diff options
Diffstat (limited to 'ggs/n3.lyx')
| -rw-r--r-- | ggs/n3.lyx | 80 |
1 files changed, 40 insertions, 40 deletions
@@ -110,7 +110,7 @@ aplicación exponencial \end_inset donde -\begin_inset Formula ${\cal D}_{p}:=\{v\in T_{p}S\mid 1\in I_{v}\}$ +\begin_inset Formula ${\cal D}_{p}\coloneqq \{v\in T_{p}S\mid 1\in I_{v}\}$ \end_inset . @@ -278,7 +278,7 @@ Como \end_inset , sea -\begin_inset Formula $\alpha(t):=tw$ +\begin_inset Formula $\alpha(t)\coloneqq tw$ \end_inset , existe @@ -377,11 +377,11 @@ entorno normal \end_inset , sean -\begin_inset Formula $v_{p}:=\exp_{p_{0}}^{-1}(p)\in{\cal U}$ +\begin_inset Formula $v_{p}\coloneqq \exp_{p_{0}}^{-1}(p)\in{\cal U}$ \end_inset y el segmento de geodésica -\begin_inset Formula $\gamma_{p}:=\gamma_{v_{p}}|_{[0,1]}:[0,1]\to V$ +\begin_inset Formula $\gamma_{p}\coloneqq \gamma_{v_{p}}|_{[0,1]}:[0,1]\to V$ \end_inset , entonces @@ -487,7 +487,7 @@ Demostración: \end_inset dada por -\begin_inset Formula $\alpha(t):=v+tw=(1+\lambda t)v$ +\begin_inset Formula $\alpha(t)\coloneqq v+tw=(1+\lambda t)v$ \end_inset , entonces @@ -512,7 +512,7 @@ Para el caso general, sea \end_inset dada por -\begin_inset Formula $\tau(s,t):=s\alpha(t):=s(v+tw)$ +\begin_inset Formula $\tau(s,t)\coloneqq s\alpha(t)\coloneqq s(v+tw)$ \end_inset , para todo @@ -591,7 +591,7 @@ Como . Proyectando el subrecubrimiento -\begin_inset Formula $A:=\bigcup_{i=1}^{k}B_{\infty}((s_{i},0),\varepsilon_{s_{i}})$ +\begin_inset Formula $A\coloneqq \bigcup_{i=1}^{k}B_{\infty}((s_{i},0),\varepsilon_{s_{i}})$ \end_inset en @@ -608,7 +608,7 @@ Como . Sea -\begin_inset Formula $\varepsilon:=\min\{\varepsilon_{s_{1}},\dots,\varepsilon_{s_{k}},\varepsilon'\}$ +\begin_inset Formula $\varepsilon\coloneqq \min\{\varepsilon_{s_{1}},\dots,\varepsilon_{s_{k}},\varepsilon'\}$ \end_inset , para @@ -632,7 +632,7 @@ luego \begin_layout Standard Sea ahora -\begin_inset Formula $\varphi:=\exp_{p}\circ\tau:(-\varepsilon,1+\varepsilon)\times(-\varepsilon,\varepsilon)\to S$ +\begin_inset Formula $\varphi\coloneqq \exp_{p}\circ\tau:(-\varepsilon,1+\varepsilon)\times(-\varepsilon,\varepsilon)\to S$ \end_inset . @@ -733,7 +733,7 @@ pues \end_inset dada por -\begin_inset Formula $\beta_{s}(t):=\exp_{p}(s\alpha(t))$ +\begin_inset Formula $\beta_{s}(t)\coloneqq \exp_{p}(s\alpha(t))$ \end_inset , @@ -909,7 +909,7 @@ Sean \end_inset tal que -\begin_inset Formula ${\cal D}(0,r):=\{v\in T_{p}S\mid \Vert v\Vert<r\}\subseteq{\cal D}_{p}$ +\begin_inset Formula ${\cal D}(0,r)\coloneqq \{v\in T_{p}S\mid \Vert v\Vert<r\}\subseteq{\cal D}_{p}$ \end_inset , llamamos @@ -925,7 +925,7 @@ disco geodésico \end_inset a -\begin_inset Formula $D(p,r):=\exp_{p}({\cal D}(0,r))$ +\begin_inset Formula $D(p,r)\coloneqq \exp_{p}({\cal D}(0,r))$ \end_inset , y si @@ -933,7 +933,7 @@ disco geodésico \end_inset cumple que -\begin_inset Formula ${\cal S}(0,r):=\{v\in T_{p}S\mid \Vert v\Vert=r\}\subseteq{\cal D}_{p}$ +\begin_inset Formula ${\cal S}(0,r)\coloneqq \{v\in T_{p}S\mid \Vert v\Vert=r\}\subseteq{\cal D}_{p}$ \end_inset , llamamos @@ -949,7 +949,7 @@ circunferencia geodésica \end_inset a -\begin_inset Formula $S(p,r):=\exp_{p}({\cal S}(0,r))$ +\begin_inset Formula $S(p,r)\coloneqq \exp_{p}({\cal S}(0,r))$ \end_inset . @@ -1042,7 +1042,7 @@ teorema Demostración: \series default Sea -\begin_inset Formula $v_{p}:=\exp_{p_{0}}^{-1}(p)$ +\begin_inset Formula $v_{p}\coloneqq \exp_{p_{0}}^{-1}(p)$ \end_inset , entonces @@ -1075,11 +1075,11 @@ Sea \begin_layout Standard Sean -\begin_inset Formula $A:=\alpha^{-1}(\{p_{0}\})$ +\begin_inset Formula $A\coloneqq \alpha^{-1}(\{p_{0}\})$ \end_inset y -\begin_inset Formula $t_{0}:=\sup A$ +\begin_inset Formula $t_{0}\coloneqq \sup A$ \end_inset , existe una sucesión @@ -1132,7 +1132,7 @@ Sean \end_inset , basta demostrar la propiedad para -\begin_inset Formula $\alpha:=\alpha'$ +\begin_inset Formula $\alpha\coloneqq \alpha'$ \end_inset . @@ -1148,7 +1148,7 @@ Sean \end_inset y -\begin_inset Formula $\tilde{\alpha}:=\exp_{p_{0}}^{-1}\circ\alpha:[0,1]\to{\cal U}$ +\begin_inset Formula $\tilde{\alpha}\coloneqq \exp_{p_{0}}^{-1}\circ\alpha:[0,1]\to{\cal U}$ \end_inset , que cumple @@ -1165,7 +1165,7 @@ Sean . Sean entonces -\begin_inset Formula $r(t):=\Vert\tilde{\alpha}(t)\Vert$ +\begin_inset Formula $r(t)\coloneqq \Vert\tilde{\alpha}(t)\Vert$ \end_inset y, para @@ -1173,7 +1173,7 @@ Sean \end_inset , -\begin_inset Formula $V(t):=\frac{\tilde{\alpha}(t)}{\Vert\tilde{\alpha}(t)\Vert}$ +\begin_inset Formula $V(t)\coloneqq \frac{\tilde{\alpha}(t)}{\Vert\tilde{\alpha}(t)\Vert}$ \end_inset , de modo que @@ -1392,7 +1392,7 @@ Finalmente, sea . En otro caso, sea -\begin_inset Formula $r^{*}:=\frac{r+\Vert v_{p}\Vert}{2}$ +\begin_inset Formula $r^{*}\coloneqq \frac{r+\Vert v_{p}\Vert}{2}$ \end_inset , de modo que @@ -1400,7 +1400,7 @@ Finalmente, sea \end_inset , y si -\begin_inset Formula $\tilde{\alpha}:=\exp_{p_{0}}^{-1}\circ\alpha$ +\begin_inset Formula $\tilde{\alpha}\coloneqq \exp_{p_{0}}^{-1}\circ\alpha$ \end_inset , como @@ -1422,7 +1422,7 @@ Finalmente, sea es \begin_inset Formula \[ -A:=\{t\in(a,b)\mid \Vert\tilde{\alpha}(t)\Vert=r^{*}\}=\{t\in[a,b]\mid \alpha(t)\in S(p_{0},r^{*})\}\neq\emptyset. +A:=\{t\in(a,b)\mid \Vert\tilde{\alpha}(t)\Vert=r^{*}\}=\{t\in[a,b]\mid\alpha(t)\in S(p_{0},r^{*})\}\neq\emptyset. \] \end_inset @@ -1436,11 +1436,11 @@ Entonces, como \end_inset también lo es y existe -\begin_inset Formula $t^{*}:=\min A$ +\begin_inset Formula $t^{*}\coloneqq \min A$ \end_inset , y llamando -\begin_inset Formula $p^{*}:=\alpha(t^{*})\in S(p_{0},r^{*})$ +\begin_inset Formula $p^{*}\coloneqq \alpha(t^{*})\in S(p_{0},r^{*})$ \end_inset , @@ -1492,7 +1492,7 @@ Sean \end_inset y -\begin_inset Formula $U:=\phi^{-1}({\cal U})$ +\begin_inset Formula $U\coloneqq \phi^{-1}({\cal U})$ \end_inset es abierto en @@ -1504,7 +1504,7 @@ Sean \end_inset dada por -\begin_inset Formula $X(u,v):=\exp_{p_{0}}(\phi(u,v))$ +\begin_inset Formula $X(u,v)\coloneqq \exp_{p_{0}}(\phi(u,v))$ \end_inset es una parametrización llamada @@ -1626,7 +1626,7 @@ Sean \end_inset , -\begin_inset Formula $\ell:=\{\lambda e_{1}\}_{\lambda\geq0}$ +\begin_inset Formula $\ell\coloneqq \{\lambda e_{1}\}_{\lambda\geq0}$ \end_inset , @@ -1642,11 +1642,11 @@ Sean \end_inset -\begin_inset Formula $V_{0}:=\exp_{p_{0}}({\cal U}\setminus\ell)$ +\begin_inset Formula $V_{0}\coloneqq \exp_{p_{0}}({\cal U}\setminus\ell)$ \end_inset y -\begin_inset Formula $U_{0}:=\phi^{-1}({\cal U}\setminus\ell)$ +\begin_inset Formula $U_{0}\coloneqq \phi^{-1}({\cal U}\setminus\ell)$ \end_inset , entonces @@ -1654,7 +1654,7 @@ Sean \end_inset dado por -\begin_inset Formula $X(r,\theta):=\exp_{p_{0}}(\phi(r,\theta))$ +\begin_inset Formula $X(r,\theta)\coloneqq \exp_{p_{0}}(\phi(r,\theta))$ \end_inset es una parametrización llamada @@ -1710,7 +1710,7 @@ teorema \begin_deeper \begin_layout Standard Sea -\begin_inset Formula $v_{\theta}:=(\cos\theta e_{1}+\sin\theta e_{2})$ +\begin_inset Formula $v_{\theta}\coloneqq (\cos\theta e_{1}+\sin\theta e_{2})$ \end_inset , de modo que @@ -1830,7 +1830,7 @@ Para un \begin_deeper \begin_layout Standard Sean -\begin_inset Formula $\overline{X}(u,v):=\exp_{p_{0}}(ue_{1}+ve_{2})$ +\begin_inset Formula $\overline{X}(u,v)\coloneqq \exp_{p_{0}}(ue_{1}+ve_{2})$ \end_inset la parametrización normal centrada en @@ -1846,7 +1846,7 @@ Sean \end_inset los parámetros de su primera forma fundamental, como -\begin_inset Formula $X(r,\theta)=\overline{X}(r_{\theta}):=\overline{X}(r\cos\theta,r\sin\theta)$ +\begin_inset Formula $X(r,\theta)=\overline{X}(r_{\theta})\coloneqq \overline{X}(r\cos\theta,r\sin\theta)$ \end_inset , se tiene @@ -1989,7 +1989,7 @@ Fijado \end_inset , sea -\begin_inset Formula $u(r):=\sqrt{G(r,\theta)}$ +\begin_inset Formula $u(r)\coloneqq \sqrt{G(r,\theta)}$ \end_inset , de modo que @@ -2212,11 +2212,11 @@ Demostración: \end_inset , -\begin_inset Formula $V_{1}:=D(p_{1},\varepsilon)$ +\begin_inset Formula $V_{1}\coloneqq D(p_{1},\varepsilon)$ \end_inset y -\begin_inset Formula $V_{2}:=D(p_{2},\varepsilon)$ +\begin_inset Formula $V_{2}\coloneqq D(p_{2},\varepsilon)$ \end_inset , entonces @@ -2252,11 +2252,11 @@ Sean ahora \end_inset una isometría lineal dada por -\begin_inset Formula $\tilde{\varphi}(e_{1}):=f_{1}$ +\begin_inset Formula $\tilde{\varphi}(e_{1})\coloneqq f_{1}$ \end_inset y -\begin_inset Formula $\tilde{\varphi}(e_{2}):=f_{2}$ +\begin_inset Formula $\tilde{\varphi}(e_{2})\coloneqq f_{2}$ \end_inset , entonces |