GGS tema 6

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diff --git a/ggs/n.lyx b/ggs/n.lyx
index fd05e1d..88cd77b 100644
--- a/ggs/n.lyx
+++ b/ggs/n.lyx
@@ -217,5 +217,19 @@ filename "n5.lyx"
 
 \end_layout
 
+\begin_layout Chapter
+Variaciones de la longitud
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n6.lyx"
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document
diff --git a/ggs/n6.lyx b/ggs/n6.lyx
new file mode 100644
index 0000000..0111c19
--- /dev/null
+++ b/ggs/n6.lyx
@@ -0,0 +1,1029 @@
+#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
+Dadas una superficie regular 
+\begin_inset Formula $S$
+\end_inset
+
+ y un 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, una 
+\series bold
+variación
+\series default
+ de un segmento de curva parametrizada 
+\begin_inset Formula $\alpha:[a,b]\to S$
+\end_inset
+
+ es una función diferenciable 
+\begin_inset Formula $\phi:[a,b]\times(-\varepsilon,\varepsilon)\to S$
+\end_inset
+
+ con 
+\begin_inset Formula $\phi_{0}(u):=\phi(u,0)=\alpha(u)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $t\in(-\varepsilon,\varepsilon)$
+\end_inset
+
+, llamamos 
+\begin_inset Formula $\alpha_{t}:=(u\mapsto\phi(u,t)):[a,b]\to S$
+\end_inset
+
+ y 
+\series bold
+curvas de la variación
+\series default
+ a 
+\begin_inset Formula $\{\alpha_{t}\}_{t\in(-\varepsilon,\varepsilon)}$
+\end_inset
+
+, con 
+\begin_inset Formula $\alpha_{0}=\alpha$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+, llamamos 
+\begin_inset Formula $\beta_{u}:=(t\mapsto\phi(u,t)):(-\varepsilon,\varepsilon)\to S$
+\end_inset
+
+ y 
+\series bold
+curvas transversales de la variación
+\series default
+ a 
+\begin_inset Formula $\{\beta_{u}\}_{u\in[a,b]}$
+\end_inset
+
+.
+ La variación es 
+\series bold
+propia
+\series default
+ o 
+\series bold
+tiene extremos fijos
+\series default
+ si 
+\begin_inset Formula $\beta_{a}$
+\end_inset
+
+ y 
+\begin_inset Formula $\beta_{b}$
+\end_inset
+
+ son constantes, es decir, 
+\begin_inset Formula $\phi(a,t)=\alpha(a)$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi(\beta,t)=\alpha(b)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $t$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+campo variacional
+\series default
+ de 
+\begin_inset Formula $\phi$
+\end_inset
+
+ a 
+\begin_inset Formula $Z:[a,b]\to\mathbb{R}^{3}$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+Z(u):=\beta'_{u}(0)=\frac{\partial\phi}{\partial t}(u,0)\in T_{\alpha(u)}S,
+\]
+
+\end_inset
+
+pues 
+\begin_inset Formula $\beta_{u}(0)=\alpha(u)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es una 
+\series bold
+variación normal
+\series default
+ si 
+\begin_inset Formula $\langle Z,\alpha'\rangle\equiv0$
+\end_inset
+
+, de modo que si 
+\begin_inset Formula $N$
+\end_inset
+
+ es una normal a 
+\begin_inset Formula $S$
+\end_inset
+
+, 
+\begin_inset Formula $Z(u)$
+\end_inset
+
+ es paralelo a 
+\begin_inset Formula $\alpha'(u)\wedge N(\alpha(u))$
+\end_inset
+
+ para 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Primera fórmula de variación del arco
+\end_layout
+
+\begin_layout Standard
+Dada una variación 
+\begin_inset Formula $\phi:[a,b]\times(-\varepsilon,\varepsilon)\to S$
+\end_inset
+
+ de la curva 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, el 
+\series bold
+funcional longitud de arco
+\series default
+ de 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es 
+\begin_inset Formula $L:(-\varepsilon,\varepsilon)\to\mathbb{R}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $L(t):=L(\alpha_{t})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ es regular, 
+\begin_inset Formula $L(t)$
+\end_inset
+
+ es diferenciable en un entorno de 
+\begin_inset Formula $t=0$
+\end_inset
+
+ y
+\begin_inset Formula 
+\begin{align*}
+L'(t) & =\int_{a}^{b}\frac{1}{\Vert\alpha'_{t}(u)\Vert}\left\langle \frac{\partial^{2}\phi}{\partial u\partial t},\frac{\partial\phi}{\partial u}\right\rangle (u,t)du\\
+ & =\int_{a}^{b}\frac{1}{\Vert\alpha'_{t}(u)\Vert}\left(\frac{d}{du}\left\langle \frac{\partial\phi}{\partial t},\frac{\partial\phi}{\partial u}\right\rangle -\left\langle \frac{\partial\phi}{\partial t},\frac{\partial^{2}\phi}{\partial u^{2}}\right\rangle \right)(u,t)du.
+\end{align*}
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $f:[a,b]\times(-\varepsilon,\varepsilon)\to\mathbb{R}$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+f(u,t):=\Vert\alpha'_{t}(u)\Vert=\left\Vert \frac{\partial\phi}{\partial u}\right\Vert (u,t)=\sqrt{\left\langle \frac{\partial\phi}{\partial u},\frac{\partial\phi}{\partial u}\right\rangle (u,t)}du,
+\]
+
+\end_inset
+
+entonces, para los 
+\begin_inset Formula $t$
+\end_inset
+
+ en que 
+\begin_inset Formula $L$
+\end_inset
+
+ es derivable,
+\begin_inset Formula 
+\[
+L'(t)=\frac{d}{dt}\int_{a}^{b}f(u,t)du=\int_{a}^{b}\frac{\partial f}{\partial t}(u,t)du,
+\]
+
+\end_inset
+
+de modo que 
+\begin_inset Formula $L'(t)$
+\end_inset
+
+ está definida si y solo si lo está 
+\begin_inset Formula $\frac{\partial f}{\partial t}(u,t)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+, si y solo si 
+\begin_inset Formula $\left\langle \frac{\partial\phi}{\partial u},\frac{\partial\phi}{\partial u}\right\rangle (u,t)>0$
+\end_inset
+
+ (pues las derivadas de 
+\begin_inset Formula $\phi$
+\end_inset
+
+ son diferenciables), si y sólo si 
+\begin_inset Formula $\left\Vert \frac{\partial\phi}{\partial u}(u,t)\right\Vert >0$
+\end_inset
+
+.
+ Ahora bien, como 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ es regular, para 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $\left\Vert \frac{\partial\phi}{\partial u}(u,0)\right\Vert =\Vert\alpha'_{0}(u)\Vert=\Vert\alpha'(u)\Vert>0$
+\end_inset
+
+, y como 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ es compacto, 
+\begin_inset Formula $\Vert\alpha'\Vert([a,b])$
+\end_inset
+
+ alcanza su máximo y su mínimo y existe 
+\begin_inset Formula $c>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall u\in[a,b],f(u,0)=\Vert\alpha'(u)\Vert\geq c>\frac{c}{2}>0$
+\end_inset
+
+.
+ Así, para 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+ existe 
+\begin_inset Formula $\delta_{u}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall t\in(-\delta_{u},\delta_{u}),f(u,t)>\frac{c}{2}$
+\end_inset
+
+.
+ Sea ahora una 
+\begin_inset Formula $\delta:[a,b]\to[0,\varepsilon)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(u,t)>\frac{c}{2}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $t\in(-\delta_{u},\delta_{u})$
+\end_inset
+
+ y tal que, para cada 
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $\delta_{u}$
+\end_inset
+
+ sea lo mayor posible.
+ Si 
+\begin_inset Formula $\varepsilon_{0}:=\inf_{u[a,b]}\delta_{u}=0$
+\end_inset
+
+, entonces existe una sucesión 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\lim_{n}\delta_{u_{n}}=0$
+\end_inset
+
+, pero como la sucesión está acotada en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, por el teorema de Bolzano-Weierstrass, existe una subsucesión convergente
+ 
+\begin_inset Formula $(u_{n_{k}})_{k}$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $\lim_{k}\delta_{u_{n_{k}}}=0$
+\end_inset
+
+.
+ Existe un 
+\begin_inset Formula $N$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $k\geq N$
+\end_inset
+
+, 
+\begin_inset Formula $\delta_{u_{n_{k}}}<\varepsilon$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f(u_{n_{k}},\delta_{u_{n_{k}}})=\frac{c}{2}$
+\end_inset
+
+, pues no puede ser menor ya que 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua y 
+\begin_inset Formula $f(u_{n_{k}},t)>\frac{c}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $t<\delta(u_{n_{k}})$
+\end_inset
+
+ y, si fuera positivo, existiría un 
+\begin_inset Formula $\delta'_{u}>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(u_{n_{k}},t)>\frac{c}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $t<\delta_{u_{n_{k}}}+\delta'_{u}$
+\end_inset
+
+, contradiciendo que 
+\begin_inset Formula $\delta_{u}$
+\end_inset
+
+ sea lo mayor posible.
+ Entonces la sucesión 
+\begin_inset Formula $((u_{n_{k}},\delta_{u_{n_{k}}}))_{k}$
+\end_inset
+
+ tiende a un cierto 
+\begin_inset Formula $(u,0)$
+\end_inset
+
+ y, por continuidad de 
+\begin_inset Formula $f$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+c\leq f(u,0)=f\left(\lim_{k}(u_{n_{k}},\delta_{u_{n_{k}}})\right)=\lim_{k}f(u_{n_{k}},\delta_{u_{n_{k}}})=\lim_{k}\frac{c}{2}=\frac{c}{2}\#.
+\]
+
+\end_inset
+
+Por tanto 
+\begin_inset Formula $m>0$
+\end_inset
+
+ y 
+\begin_inset Formula $f(u,t)>\frac{c}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $(u,t)\in[a,b]\times(-\varepsilon_{0},\varepsilon_{0})$
+\end_inset
+
+.
+ En este intervalo, 
+\begin_inset Formula $L'(t)$
+\end_inset
+
+ está definida, y para 
+\begin_inset Formula $t\in(-\varepsilon_{0},\varepsilon_{0})$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{align*}
+L'(t) & =\int_{a}^{b}\frac{\partial f}{\partial t}(u,t)du=\int_{a}^{b}\frac{\partial}{\partial t}\left(\sqrt{\left\langle \frac{\partial\phi}{\partial u},\frac{\partial\phi}{\partial u}\right\rangle (u,t)}\right)du=\\
+ & =\int_{a}^{b}-\frac{2\left\langle \frac{\partial^{2}\phi}{\partial u\partial t},\frac{\partial\phi}{\partial t}\right\rangle }{2\sqrt{\left\langle \frac{\partial\phi}{\partial u},\frac{\partial\phi}{\partial u}\right\rangle }}(u,t)du=\int_{a}^{b}\frac{1}{\Vert\alpha'_{t}(u)\Vert}\left\langle \frac{\partial^{2}\phi}{\partial t\partial u},\frac{\partial\phi}{\partial t}\right\rangle (u,t)du,
+\end{align*}
+
+\end_inset
+
+pero
+\begin_inset Formula 
+\[
+\frac{\partial}{\partial u}\left(\left\langle \frac{\partial\phi}{\partial t},\frac{\partial\phi}{\partial u}\right\rangle \right)=\left\langle \frac{\partial^{2}\phi}{\partial t\partial u},\frac{\partial^{2}\phi}{\partial u}\right\rangle +\left\langle \frac{\partial\phi}{\partial t},\frac{\partial^{2}\phi}{\partial u^{2}}\right\rangle ,
+\]
+
+\end_inset
+
+y despejando y sustituyendo se obtiene el resultado.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Primera fórmula de variación del arco:
+\series default
+ Si 
+\begin_inset Formula $\alpha:[a,b]\to S$
+\end_inset
+
+ es un segmento de curva regular p.p.a.
+ con 
+\begin_inset Formula $a<b$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es una variación de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ con campo variacional 
+\begin_inset Formula $Z$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+L'(0)=\langle Z(b),\alpha'(b)\rangle-\langle Z(a),\alpha'(a)\rangle-\int_{a}^{b}\left\langle Z,\frac{D\alpha'}{ds}\right\rangle ,
+\]
+
+\end_inset
+
+por lo que si además la variación es propia o normal,
+\begin_inset Formula 
+\[
+L'(0)=-\int_{a}^{b}\left\langle Z,\frac{D\alpha'}{ds}\right\rangle .
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Usando la fórmula anterior y que 
+\begin_inset Formula $\Vert\alpha'\Vert\equiv1$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{align*}
+L'(0) & =\int_{a}^{b}\left(\frac{d}{ds}\left\langle \frac{\partial\phi}{\partial t},\frac{\partial\phi}{\partial s}\right\rangle -\left\langle \frac{\partial\phi}{\partial t},\frac{\partial^{2}\phi}{\partial s^{2}}\right\rangle \right)(s,0)ds\\
+ & =\left[\left\langle \frac{\partial\phi}{\partial t},\frac{\partial\phi}{\partial s}\right\rangle (s,0)\right]_{s=a}^{b}-\int_{a}^{b}\left\langle \frac{\partial\phi}{\partial t},\frac{\partial^{2}\phi}{\partial s^{2}}\right\rangle (s,0)ds\\
+ & =\left[\langle Z(s),\alpha'(s)\rangle\right]_{s=a}^{b}-\int_{a}^{b}\left\langle Z,\alpha''\right\rangle (s,0)ds,
+\end{align*}
+
+\end_inset
+
+pero como, para 
+\begin_inset Formula $s\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $Z(s)\in T_{\alpha(s)}S$
+\end_inset
+
+, 
+\begin_inset Formula $\langle Z(s),\alpha''(s)\rangle=\langle Z(s),\frac{D\alpha'}{ds}(s)\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Caracterización variaciones de las geodésicas:
+\series default
+ Si 
+\begin_inset Formula $\alpha:[a,b]\to S$
+\end_inset
+
+ es un segmento de curva regular p.p.a., 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ es un segmento de geodésica si y sólo si 
+\begin_inset Formula $L'(0)=0$
+\end_inset
+
+ para toda variación propia de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $L'(0)=0$
+\end_inset
+
+ para toda variación normal de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2,3]$
+\end_inset
+
+ Como 
+\begin_inset Formula $\frac{D\alpha'}{ds}\equiv0$
+\end_inset
+
+, por la primera fórmula de variación del arco para variaciones propias
+ o normales, para todas estas variaciones es 
+\begin_inset Formula $L'(0)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2,3\implies1]$
+\end_inset
+
+ Suponemos que 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ no es geodésica y encontramos una variación normal y propia con 
+\begin_inset Formula $L'(0)\neq0$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ no es geodésica, existe 
+\begin_inset Formula $s_{0}\in[a,b]$
+\end_inset
+
+ con 
+\begin_inset Formula $\frac{D\alpha'}{ds}(s_{0})\neq0$
+\end_inset
+
+, y podemos suponer que 
+\begin_inset Formula $s_{0}\in(a,b)$
+\end_inset
+
+ ya que en otro caso habrá un 
+\begin_inset Formula $s\in(a,b)$
+\end_inset
+
+ con 
+\begin_inset Formula $\frac{D\alpha'}{ds}(s_{0})\neq0$
+\end_inset
+
+ por continuidad.
+ Entonces existe un 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\left\Vert \frac{D\alpha'}{ds}\right\Vert >0$
+\end_inset
+
+ para todo 
+\begin_inset Formula $s\in(s_{0}-\delta,s_{0}+\delta)$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $Z:[a,b]\to\mathbb{R}^{3}$
+\end_inset
+
+ el campo tangente dado por 
+\begin_inset Formula $Z(s):=-(s^{2}-s(a+b)+ab)\frac{D\alpha'}{ds}(s)$
+\end_inset
+
+, si existe una variación 
+\begin_inset Formula $\phi:[a,b]\times(-\varepsilon,\varepsilon)\to S$
+\end_inset
+
+ de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ con campo variacional 
+\begin_inset Formula $Z$
+\end_inset
+
+, 
+\begin_inset Formula $\phi$
+\end_inset
+
+ sería normal porque, al ser 
+\begin_inset Formula $\langle\frac{D\alpha'}{ds},\alpha'\rangle\equiv0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\langle Z,\alpha'\rangle\equiv0$
+\end_inset
+
+.
+ Además, 
+\begin_inset Formula $f(s):=s^{2}-s(a+b)+ab$
+\end_inset
+
+ es una parábola que vale 0 en 
+\begin_inset Formula $s=a,b$
+\end_inset
+
+, cuyo pico está en 
+\begin_inset Formula $s=\frac{a+b}{2}$
+\end_inset
+
+ y que cumple que 
+\begin_inset Formula 
+\[
+f(\frac{a+b}{2})=\frac{(a+b)^{2}}{4}-\frac{(a+b)^{2}}{2}+ab=\frac{-(a+b)^{2}+4ab}{4}=\frac{-(a-b)^{2}}{4}\overset{a\neq b}{<}0,
+\]
+
+\end_inset
+
+de modo que 
+\begin_inset Formula $f(s)<0$
+\end_inset
+
+ para todo 
+\begin_inset Formula $s\in(a,b)$
+\end_inset
+
+ y
+\begin_inset Formula 
+\begin{align*}
+L'(0) & =-\int_{a}^{b}\left\langle -f\frac{D\alpha'}{ds},\frac{D\alpha'}{ds}\right\rangle =\int_{a}^{b}f\left\Vert \frac{D\alpha'}{ds}\right\Vert \leq\\
+ & \leq\int_{s_{0}-\delta}^{s_{0}+\delta}f\left\Vert \frac{D\alpha'}{ds}\right\Vert =2\delta f(\xi)\left\Vert \frac{D\alpha'}{ds}(\xi)\right\Vert <0,
+\end{align*}
+
+\end_inset
+
+donde 
+\begin_inset Formula $\xi\in(s_{0}-\delta,s_{0}+\delta)$
+\end_inset
+
+ viene dado por el teorema del punto medio.
+ Queda ver que tal variación existe y que, además de normal, es propia.
+ Para 
+\begin_inset Formula $s\in[a,b]$
+\end_inset
+
+, como 
+\begin_inset Formula $Z(s)\in T_{\alpha(s)}S$
+\end_inset
+
+, existe una geodésica 
+\begin_inset Formula $\gamma_{Z(s)}:I_{Z(s)}\to S$
+\end_inset
+
+, y como 
+\begin_inset Formula $0\in I_{Z(s)}$
+\end_inset
+
+, existe 
+\begin_inset Formula $\varepsilon_{s}>0$
+\end_inset
+
+ con 
+\begin_inset Formula $(-\varepsilon_{s},\varepsilon_{s})\subseteq I_{Z(s)}$
+\end_inset
+
+.
+ Por la forma en que se obtiene 
+\begin_inset Formula $\gamma_{Z(s)}$
+\end_inset
+
+ y por el teorema de dependencia de una solución de una e.d.o.
+ respecto a un parámetro
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+No recuerdo haber visto este teorema.
+\end_layout
+
+\end_inset
+
+, 
+\begin_inset Formula $s\mapsto\varepsilon_{s}$
+\end_inset
+
+ es continua, de modo que por compacidad de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ existe 
+\begin_inset Formula $\varepsilon:=\min_{s\in[a,b]}\varepsilon_{s}>0$
+\end_inset
+
+, 
+\begin_inset Formula $(-\varepsilon,\varepsilon)\subseteq I_{Z(s)}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $s\in[a,b]$
+\end_inset
+
+ y podemos definir 
+\begin_inset Formula $\phi:[a,b]\times(-\varepsilon,\varepsilon)\to S$
+\end_inset
+
+ como 
+\begin_inset Formula $\phi(s,t):=\gamma_{Z(s)}(t)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es diferenciable por el mismo teorema de dependencia y su campo variacional
+ es
+\begin_inset Formula 
+\[
+\frac{\partial\phi}{\partial t}(s,0)=\frac{d}{dt}(\gamma_{Z(s)}(t))(0)=\gamma'_{Z(s)}(0)=Z(s).
+\]
+
+\end_inset
+
+Finalmente, como 
+\begin_inset Formula $f(a)=f(b)=0$
+\end_inset
+
+, para todo 
+\begin_inset Formula $t$
+\end_inset
+
+, 
+\begin_inset Formula $\phi(a,t)=\gamma_{Z(a)}(t)=\gamma_{0}(t)=\exp_{\alpha(a)}(0)=\alpha(a)$
+\end_inset
+
+, y análogamente 
+\begin_inset Formula $\phi(b,t)=\alpha(b)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Segunda fórmula de variación del arco
+\end_layout
+
+\begin_layout Standard
+Esta afirma que, si 
+\begin_inset Formula $S$
+\end_inset
+
+ es una superficie regular, 
+\begin_inset Formula $\gamma:[a,b]\to S$
+\end_inset
+
+ es un segmento de geodésica p.p.a.
+ y 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es una variación normal y propia de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ con campo variacional 
+\begin_inset Formula $Z$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+L''(0)=\int_{a}^{b}\left(\left\Vert \frac{DZ}{ds}(s)\right\Vert ^{2}-K(\gamma(s))\Vert Z(s)\Vert^{2}\right)ds=-\int_{a}^{b}\left\langle \frac{D^{2}Z}{ds^{2}}(s)+K(\gamma(s))Z(s),Z(s)\right\rangle ds,
+\]
+
+\end_inset
+
+donde
+\begin_inset Formula 
+\[
+\frac{D^{2}Z}{ds^{2}}(s)=\frac{D}{ds}\left(\frac{DZ}{ds}\right)(s)
+\]
+
+\end_inset
+
+y 
+\begin_inset Formula $K$
+\end_inset
+
+ es la curvatura de Gauss de 
+\begin_inset Formula $S$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un segmento de geodésica 
+\begin_inset Formula $\gamma:[a,b]\to S$
+\end_inset
+
+ con 
+\begin_inset Formula $K\circ\gamma\leq0$
+\end_inset
+
+, toda variación de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ normal y propia con campo variacional no paralelo de cumple 
+\begin_inset Formula $L''(0)>0$
+\end_inset
+
+, por lo que en una superficie llana todo segmento de geodésica de 
+\begin_inset Formula $S$
+\end_inset
+
+ es un mínimo del funcional longitud de arco.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $\phi$
+\end_inset
+
+ el campo variacional, por la segunda fórmula de variación,
+\begin_inset Formula 
+\[
+L''(0)=\int_{a}^{b}\left(\left\Vert \frac{DZ}{ds}(s)\right\Vert ^{2}-K(\gamma(s))\Vert Z(s)\Vert^{2}\right)ds\overset{K\circ\gamma\leq0}{\geq}\int_{a}^{b}\left\Vert \frac{DZ}{ds}(s)\right\Vert ^{2}ds\geq0,
+\]
+
+\end_inset
+
+pero si 
+\begin_inset Formula $L''(0)=0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\left\Vert \frac{DZ}{ds}\right\Vert ^{2}\equiv0$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{DZ}{ds}\equiv0$
+\end_inset
+
+ y 
+\begin_inset Formula $Z$
+\end_inset
+
+ es paralelo a lo largo de 
+\begin_inset Formula $\gamma$
+\end_inset
+
+, luego en este caso 
+\begin_inset Formula $L''(0)>0$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document