX_{u}(u_{0},v_{0})+bX_{v}(u_{0},v_{0})=v$ +\end_inset + +, de modo que +\begin_inset Formula $\alpha\in{\cal J}_{p,v}\neq\emptyset$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Sean ahora +\begin_inset Formula $(I_{1},\alpha_{1}),(I_{2},\alpha_{2})\in{\cal J}_{p,v}$ +\end_inset + +, y queremos ver que +\begin_inset Formula $\alpha_{1}(t)=\alpha_{2}(t)$ +\end_inset + + para todo +\begin_inset Formula $t\in I_{1}\cap I_{2}$ +\end_inset + +. + Como +\begin_inset Formula $0\in I_{1}\cap I_{2}$ +\end_inset + + e +\begin_inset Formula $I_{1}$ +\end_inset + + e +\begin_inset Formula $I_{2}$ +\end_inset + + son abiertos conexos, +\begin_inset Formula $I_{1}\cap I_{2}$ +\end_inset + + es abierto y, por el teorema del peine, también conexo, luego es un intervalo. + Sea +\begin_inset Formula $A:=\{t\in I_{1}\cap I_{2}:\alpha_{1}(t)=\alpha_{2}(t),\alpha'_{1}(t)=\alpha'_{2}(t)\}$ +\end_inset + +, y queremos ver que +\begin_inset Formula $A$ +\end_inset + + es abierto y cerrado en +\begin_inset Formula $I_{1}\cap I_{2}$ +\end_inset + + y no vacío y por tanto +\begin_inset Formula $A=I_{1}\cap I_{2}$ +\end_inset + +. + Claramente es no vacío, pues +\begin_inset Formula $\alpha_{1}(0)=\alpha_{2}(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'_{1}(0)=\alpha'_{2}(0)=v$ +\end_inset + +, y es cerrado por ser la anti-imagen del 0 por la función continua +\begin_inset Formula $F(t):=\Vert\alpha_{1}(t)-\alpha_{2}(t)\Vert+\Vert\alpha'_{1}(t)+\alpha'_{2}(t)\Vert$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Sean ahora +\begin_inset Formula $t_{0}\in A$ +\end_inset + + y +\begin_inset Formula $(X,U)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $\alpha_{1}(t_{0})=\alpha_{2}(t_{0})$ +\end_inset + +, existen +\begin_inset Formula $\varepsilon_{1}>0$ +\end_inset + + tal que para +\begin_inset Formula $t\in(t_{0}-\varepsilon_{1},t_{0}+\varepsilon_{1})$ +\end_inset + + es +\begin_inset Formula $\alpha_{1}(t)\in X(U)$ +\end_inset + + y +\begin_inset Formula $\varepsilon_{2}>0$ +\end_inset + + tal que para +\begin_inset Formula $t\in(t_{0}-\varepsilon_{2},t_{0}+\varepsilon_{2})$ +\end_inset + + es +\begin_inset Formula $\alpha_{2}(t)\in X(U)$ +\end_inset + +, y si +\begin_inset Formula $\varepsilon:=\min\{\varepsilon_{1},\varepsilon_{2}\}$ +\end_inset + +, +\begin_inset Formula $(u_{1},v_{1}):=X^{-1}\circ\alpha_{1}$ +\end_inset + + y +\begin_inset Formula $(u_{2},v_{2}):=X^{-1}\circ\alpha_{2}$ +\end_inset + +, entonces +\begin_inset Formula $(u_{1},v_{1})$ +\end_inset + + y +\begin_inset Formula $(u_{2},v_{2})$ +\end_inset + + son soluciones de la e.d.o. + intrínseca de las geodésicas con las mismas condiciones iniciales en +\begin_inset Formula $t_{0}$ +\end_inset + +. + Por el teorema de Picard, la e.d.o. + tiene solución única local para cualesquiera +\begin_inset Formula $(u,v)(t_{0})\in U$ +\end_inset + + y +\begin_inset Formula $(u',v')(t_{0})\in\mathbb{R}^{2}$ +\end_inset + +, por lo que +\begin_inset Formula $(u_{1},v_{1})$ +\end_inset + + y +\begin_inset Formula $(u_{2},v_{2})$ +\end_inset + + coinciden en todo +\begin_inset Formula $(t_{0}-\varepsilon,t_{0}+\varepsilon)$ +\end_inset + + y +\begin_inset Formula $A$ +\end_inset + + es abierto. +\end_layout + +\begin_layout Standard +Así, +\begin_inset Formula $A=I_{1}\cap I_{2}$ +\end_inset + +. + Sea entonces +\begin_inset Formula $I_{v}:=\bigcup_{(I,\alpha)\in{\cal J}_{p,v}}I$ +\end_inset + +, +\begin_inset Formula $I_{v}$ +\end_inset + + es un intervalo abierto por ser unión de intervalos abiertos que contienen + al 0, y definiendo +\begin_inset Formula $\gamma_{v}:I_{v}\to S$ +\end_inset + + como +\begin_inset Formula $\gamma_{v}(t)=\alpha(t)$ +\end_inset + + para +\begin_inset Formula $(I,\alpha)\in{\cal J}_{p,v}$ +\end_inset + + con +\begin_inset Formula $t\in I$ +\end_inset + +, entonces +\begin_inset Formula $\gamma_{v}$ +\end_inset + + está bien definido por lo anterior y cumple las propiedades. +\end_layout + +\begin_layout Section +Ecuaciones diferenciales lineales +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{EDO} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $T\in{\cal L}(\mathbb{R}^{n})$ +\end_inset + + [...], el problema de Cauchy +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =Tx\\ +x(t_{0}) & =x_{0} +\end{aligned} +\right. +\] + +\end_inset + +tiene solución única definida en todo +\begin_inset Formula $\mathbb{R}$ +\end_inset + + y dada por +\begin_inset Formula $x(t)=e^{(t-t_{0})T}x_{0}$ +\end_inset + +. + [...] +\end_layout + +\begin_layout Standard + +\series bold +Cálculo de +\begin_inset Formula $e^{At}$ +\end_inset + + +\series default + [...] Si el polinomio característico de +\begin_inset Formula $T\in{\cal L}(E)$ +\end_inset + +, con +\begin_inset Formula $E$ +\end_inset + + real o complejo, es [...] +\begin_inset Formula $\prod_{k=1}^{p}(t-\lambda_{k})^{n_{k}}$ +\end_inset + +, [...] +\begin_inset Formula $E(T,\lambda_{k}):=\ker(T-\lambda_{k}I)^{n_{k}}$ +\end_inset + +, y [...] +\begin_inset Formula $E=E(T,\lambda_{1})\oplus\dots\oplus E(T,\lambda_{p})$ +\end_inset + + [...]. + [...] +\end_layout + +\begin_layout Enumerate +Hallar los valores propios +\begin_inset Formula $\lambda_{1},\dots,\lambda_{r},a_{1}+ib_{1},a_{1}-ib_{1},\dots,a_{s}+ib_{s},a_{s}-ib_{s}$ +\end_inset + + [ +\begin_inset Formula $\lambda_{i},a_{i},b_{i}\in\mathbb{R}$ +\end_inset + +] de +\begin_inset Formula $A_{\mathbb{C}}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Hallar bases +\begin_inset Formula $(w_{k1},\dots,w_{kp_{k}})$ +\end_inset + + de +\begin_inset Formula $\mathbb{R}^{n}(A,\lambda_{k})$ +\end_inset + + y +\begin_inset Formula $(u_{k1}+iv_{k1},\dots,u_{kq_{k}}+v_{kq_{k}})$ +\end_inset + + de +\begin_inset Formula $\mathbb{C}^{n}(A_{\mathbb{C}},a_{k}+ib_{k})$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Respecto de la base +\begin_inset ERT +status open + +\begin_layout Plain Layout + +{ +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +{\cal B}:= & (w_{11},\dots,w_{1p_{1}},\dots,w_{r1},\dots,w_{rp_{r}},\\ + & \,v_{11},u_{11},\dots,v_{1q_{1}},u_{1q_{1}},\dots,v_{s1},u_{s1},\dots,v_{sq_{s}},u_{sq_{s}}), +\end{align*} + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{EDO} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout + +\end_layout + +\end_inset + +la matriz semisimple es +\begin_inset Formula +\[ +S_{0}:=\begin{pmatrix}\boxed{D_{1}}\\ + & \ddots\\ + & & \boxed{D_{r}}\\ + & & & \boxed{M_{1}}\\ + & & & & \ddots\\ + & & & & & \boxed{M_{s}} +\end{pmatrix}, +\] + +\end_inset + +donde +\begin_inset ERT +status open + +\begin_layout Plain Layout + +{ +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +D_{k} & =\begin{pmatrix}\lambda_{k}\\ + & \ddots\\ + & & \lambda_{k} +\end{pmatrix}, & M_{k} & :=\begin{pmatrix}a_{k} & -b_{k}\\ +b_{k} & a_{k}\\ + & & \ddots\\ + & & & a_{k} & -b_{k}\\ + & & & b_{k} & a_{k} +\end{pmatrix}. +\end{align*} + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +4. +\end_layout + +\end_inset + +Sea +\begin_inset Formula $P:=M_{{\cal CB}}$ +\end_inset + +, entonces la parte semisimple es +\begin_inset Formula $S:=PS_{0}P^{-1}$ +\end_inset + + y la nilpotente es +\begin_inset Formula $N:=A-S$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +5. +\end_layout + +\end_inset + +Finalmente, +\begin_inset Formula +\[ +e^{At}=Pe^{S_{0}t}P^{-1}\sum_{k=1}^{n}\frac{N^{k}t^{k}}{k!}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +[...] Sea +\begin_inset Formula $E$ +\end_inset + + un +\begin_inset Formula $\mathbb{R}$ +\end_inset + +-espacio vectorial y +\begin_inset Formula $T\in{\cal L}(E)$ +\end_inset + +, existe una base de +\begin_inset Formula $E$ +\end_inset + + respecto a la que +\begin_inset Formula $T$ +\end_inset + + tiene una matriz compuesta de bloques diagonales de la forma +\begin_inset ERT +status open + +\begin_layout Plain Layout + +{ +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +\begin{pmatrix}\lambda\\ +1 & \ddots\\ + & \ddots & \ddots\\ + & & 1 & \lambda +\end{pmatrix} & & \text{ó} & & \begin{pmatrix}\boxed{D} & \ddots\\ +\boxed{I_{2}} & \ddots & \ddots\\ + & & \boxed{I_{2}} & \boxed{D} +\end{pmatrix}, & & D & =\begin{pmatrix}a & -b\\ +b & a +\end{pmatrix} +\end{align*} + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + +Si +\begin_inset Formula $A$ +\end_inset + + es [de la primera forma][...], [...] +\begin_inset Formula +\[ +e^{tA}[...]=e^{t\lambda}\begin{pmatrix}1\\ +t & 1\\ +\frac{t^{2}}{2} & t & 1\\ +\vdots & \ddots & \ddots & \ddots\\ +\frac{t^{n-1}}{(n-1)!} & \cdots & \frac{t^{2}}{2} & t & 1 +\end{pmatrix} +\] + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{EDO} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si [es de la segunda][...], +\begin_inset Formula +\[ +[e^{tA}]=e^{at}\begin{pmatrix}\tilde{D}\\ +t\tilde{D} & \tilde{D}\\ +\frac{t^{2}}{2}\tilde{D} & t\tilde{D} & \tilde{D}\\ +\vdots & \ddots & \ddots & \ddots\\ +\frac{t^{m-1}}{(m-1)!}\tilde{D} & \cdots & \frac{t^{2}}{2}\tilde{D} & t\tilde{D} & \tilde{D} +\end{pmatrix} +\] + +\end_inset + +[...] +\begin_inset Formula +\[ +\tilde{D}=\begin{pmatrix}\cos(bt) & -\sin(bt)\\ +\sin(bt) & \cos(bt) +\end{pmatrix} +\] + +\end_inset + +[...] Llamamos +\series bold +base de soluciones +\series default + de +\begin_inset Formula $x^{(n)}+a_{1}(t)x^{(n-1)}+\dots+a_{1}(t)x=0$ +\end_inset + + a una familia +\begin_inset Formula $x_{1},\dots,x_{n}$ +\end_inset + + de soluciones linealmente independiente. + +\end_layout + +\begin_layout Standard +[...] Dada la ecuación homogénea +\begin_inset Formula $x^{(n)}+a_{1}x^{(n-1)}+\dots+a_{n}x=0$ +\end_inset + +, una combinación lineal de soluciones de esta ecuación es también solución, + así como la derivada de una solución. +\end_layout + +\begin_layout Standard +La matriz de la ecuación vectorial asociada [con coeficientes +\begin_inset Formula $(x,\dot{x},\dots,x^{(n-1)})$ +\end_inset + +] es +\begin_inset Formula +\[ +\begin{pmatrix} & 1\\ + & & \ddots\\ + & & & 1\\ +-a_{n} & \cdots & \cdots & -a_{1} +\end{pmatrix}, +\] + +\end_inset + +que llamamos +\series bold +asociada +\series default + al polinomio +\begin_inset Formula $p(\lambda)=(-1)^{n}(\lambda^{n}+a_{1}\lambda^{n-1}+\dots+a_{n-1}\lambda+a_{n})$ +\end_inset + +, [...] el polinomio característico de la matriz. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Superficies geodésicamente completas +\end_layout + +\begin_layout Standard +Una superficie regular +\begin_inset Formula $S$ +\end_inset + + es +\series bold +geodésicamente completa +\series default + en un +\begin_inset Formula $p\in S$ +\end_inset + + si para +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + es +\begin_inset Formula $I_{v}=\mathbb{R}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +, y es geodésicamente completa si lo es en todo +\begin_inset Formula $p\in S$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Dado el plano +\begin_inset Formula $S=\{p\in\mathbb{R}^{3}:\langle p,a\rangle=c\}$ +\end_inset + +, la geodésica maximal de +\begin_inset Formula $S$ +\end_inset + + con condiciones iniciales +\begin_inset Formula $p\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + es la recta +\begin_inset Formula $\gamma:\mathbb{R}\to S$ +\end_inset + + dada por +\begin_inset Formula $\gamma(t):=p+tv$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Tomando la normal +\begin_inset Formula $N(p):=a$ +\end_inset + +, como +\begin_inset Formula $N$ +\end_inset + + es constante, debe ser +\begin_inset Formula +\[ +0=\gamma''(t)+\langle\gamma'(t),(N\circ\gamma)'(t))\rangle N(\gamma(t))=\gamma''(t), +\] + +\end_inset + +de modo que +\begin_inset Formula $\gamma$ +\end_inset + + es de la forma +\begin_inset Formula $\gamma(t)=a+bt$ +\end_inset + +, pero +\begin_inset Formula $p=\gamma(0)=a$ +\end_inset + + y +\begin_inset Formula $v=\gamma'(0)=b$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Dado +\begin_inset Formula $r>0$ +\end_inset + +, la geodésica maximal de la esfera +\begin_inset Formula $S:=\mathbb{S}^{2}(r)$ +\end_inset + + con condiciones iniciales +\begin_inset Formula $p\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p}S\setminus0$ +\end_inset + + es el círculo máximo +\begin_inset Formula $\gamma:\mathbb{R}\to S$ +\end_inset + + dado por +\begin_inset Formula +\[ +\gamma(t)=\cos\left(\frac{\Vert v\Vert}{r}t\right)p+\frac{r}{\Vert v\Vert}\sin\left(\frac{\Vert v\Vert}{r}t\right)v. +\] + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Standard +Tomando la normal +\begin_inset Formula $N(p):=\frac{p}{r}$ +\end_inset + + y llamando +\begin_inset Formula $N(t):=N(\gamma(t))$ +\end_inset + +, +\begin_inset Formula $N(t)=\frac{\gamma(t)}{r}$ +\end_inset + + y +\begin_inset Formula $N'(t)=\frac{1}{r}\gamma'(t)$ +\end_inset + +, y debe ser +\begin_inset Formula +\[ +0=\gamma''(t)+\left\langle \gamma'(t),\frac{1}{r}\gamma'(t)\right\rangle \frac{1}{r}\gamma(t)=\gamma''(t)+\frac{1}{r^{2}}\Vert\gamma'(t)\Vert^{2}\gamma(t)\overset{\Vert\gamma'(t)\Vert=\Vert\gamma'(0)\Vert}{=}\gamma''(t)+\frac{\Vert v\Vert^{2}}{r^{2}}\gamma(t), +\] + +\end_inset + +Si +\begin_inset Formula $c:=\frac{\Vert v\Vert^{2}}{r^{2}}=0$ +\end_inset + +, +\begin_inset Formula $v=0$ +\end_inset + +, y en otro caso, en cada coordenada, el polinomio asociado a la ecuación + lineal homogénea +\begin_inset Formula $p(\lambda)=\lambda^{2}+c$ +\end_inset + +, los valores propios son +\begin_inset Formula $\pm\sqrt{c}i$ +\end_inset + + y una base de soluciones es pues +\begin_inset Formula $\{\cos(\sqrt{c}t),\sin(\sqrt{c}t)\}$ +\end_inset + +. + Por tanto existen +\begin_inset Formula $a_{i},b_{i}\in\mathbb{R}$ +\end_inset + + con +\begin_inset Formula $\gamma_{i}(t)=a_{i}\cos(\sqrt{c}t)+b_{i}\sin(\sqrt{c}t)$ +\end_inset + +, pero +\begin_inset Formula +\begin{align*} +p_{i} & =\gamma_{i}(0)=a_{i}, & v_{i} & =\gamma'_{i}(0)=b_{i}\sqrt{c}, +\end{align*} + +\end_inset + +luego en resumen +\begin_inset Formula $\gamma(t)=p\cos(\sqrt{c}t)+\frac{v}{\sqrt{C}}\sin(\sqrt{c}t)$ +\end_inset + +, y +\begin_inset Formula $\sqrt{c}=\frac{\Vert v\Vert}{r}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Sean +\begin_inset Formula $r>0$ +\end_inset + +, +\begin_inset Formula $S:=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}=r^{2}\}$ +\end_inset + + un cilindro, +\begin_inset Formula $p\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p}S$ +\end_inset + +, la geodésica maximal de +\begin_inset Formula $S$ +\end_inset + + con condiciones iniciales +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $v$ +\end_inset + + es la recta +\begin_inset Formula $\gamma:\mathbb{R}\to S$ +\end_inset + + dada por +\begin_inset Formula +\[ +\gamma(t):=p+tv +\] + +\end_inset + +si +\begin_inset Formula $v_{1}=v_{2}=0$ +\end_inset + + o la hélice +\begin_inset Formula $\gamma:\mathbb{R}\to S$ +\end_inset + + dada por +\begin_inset Formula +\[ +\gamma(t):=\begin{pmatrix}{\displaystyle p_{1}\cos(ct)+\frac{v_{1}}{c}\sin(ct)}\\ +{\displaystyle p_{2}\cos(ct)+\frac{v_{2}}{c}\sin(ct)}\\ +p_{3}+tv_{3} +\end{pmatrix} +\] + +\end_inset + +en otro caso, donde +\begin_inset Formula $c:=\frac{\sqrt{\Vert v\Vert^{2}-v_{3}^{2}}}{r}$ +\end_inset + +, que es una circunferencia horizontal si +\begin_inset Formula $v_{3}=0$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(x,y,z)=x^{2}+y^{2}$ +\end_inset + +, como +\begin_inset Formula $f'(x,y,z)=(2x,2y,0)$ +\end_inset + +, los puntos críticos de +\begin_inset Formula $f$ +\end_inset + + son aquellos con +\begin_inset Formula $z=0$ +\end_inset + +, el único valor crítico es 0 y +\begin_inset Formula $r^{2}$ +\end_inset + + es un valor regular, de modo que +\begin_inset Formula $S=\{f(x,y,z)=r^{2}\}$ +\end_inset + + es una superficie de nivel con normal +\begin_inset Formula +\[ +N(x,y,z)=\frac{\nabla f}{\Vert\nabla f\Vert}=\frac{(2x,2y,0)}{2\sqrt{x^{2}+y^{2}}}=\frac{1}{r}(x,y,0). +\] + +\end_inset + +Entonces, sean +\begin_inset Formula $N(t):=N(\gamma(t))$ +\end_inset + + y +\begin_inset Formula $\gamma(t)=:(x(t),y(t),z(t))$ +\end_inset + +, +\begin_inset Formula $N'(t)=\frac{1}{r}(x'(t),y'(t),0)$ +\end_inset + + y +\begin_inset Formula $\gamma$ +\end_inset + + debe cumplir +\begin_inset Formula +\[ +\gamma''(t)+\langle\gamma'(t),N'(t)\rangle N(t)=\begin{pmatrix}x''(t)\\ +y''(t)\\ +z''(t) +\end{pmatrix}+\frac{1}{r^{2}}(x'(t)^{2}+y'(t)^{2})\begin{pmatrix}x'(t)\\ +y'(t)\\ +0 +\end{pmatrix}=0. +\] + +\end_inset + +Así, +\begin_inset Formula $z''(t)=0$ +\end_inset + + y por tanto +\begin_inset Formula $z(t)=a+bt$ +\end_inset + + para ciertos +\begin_inset Formula $a,b\in\mathbb{R}$ +\end_inset + +, con +\begin_inset Formula $p_{3}=z(0)=a$ +\end_inset + + y +\begin_inset Formula $v_{3}=z'(0)=b$ +\end_inset + +. + Si +\begin_inset Formula $v_{1}=v_{2}=0$ +\end_inset + + entonces +\begin_inset Formula $x$ +\end_inset + + es constante en +\begin_inset Formula $p_{1}$ +\end_inset + + e +\begin_inset Formula $y$ +\end_inset + + lo es en +\begin_inset Formula $p_{2}$ +\end_inset + +. + En otro caso +\begin_inset Formula $c>0$ +\end_inset + +, y como +\begin_inset Formula $z'$ +\end_inset + + es constante en +\begin_inset Formula $v_{3}$ +\end_inset + + y +\begin_inset Formula $\Vert\gamma'\Vert$ +\end_inset + + lo es en +\begin_inset Formula $\Vert v\Vert$ +\end_inset + +, se tiene +\begin_inset Formula +\[ +x'(t)^{2}+y'(t)^{2}=\Vert\gamma'(t)\Vert^{2}-z'(t)^{2}=\Vert v\Vert^{2}-v_{3}^{2} +\] + +\end_inset + + y +\begin_inset Formula $\frac{x'(t)^{2}+y'(t)^{2}}{r^{2}}=c^{2}$ +\end_inset + +, y queda +\begin_inset Formula +\[ +(x''(t),y''(t))+c^{2}(x'(t),y'(t))=0. +\] + +\end_inset + +Para la coordenada +\begin_inset Formula $x$ +\end_inset + +, el polinomio asociado es +\begin_inset Formula $p(\lambda)=\lambda^{2}+c^{2}$ +\end_inset + + y los valores propios son +\begin_inset Formula $\pm ci$ +\end_inset + +, de modo que una base de soluciones es +\begin_inset Formula $\{\cos(ct),\sin(ct)\}$ +\end_inset + + y existen +\begin_inset Formula $a,b\in\mathbb{R}$ +\end_inset + + tales que +\begin_inset Formula $x(t)=a\cos(ct)+b\sin(ct)$ +\end_inset + +, pero +\begin_inset Formula +\begin{align*} +p_{1} & =x(0)=a, & v_{1} & =x'(0)=bc, +\end{align*} + +\end_inset + +de modo que +\begin_inset Formula $x(t)=p_{1}\cos(ct)+\frac{v_{1}}{c}\sin(ct)$ +\end_inset + +, y análogamente +\begin_inset Formula $y(t)=p_{2}\cos(ct)+\frac{v_{2}}{c}\sin(ct)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Así, el plano, la esfera y el cilindro son geodésicamente completos; de + hecho toda superficie de nivel de una función +\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ +\end_inset + + lo es. +\end_layout + +\begin_layout Section +Pregeodésicas +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{GCS} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada por +\begin_inset Formula $N$ +\end_inset + + y +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva, [...] +\begin_inset Formula +\[ +\alpha''(t)=\frac{D\alpha'}{dt}(t)+\langle\alpha''(t),N(\alpha(t))\rangle N(\alpha(t)). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva parametrizada por [...] arco, el +\series bold +triedro de Darboux +\series default + es la base [...] +\begin_inset Formula $(\alpha'(s),J\alpha'(s):=\alpha'(s)\wedge N(\alpha(s)),N(\alpha(s))\rangle$ +\end_inset + +. + Entonces +\begin_inset Formula +\[ +\frac{D\alpha'}{ds}(s)=\kappa_{g}(s)J\alpha'(s), +\] + +\end_inset + +donde +\begin_inset Formula $\kappa_{g}:=\langle\alpha'',J\alpha'\rangle:I\to\mathbb{R}$ +\end_inset + +, es la +\series bold +curvatura geodésica +\series default + de +\begin_inset Formula $\alpha$ +\end_inset + +, cuyo signo depende de +\begin_inset Formula $N$ +\end_inset + +[, y +\begin_inset Formula $\kappa_{n}:=\langle\alpha'',N(\alpha)\rangle$ +\end_inset + + es la +\series bold +curvatura normal +\series default + de +\begin_inset Formula $\alpha$ +\end_inset + +]. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + p.p.a. + es una geodésica si y sólo si +\begin_inset Formula $\kappa_{g}\equiv0$ +\end_inset + +, pues +\begin_inset Formula $\frac{D\alpha'}{ds}(s)=0$ +\end_inset + + si y sólo si +\begin_inset Formula $\kappa_{g}(s)J\alpha'(s)=0$ +\end_inset + +, pero +\begin_inset Formula $J\alpha'(s)\neq0$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $S$ +\end_inset + + es una superficie regular, +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + es una curva y +\begin_inset Formula $h:J\to I$ +\end_inset + + es un cambio de parámetro que conserva la orientación con +\begin_inset Formula $\beta:=\alpha\circ h$ +\end_inset + + p.p.a., la curvatura geodésica de +\begin_inset Formula $\alpha$ +\end_inset + + es +\begin_inset Formula +\[ +\kappa_{g}^{\alpha}(t):=\kappa_{g}^{\beta}(h^{-1}(t))=\frac{\langle\alpha''(t),J\alpha'(t)\rangle}{\Vert\alpha'(t)\Vert^{3}}. +\] + +\end_inset + + +\series bold +Demostración: +\series default + +\begin_inset Formula $1=\Vert\beta'(s)\Vert=h'(s)\Vert\alpha'(h(s))\Vert$ +\end_inset + +, luego +\begin_inset Formula $h'(s)=\frac{1}{\Vert\alpha'(h(s))\Vert}$ +\end_inset + + y para +\begin_inset Formula $t\in I$ +\end_inset + +, sea +\begin_inset Formula $s:=h^{-1}(t)$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +\kappa_{g}^{\alpha}(t) & =\kappa_{g}^{\beta}(s)=\langle\beta''(s),J\beta'(s)\rangle=\langle h''(s)\alpha'(h(s))+h'(s)^{2}\alpha''(h(s)),h'(s)J\alpha'(h(s))\rangle\\ + & =h'(s)^{3}\langle\alpha''(h(s)),J\alpha'(h(s))\rangle=\frac{\langle\alpha''(t),J\alpha'(t)\rangle}{\Vert\alpha'(t)\Vert^{3}}, +\end{align*} + +\end_inset + +donde en la penúltima igualdad se usa que +\begin_inset Formula $\langle\alpha'(h(s)),J\alpha'(h(s))\rangle=0$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + es una +\series bold +pregeodésica +\series default + de +\begin_inset Formula $S$ +\end_inset + + si existe un cambio de parámetro +\begin_inset Formula $h:J\to I$ +\end_inset + + tal que +\begin_inset Formula $\beta:=\alpha\circ h$ +\end_inset + + es una geodésica de +\begin_inset Formula $S$ +\end_inset + +, si y sólo si +\begin_inset Formula $\kappa_{g}^{\alpha}\equiv0$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Sea +\begin_inset Formula $h$ +\end_inset + + un cambio de parámetro tal que +\begin_inset Formula $\beta:=\alpha\circ h$ +\end_inset + + es una geodésica, entonces +\begin_inset Formula $\Vert\beta'\Vert$ +\end_inset + + es constante en algún +\begin_inset Formula $c>0$ +\end_inset + +, luego +\begin_inset Formula $\gamma(s):=\beta(\frac{s}{c})$ +\end_inset + + es una geodésica y es p.p.a. + al ser +\begin_inset Formula $\Vert\gamma'(s)\Vert=\Vert\frac{1}{c}\beta'(s)\Vert=1$ +\end_inset + +. + Sea entonces +\begin_inset Formula $\tilde{h}(s):=h(\frac{s}{c})$ +\end_inset + +, entonces +\begin_inset Formula $\gamma=\alpha\circ\tilde{h}$ +\end_inset + + y +\begin_inset Formula $\kappa_{g}^{\alpha}(t)=\kappa_{g}^{\gamma}(\tilde{h}^{-1}(t))=0$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Sea +\begin_inset Formula $\beta=\alpha\circ h$ +\end_inset + + la reparametrización por arco de +\begin_inset Formula $\alpha$ +\end_inset + +, como +\begin_inset Formula $\kappa_{g}^{\alpha}(t)=\kappa_{g}^{\beta}(h^{-1}(t))$ +\end_inset + +, +\begin_inset Formula $\kappa_{g}^{\beta}(s)=\kappa_{g}^{\alpha}(h(s))=0$ +\end_inset + +, luego +\begin_inset Formula $\beta$ +\end_inset + + es una geodésica y por tanto +\begin_inset Formula $\alpha$ +\end_inset + + es una pregeodésica. +\end_layout + +\end_body +\end_document |