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| -rw-r--r-- | gcs/n3.lyx | 378 |
1 files changed, 374 insertions, 4 deletions
@@ -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 |