Terminado GCS

1 files changed, 374 insertions, 4 deletions

diff --git a/gcs/n3.lyx b/gcs/n3.lyx
index 90cb5bf..a3246ce 100644
--- a/gcs/n3.lyx
+++ b/gcs/n3.lyx
@@ -4008,11 +4008,31 @@ ecuación de Gauss
  es
 \begin_inset Formula 
 \[
-\Gamma_{11}^{1}\Gamma_{12}^{2}+(\Gamma_{11}^{2})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{2}-\Gamma_{12}^{1}\Gamma_{11}^{2}-(\Gamma_{12}^{2})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{2}=EK.
+\Gamma_{11}^{1}\Gamma_{12}^{2}+(\Gamma_{11}^{2})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{2}-\Gamma_{12}^{1}\Gamma_{11}^{2}-(\Gamma_{12}^{2})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{2}=EK,
 \]
 
 \end_inset
 
+la primera 
+\series bold
+ecuación de Mainardi-Codazzi
+\series default
+ es
+\begin_inset Formula 
+\[
+e\Gamma_{12}^{1}+f(\Gamma_{12}^{2}-\Gamma_{11}^{1})-g\Gamma_{11}^{2}=e_{v}-f_{u}
+\]
+
+\end_inset
+
+y, además,
+\begin_inset Formula 
+\begin{align*}
+(\Gamma_{11}^{1})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{1}-(\Gamma_{12}^{1})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{1} & =-FK.
+\end{align*}
+
+\end_inset
+
  
 \series bold
 Demostración:
@@ -4075,11 +4095,37 @@ Como
 \end_inset
 
 
+\begin_inset Formula $C_{1}=0$
+\end_inset
+
+ nos da
+\begin_inset Formula 
+\begin{multline*}
+\Gamma_{11}^{1}f+\Gamma_{11}^{2}g-\Gamma_{12}^{1}e-\Gamma_{12}^{1}f+e_{v}-f_{u}=0,
+\end{multline*}
+
+\end_inset
+
+de donde se obtiene directamente la primera ecuación de Mainardi-Codazzi,
+ y 
+\begin_inset Formula $A_{1}=0$
+\end_inset
+
+ nos da 
+\begin_inset Formula 
+\begin{multline*}
+(\Gamma_{11}^{1})_{v}+\Gamma_{11}^{1}\Gamma_{12}^{1}+\Gamma_{11}^{2}\Gamma_{22}^{1}-(\Gamma_{12}^{1})_{u}-\Gamma_{12}^{1}\Gamma_{11}^{1}-\Gamma_{12}^{2}\Gamma_{12}^{1}=(\Gamma_{11}^{1})_{v}+\Gamma_{11}^{2}\Gamma_{22}^{1}-(\Gamma_{12}^{1})_{u}-\Gamma_{12}^{2}\Gamma_{12}^{1}=\\
+=fa_{11}-ea_{12}=f\frac{fF-eG}{EG-F^{2}}-e\frac{gF-fG}{EG-F^{2}}=\frac{f^{2}F-egF}{EG-F^{2}}=-FK.
+\end{multline*}
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
-Esto implica la curvatura de Gauss depende solo de la primera forma fundamental,
- pues los símbolos de Christoffel solo dependen de esta y, como 
+La curvatura de Gauss depende solo de la primera forma fundamental, pues
+ como 
 \begin_inset Formula $EG-F^{2}>0$
 \end_inset
 
@@ -4087,20 +4133,344 @@ Esto implica la curvatura de Gauss depende solo de la primera forma fundamental,
 \begin_inset Formula $E\neq0$
 \end_inset
 
+ y por la ecuación de Gauss 
+\begin_inset Formula $K$
+\end_inset
+
+ se puede obtener de 
+\begin_inset Formula $E$
+\end_inset
+
+ y los símbolos de Christoffel, que dependen solo de la primera forma fundamenta
+l.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+\lang latin
+Theorema Egregium
+\lang spanish
+ de Gauss:
+\series default
+ La curvatura de Gauss de una superficie regular es invariante por isometrías
+ locales.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $\phi:S_{1}\to S_{2}$
+\end_inset
+
+ una isometría local entre superficies regulares, 
+\begin_inset Formula $p\in S_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $(U,X)$
+\end_inset
+
+ una parametrización de 
+\begin_inset Formula $S_{1}$
+\end_inset
+
+ en 
+\begin_inset Formula $p$
+\end_inset
+
+ con 
+\begin_inset Formula $U$
+\end_inset
+
+ lo suficientemente pequeña para que 
+\begin_inset Formula $\phi|_{V:=X(U)}:V\to\phi(V)$
+\end_inset
+
+ sea un difeomorfismo, entonces 
+\begin_inset Formula $(U,\overline{X}:=\phi\circ X)$
+\end_inset
+
+ es una parametrización de 
+\begin_inset Formula $S_{2}$
+\end_inset
+
+ en 
+\begin_inset Formula $\phi(p)$
+\end_inset
+
+.
+ Entonces, como los coeficientes de la primera forma fundamental son los
+ mismos para ambas parametrizaciones y la curvatura de Gauss solo depende
+ de estos, las curvaturas de Gauss coinciden para el mismo punto de 
+\begin_inset Formula $U$
+\end_inset
+
+ y en particular 
+\begin_inset Formula $K_{1}(p)=K_{2}(\phi(p))$
+\end_inset
+
+, donde 
+\begin_inset Formula $K_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $K_{2}$
+\end_inset
+
+ son las curvaturas de Gauss respectivas de 
+\begin_inset Formula $S_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $S_{2}$
+\end_inset
+
 .
 \end_layout
 
 \begin_layout Standard
-\begin_inset Note Note
+En general un difeomorfismo local que conserva la curvatura no es una isometría
+ local.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $S_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $S_{2}$
+\end_inset
+
+ parametrizadas por 
+\begin_inset Formula $X(u,v):=(u\cos v,u\sin v,\log u)$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{X}(u,v):=(u\cos v,u\sin v,v)$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\begin{align*}
+X_{u} & =(\cos v,\sin v,\tfrac{1}{u}), & \overline{X}_{u} & =(\cos v,\sin v,0),\\
+X_{v} & =(-u\sin v,u\cos v,0), & \overline{X}_{v} & =(-u\sin v,u\cos v,1),\\
+N & =\frac{(-\cos v,-\sin v,u)}{\sqrt{1+u^{2}}}, & \overline{N} & =\frac{(\sin v,-\cos v,u)}{\sqrt{1+u^{2}}},
+\end{align*}
+
+\end_inset
+
+luego 
+\begin_inset Formula $N$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{N}$
+\end_inset
+
+ se diferencian en alguna transformación ortogonal.
+ Si 
+\begin_inset Formula $\overline{N}=O\circ N$
+\end_inset
+
+ para una transformación ortogonal 
+\begin_inset Formula $O$
+\end_inset
+
+, entonces 
+\begin_inset Formula $d\overline{N}_{q}=dO_{N(q)}\circ dN_{q}=O\circ dN_{q}$
+\end_inset
+
+, luego 
+\begin_inset Formula $d\overline{N}_{q}$
+\end_inset
+
+ y 
+\begin_inset Formula $dN_{q}$
+\end_inset
+
+ se diferencian por 
+\begin_inset Formula $O$
+\end_inset
+
+ y por tanto tienen igual determinante, que será la curvatura de Gauss.
+ Sin embargo, 
+\begin_inset Formula $\phi:=\overline{X}\circ X^{-1}=((x,y,z)\mapsto(x,y,e^{z}))$
+\end_inset
+
+ no es una isometría.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\begin{align*}
+a_{11} & =\frac{fF-eG}{EG-F^{2}}, & a_{12} & =\frac{gF-fG}{EG-F^{2}}, & a_{21} & =\frac{eF-fE}{EG-F^{2}}, & a_{22} & =\frac{fF-gE}{EG-F^{2}}.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La segunda 
+\series bold
+ecuación de Mainardi-Codazzi
+\series default
+ es
+\begin_inset Formula 
+\[
+f_{v}-g_{u}=e\Gamma_{22}^{1}+f(\Gamma_{22}^{2}-\Gamma_{12}^{1})-g\Gamma_{12}^{2}.
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Como 
+\begin_inset Formula $X_{vvu}=X_{vuv}$
+\end_inset
+
+, aplicando las fórmulas de Gauss,
+\begin_inset Note Comment
 status open
 
 \begin_layout Plain Layout
 
+\lang english
+\begin_inset Formula $gN_{u}$
+\end_inset
+
+ is not Unix.
 \end_layout
 
 \end_inset
 
 
+\begin_inset Formula 
+\begin{multline*}
+0=X_{vvu}-X_{vuv}=(\Gamma_{22}^{1})_{u}X_{u}+\Gamma_{22}^{1}X_{uu}+(\Gamma_{22}^{2})_{u}X_{v}+\Gamma_{22}^{2}X_{vu}+g_{u}N+gN_{u}-\\
+-(\Gamma_{21}^{1})_{v}X_{u}-\Gamma_{21}^{1}X_{uv}-(\Gamma_{21}^{2})_{v}X_{v}-\Gamma_{21}^{2}X_{vv}-f_{v}N-fN_{v},
+\end{multline*}
+
+\end_inset
+
+y sustituyendo de nuevo,
+\begin_inset Formula 
+\begin{multline*}
+0=(\Gamma_{22}^{1})_{u}X_{u}+\Gamma_{22}^{1}(\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN)+(\Gamma_{22}^{2})_{u}X_{v}+\Gamma_{22}^{2}(\Gamma_{12}^{2}X_{u}+\Gamma_{12}^{2}X_{v}+fN)-\\
+-(\Gamma_{12}^{1})_{v}X_{u}-\Gamma_{12}^{1}(\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN)-(\Gamma_{12}^{2})_{v}X_{v}-\Gamma_{12}^{2}(\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN)+\\
++g_{u}N+g(a_{11}X_{u}+a_{21}X_{v})-f_{v}N-f(a_{12}X_{u}+a_{22}X_{v})=:A_{2}X_{u}+B_{2}X_{v}+C_{2}N.
+\end{multline*}
+
+\end_inset
+
+Como antes, 
+\begin_inset Formula $A_{2},B_{2},C_{2}=0$
+\end_inset
+
+, luego como 
+\begin_inset Formula $C_{2}=0$
+\end_inset
+
+, 
+\begin_inset Formula $e\Gamma_{22}^{1}+f\Gamma_{22}^{2}-f\Gamma_{12}^{1}-g\Gamma_{12}^{2}=f_{v}-g_{u}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Las 
+\series bold
+ecuaciones de compatibilidad
+\series default
+ son la ecuación de Gauss y las dos ecuaciones de Mainardi-Codazzi.
+ 
+\series bold
+Teorema de Bonnet:
+\series default
+ Sean 
+\begin_inset Formula $E,F,G,e,f,g:V\to\mathbb{R}$
+\end_inset
+
+ funciones diferenciables en un abierto 
+\begin_inset Formula $V\subseteq\mathbb{R}^{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $E>0$
+\end_inset
+
+, 
+\begin_inset Formula $G>0$
+\end_inset
+
+, 
+\begin_inset Formula $EG-F^{2}>0$
+\end_inset
+
+ y que verifican las ecuaciones de compatibilidad, entonces existen un abierto
+ 
+\begin_inset Formula $U\subseteq V$
+\end_inset
+
+ y un difeomorfismo 
+\begin_inset Formula $X:U\to X(U)\subseteq\mathbb{R}^{3}$
+\end_inset
+
+ tales que 
+\begin_inset Formula $(U,X)$
+\end_inset
+
+ es una parametrización de la superficie regular 
+\begin_inset Formula $X(U)$
+\end_inset
+
+ en la que los coeficientes de la primera y segunda formas fundamentales
+ son 
+\begin_inset Formula $E,F,G$
+\end_inset
+
+ y 
+\begin_inset Formula $e,f,g$
+\end_inset
+
+, respectivamente, y si 
+\begin_inset Formula $U$
+\end_inset
+
+ es conexo y 
+\begin_inset Formula $\overline{X}:U\to\overline{X}(U)$
+\end_inset
+
+ es otro difeomorfismo con los mismos coeficientes de las formas fundamentales
+ primera y segunda, entonces existe un movimiento rígido 
+\begin_inset Formula $M$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{R}^{3}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\overline{X}=M\circ X$
+\end_inset
+
+.
 \end_layout
 
 \end_body