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\input{../defs}
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\index Index
\shortcut idx
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\end_header

\begin_body

\begin_layout Standard
Dada una superficie regular 
\begin_inset Formula $S$
\end_inset

, un 
\series bold
segmento de curva diferenciable
\series default
 en 
\begin_inset Formula $S$
\end_inset

 es una función 
\begin_inset Formula $\alpha:[a,b]\to S$
\end_inset

 para la que existen 
\begin_inset Formula $\varepsilon>0$
\end_inset

 y una curva diferenciable (no necesariamente regular) 
\begin_inset Formula $\beta:(a-\varepsilon,b+\varepsilon)\to S$
\end_inset

 de modo que 
\begin_inset Formula $\beta|_{[a,b]}=\alpha$
\end_inset

.
\end_layout

\begin_layout Standard
Un 
\series bold
segmento de curva diferenciable a trozos
\series default
 en 
\begin_inset Formula $S$
\end_inset

 es una función continua 
\begin_inset Formula $\alpha:[a,b]\to S$
\end_inset

 para la que existe una partición 
\begin_inset Formula $a=t_{0}<\dots<t_{k}=b$
\end_inset

 tal que, para 
\begin_inset Formula $i\in\{1,\dots,k\}$
\end_inset

, 
\begin_inset Formula $\alpha_{i}\coloneqq \alpha|_{[t_{i-1},t_{i}]}$
\end_inset

 es un segmento de curva diferenciable.
 Entonces, para 
\begin_inset Formula $i\in\{1,\dots,k-1\}$
\end_inset

, llamamos 
\begin_inset Formula $\alpha'_{-}(t_{i})=\lim_{t\to t_{i}^{-}}\alpha'(t)=\alpha'_{i}(t_{i})$
\end_inset

 y 
\begin_inset Formula $\alpha'_{+}(t_{i})=\lim_{t\to t_{i}^{+}}\alpha'(t)=\alpha'_{i+1}(t_{i})$
\end_inset

.
 Entonces 
\begin_inset Formula $\alpha(t_{i})$
\end_inset

 es un 
\series bold
vértice
\series default
 de 
\begin_inset Formula $\alpha$
\end_inset

 si 
\begin_inset Formula $\alpha'_{-}(t_{i})\neq\alpha'_{+}(t_{i})$
\end_inset

.
\end_layout

\begin_layout Section
Distancia
\end_layout

\begin_layout Standard
Si 
\begin_inset Formula $S$
\end_inset

 es una superficie regular conexa, dados 
\begin_inset Formula $p,q\in S$
\end_inset

, llamamos 
\begin_inset Formula $\Omega(p,q)$
\end_inset

 al conjunto de segmentos de curvas diferenciables a trozos 
\begin_inset Formula $\alpha:[a,b]\to S$
\end_inset

 con 
\begin_inset Formula $\alpha(a)=p$
\end_inset

 y 
\begin_inset Formula $\alpha(b)=q$
\end_inset

, que no es vacío, y llamamos 
\series bold
distancia intrínseca
\series default
 en 
\begin_inset Formula $S$
\end_inset

 a la 
\begin_inset Formula $d:S\times S\to\mathbb{R}$
\end_inset

 dada por
\begin_inset Formula 
\[
d(p,q):=\inf_{\alpha\in\Omega(p,q)}L(\alpha),
\]

\end_inset

siendo 
\begin_inset Formula $L(\alpha)$
\end_inset

 la longitud de 
\begin_inset Formula $\alpha$
\end_inset

, que es una distancia.
 Además, 
\begin_inset Formula $\Vert p-q\Vert\leq d(p,q)$
\end_inset

 para 
\begin_inset Formula $p,q\in S$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Demostración:
\series default
 Primero hay que ver que está bien definida, es decir, que para 
\begin_inset Formula $p,q\in S$
\end_inset

, 
\begin_inset Formula $\{L(\alpha)\}_{\alpha\in\Omega(p,q)}$
\end_inset

 tiene ínfimo.
 
\end_layout

\begin_layout Standard
Primero vemos que 
\begin_inset Formula $A\coloneqq \{q\in S\mid \Omega(p,q)\neq\emptyset\}=S$
\end_inset

 viendo que es abierto, cerrado y no vacío.
 La curva constante en 
\begin_inset Formula $p$
\end_inset

 está en 
\begin_inset Formula $\Omega(p,p)$
\end_inset

, luego 
\begin_inset Formula $p\in A\neq\emptyset$
\end_inset

.
 
\end_layout

\begin_layout Standard
Para ver que 
\begin_inset Formula $A$
\end_inset

 es abierto, sea 
\begin_inset Formula $q\in A$
\end_inset

, existe 
\begin_inset Formula $\varepsilon>0$
\end_inset

 tal que 
\begin_inset Formula $D(q,\varepsilon)$
\end_inset

 es un entorno normal de 
\begin_inset Formula $q$
\end_inset

, de modo que si 
\begin_inset Formula $\alpha\in\Omega(p,q)$
\end_inset

, para 
\begin_inset Formula $q'\in D(q,\varepsilon)$
\end_inset

, sea 
\begin_inset Formula $\gamma_{q'}$
\end_inset

 el segmento de geodésica que une 
\begin_inset Formula $q$
\end_inset

 con 
\begin_inset Formula $q'$
\end_inset

, entonces 
\begin_inset Formula $\alpha\wedge\gamma_{q'}\in\Omega(p,q')$
\end_inset

, 
\begin_inset Formula $q'\in A$
\end_inset

 y, como 
\begin_inset Formula $q'$
\end_inset

 es arbitrario, 
\begin_inset Formula $D(q,\varepsilon)\subseteq A$
\end_inset

.
 
\end_layout

\begin_layout Standard
Para ver que 
\begin_inset Formula $A$
\end_inset

 es cerrado, vemos que 
\begin_inset Formula $A^{\complement}=S\setminus A$
\end_inset

 es abierto.
 Sea 
\begin_inset Formula $q\in A^{\complement}$
\end_inset

 y 
\begin_inset Formula $\varepsilon>0$
\end_inset

 tal que 
\begin_inset Formula $D(q,\varepsilon)$
\end_inset

 es un entorno normal de 
\begin_inset Formula $q$
\end_inset

, entonces 
\begin_inset Formula $D(q,\varepsilon)\subseteq A^{\complement}$
\end_inset

, pues si hubiera 
\begin_inset Formula $q'\in D(q,\varepsilon)\cap A$
\end_inset

, sea 
\begin_inset Formula $\beta\in\Omega(p,q')$
\end_inset

 y 
\begin_inset Formula $\gamma_{q'}$
\end_inset

 el segmento de geodésica que une 
\begin_inset Formula $q$
\end_inset

 con 
\begin_inset Formula $q'$
\end_inset

, entonces 
\begin_inset Formula $\beta\wedge\overline{\gamma_{q'}}\in\Omega(p,q)\#$
\end_inset

.
 Como 
\begin_inset Formula $A$
\end_inset

 es abierto, cerrado y no vacío en el conexo 
\begin_inset Formula $S$
\end_inset

, 
\begin_inset Formula $A=S$
\end_inset

.
\end_layout

\begin_layout Standard
Con esto, como 
\begin_inset Formula $\Omega(p,q)\neq\emptyset$
\end_inset

 y 
\begin_inset Formula $\{L(\alpha)\}_{\alpha\in\Omega(p,q)}$
\end_inset

 está acotado inferiormente por 0, el ínfimo existe.
 Queda ver que 
\begin_inset Formula $d$
\end_inset

 es una distancia.
 Sean 
\begin_inset Formula $p,q,r\in S$
\end_inset

:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $d(p,q)\geq0$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $d(p,q)=0\iff p=q$
\end_inset

.
\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

Por la desigualdad de Cauchy-Schwarz, sean 
\begin_inset Formula $\alpha:[a,b]\to S$
\end_inset

 en 
\begin_inset Formula $\Omega(p,q)$
\end_inset

 y 
\begin_inset Formula $v\coloneqq \frac{\overrightarrow{q-p}}{\Vert\overrightarrow{q-p}\Vert}$
\end_inset

, entonces
\begin_inset Formula 
\begin{align*}
\Vert p-q\Vert & =\langle q-p,v\rangle=\langle q,v\rangle-\langle p,v\rangle=\langle\alpha(b),v\rangle-\langle\alpha(a),v\rangle=\\
 & =\int_{a}^{b}\langle\alpha'(t),v\rangle dt\leq\int_{a}^{b}|\langle\alpha'(t),v\rangle|dt=\int_{a}^{b}\Vert\alpha'(t)\Vert dt=L(\alpha),
\end{align*}

\end_inset

y tomando el ínfimo, 
\begin_inset Formula $\Vert p-q\Vert\leq\inf_{\alpha\in\Omega(p,q)}L(\alpha)=d(p,q)=0$
\end_inset

, luego 
\begin_inset Formula $p=q$
\end_inset

.
\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

Basta tomar la curva constante, de longitud 0.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $d(p,q)=d(q,p)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula $\Omega(q,p)=\{\overline{\alpha}\}_{\alpha\in\Omega(p,q)}$
\end_inset

, pero 
\begin_inset Formula $L(\overline{\alpha})=L(\alpha)$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $d(p,q)\leq d(p,r)+d(r,q)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Para 
\begin_inset Formula $\alpha\in\Omega(p,r)$
\end_inset

 y 
\begin_inset Formula $\beta\in\Omega(r,q)$
\end_inset

, 
\begin_inset Formula $\alpha\wedge\beta\in\Omega(p,q)$
\end_inset

, luego
\begin_inset Formula 
\[
d(p,q)=\inf_{\gamma\in\Omega(p,q)}L(\gamma)\leq L(\alpha\wedge\beta)=L(\alpha)+L(\beta).
\]

\end_inset

Entonces 
\begin_inset Formula $d(p,q)-L(\beta)\leq L(\alpha)$
\end_inset

 y tomando el ínfimo 
\begin_inset Formula $d(p,q)-L(\beta)\leq d(p,r)$
\end_inset

, luego 
\begin_inset Formula $d(p,q)-d(p,r)\leq L(\beta)$
\end_inset

 y tomando el ínfimo 
\begin_inset Formula $d(p,q)-d(p,r)\leq d(r,q)$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Section
Propiedades
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $S$
\end_inset

 una superficie regular conexa y 
\begin_inset Formula $p\in S$
\end_inset

 y 
\begin_inset Formula $r>0$
\end_inset

 con 
\begin_inset Formula ${\cal D}(0_{p},r)\subseteq{\cal D}_{p}$
\end_inset

:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $D(p,r)\subseteq B_{d}(p,r)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Para 
\begin_inset Formula $q\in D(p,r)=\exp_{p}({\cal D}(0_{p},r))$
\end_inset

, existe 
\begin_inset Formula $v\in{\cal D}(0_{p},r)$
\end_inset

, no necesariamente único, con 
\begin_inset Formula $q=\exp_{p}(v)$
\end_inset

.
 Sea entonces 
\begin_inset Formula $\gamma:[0,1]\to D(p,r)$
\end_inset

 el segmento de geodésica radial de 
\begin_inset Formula $p$
\end_inset

 a 
\begin_inset Formula $q$
\end_inset

, como 
\begin_inset Formula $\gamma\in\Omega(p,q)$
\end_inset

,
\begin_inset Formula 
\[
d(p,q)\leq L(\gamma)=\int_{0}^{1}\Vert\gamma'(t)\Vert dt=\Vert\gamma'(0)\Vert=\Vert v\Vert<r,
\]

\end_inset

luego 
\begin_inset Formula $q\in B_{d}(p,r)$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Existe 
\begin_inset Formula $\delta>0$
\end_inset

 tal que, para 
\begin_inset Formula $r\in(0,\delta)$
\end_inset

, 
\begin_inset Formula $D(p,r)$
\end_inset

 es un entorno normal de 
\begin_inset Formula $p$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Existe un entorno 
\begin_inset Formula ${\cal U}$
\end_inset

 estrellado respecto al 0 con 
\begin_inset Formula $\exp_{p}:{\cal U}\to(V\coloneqq \exp_{p}({\cal U}))$
\end_inset

 difeomorfismo, luego existe 
\begin_inset Formula $\delta>0$
\end_inset

 con 
\begin_inset Formula ${\cal D}(0,\delta)\subseteq{\cal U}$
\end_inset

 y, para 
\begin_inset Formula $r<\delta$
\end_inset

, 
\begin_inset Formula ${\cal D}(0,r)\subseteq{\cal U}$
\end_inset

 y 
\begin_inset Formula $\exp_{p}:{\cal D}(0,r)\to D(0,r)$
\end_inset

 es un difeomorfismo.
\end_layout

\end_deeper
\begin_layout Enumerate
Si 
\begin_inset Formula $D(p,r)$
\end_inset

 es normal, 
\begin_inset Formula $D(p,r)=B_{d}(p,r)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Queremos ver que 
\begin_inset Formula $B_{d}(p,r)\subseteq D(p,r)$
\end_inset

.
 Supongamos que esto no ocurre, con lo que existe 
\begin_inset Formula $q\in B_{d}(p,r)\setminus D(p,r)$
\end_inset

.
 Entonces, para 
\begin_inset Formula $\alpha:[a,b]\to S$
\end_inset

 en 
\begin_inset Formula $\Omega(p,q)$
\end_inset

 y 
\begin_inset Formula $r^{*}\in(0,r)$
\end_inset

, como 
\begin_inset Formula $q\notin D(p,r^{*})$
\end_inset

, existe 
\begin_inset Formula $t^{*}\coloneqq \inf\{t\in[a,b]\mid \alpha(t)\notin D(p,r^{*})\}$
\end_inset

, pero 
\begin_inset Formula $t\neq a$
\end_inset

, por tanto 
\begin_inset Formula $t>a$
\end_inset

, existe una sucesión creciente 
\begin_inset Formula $\{t_{n}\}_{n}\subseteq(a,t)$
\end_inset

 que tiende a 
\begin_inset Formula $t$
\end_inset

 y, por continuidad, 
\begin_inset Formula 
\[
p^{*}:=\alpha(t^{*})=\lim_{n}\alpha(t_{n})\in\overline{D(p,r^{*})},
\]

\end_inset

de modo que 
\begin_inset Formula $p^{*}\in\partial D(p,r^{*})=S(p,r^{*})$
\end_inset

.
 Entonces existe 
\begin_inset Formula $v^{*}\in{\cal S}(0,r)$
\end_inset

 con 
\begin_inset Formula $p^{*}=\exp_{p}(v^{*})$
\end_inset

 y 
\begin_inset Formula $\Vert v^{*}\Vert=r^{*}$
\end_inset

.
 Con esto, 
\begin_inset Formula $L(\alpha)\geq L(\alpha|_{[a,t^{*}]})\geq L(\gamma_{p^{*}})=\Vert v^{*}\Vert=r^{*}$
\end_inset

, pero como 
\begin_inset Formula $\alpha\in\Omega(p,q)$
\end_inset

 es arbitrario, 
\begin_inset Formula $d(p,q)\geq r^{*}$
\end_inset

, y como 
\begin_inset Formula $r^{*}\in(0,r)$
\end_inset

 es arbitrario, 
\begin_inset Formula $d(p,q)\geq r\#$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
La topología inducida en 
\begin_inset Formula $S$
\end_inset

 por la usual en 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 coincide con la inducida por la distancia intrínseca en 
\begin_inset Formula $S$
\end_inset

.
 
\series bold
Demostración:
\series default
 Sean 
\begin_inset Formula ${\cal T}_{S}$
\end_inset

 y 
\begin_inset Formula ${\cal T}_{d}$
\end_inset

 respectivamente estas topologías: 
\end_layout

\begin_layout Itemize
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\subseteq]$
\end_inset


\end_layout

\end_inset

Para 
\begin_inset Formula $A\in{\cal T}_{S}$
\end_inset

 y 
\begin_inset Formula $p\in A$
\end_inset

, existe 
\begin_inset Formula $\delta>0$
\end_inset

 con 
\begin_inset Formula $B_{d_{\mathbb{R}^{3}}}(p,\delta)\cap S\subseteq A$
\end_inset

, pero para 
\begin_inset Formula $q\in B_{d}(p,\delta)$
\end_inset

 es 
\begin_inset Formula $\Vert p-q\Vert\leq d(p,q)<\delta$
\end_inset

 y por tanto 
\begin_inset Formula $q\in B_{d_{\mathbb{R}^{3}}}(p,\delta)\cap S\subseteq A$
\end_inset

, luego 
\begin_inset Formula $B_{d}(p,\delta)\subseteq A$
\end_inset

 y, como 
\begin_inset Formula $p$
\end_inset

 es arbitrario, 
\begin_inset Formula $A\in{\cal T}_{d}$
\end_inset

.
\end_layout

\begin_layout Itemize
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\supseteq]$
\end_inset


\end_layout

\end_inset

Para 
\begin_inset Formula $A\in{\cal T}_{d}$
\end_inset

 y 
\begin_inset Formula $p\in A$
\end_inset

, existe 
\begin_inset Formula $\delta_{p}>0$
\end_inset

 con 
\begin_inset Formula $B_{d}(p,\delta_{p})\subseteq A$
\end_inset

, y haciendo 
\begin_inset Formula $\delta$
\end_inset

 suficientemente pequeño, 
\begin_inset Formula $D(p,\delta_{p})$
\end_inset

 es normal e igual a 
\begin_inset Formula $B_{d}(p,\delta_{p})$
\end_inset

, pero 
\begin_inset Formula $D(p,\delta_{p})$
\end_inset

 es abierto en 
\begin_inset Formula ${\cal T}_{S}$
\end_inset

 ya que 
\begin_inset Formula $\exp_{p}:{\cal D}(0_{p},\delta_{p})\to D(p,\delta_{p})$
\end_inset

 es un difeomorfismo, de modo que 
\begin_inset Formula $A=\bigcup_{p\in A}D(p,\delta_{p})\in{\cal T}_{S}$
\end_inset

 por ser unión de abiertos.
\end_layout

\end_body
\end_document