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@@ -2051,5 +2051,937 @@ Si Curvaturas principales \end_layout +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{AAlG} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Toda matriz simétrica real +\begin_inset Formula $A\in{\cal M}_{m}(\mathbb{R})$ +\end_inset + + admite una matriz ortogonal +\begin_inset Formula $P$ +\end_inset + + tal que +\begin_inset Formula $P^{-1}AP=P^{t}AP$ +\end_inset + + es diagonal. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dados una superficie regular +\begin_inset Formula $S$ +\end_inset + + orientada y +\begin_inset Formula $p\in S$ +\end_inset + +, existe una base ortonormal +\begin_inset Formula $(e_{1},e_{2})$ +\end_inset + + en la que +\begin_inset Formula $A_{p}$ +\end_inset + + es diagonal, pues +\begin_inset Formula $A_{p}$ +\end_inset + + es simétrica. + Si +\begin_inset Formula $\kappa_{1}(p)$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)$ +\end_inset + + son los valores propios asociados respectivamente a +\begin_inset Formula $e_{1}$ +\end_inset + + y +\begin_inset Formula $e_{2}$ +\end_inset + +, podemos suponer que +\begin_inset Formula $\kappa_{1}(p)\leq\kappa_{2}(p)$ +\end_inset + +, y llamamos +\series bold +curvaturas principales +\series default + de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + a +\begin_inset Formula $\kappa_{1}(p)$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)$ +\end_inset + + y +\series bold +direcciones principales +\series default + a +\begin_inset Formula $e_{1}$ +\end_inset + + y +\begin_inset Formula $e_{2}$ +\end_inset + +, o a todos los vectores unitarios de +\begin_inset Formula $T_{p}S$ +\end_inset + + si +\begin_inset Formula $\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +, pues en tal caso todos los vectores no nulos son propios al ser +\begin_inset Formula $A_{p}$ +\end_inset + + una homotecia. + Se tiene +\begin_inset Formula $\kappa_{1}(p)=\kappa_{n}(e_{1},p)$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)=\kappa_{n}(e_{2},p)$ +\end_inset + +, pues +\begin_inset Formula $\kappa_{n}(e_{i},p)=\langle A_{p}e_{i},e_{i}\rangle=\langle\kappa_{i}(p)e_{i},e_{i}\rangle=\kappa_{i}(p)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Ejemplos: +\end_layout + +\begin_layout Enumerate +Todas las direcciones del plano y la esfera son principales. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $\kappa_{n}$ +\end_inset + + es constante, +\begin_inset Formula $\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +El cilindro +\begin_inset Formula $\{x^{2}+y^{2}=r^{2}\}$ +\end_inset + + tiene como curvaturas principales +\begin_inset Formula $-\frac{1}{r}$ +\end_inset + + y 0. +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $C:=\{x^{2}+y^{2}=r^{2}\}=\{X(u,v):=(r\cos u,r\sin u,v)\}_{u,v\in\mathbb{R}}$ +\end_inset + +, +\begin_inset Formula $p=(x,y,z)\in C$ +\end_inset + + y la orientación +\begin_inset Formula $N(p):=\frac{1}{r}(x,y,0)$ +\end_inset + +, entonces +\begin_inset Formula $X_{u}=(-r\sin u,r\cos u,0)$ +\end_inset + +, +\begin_inset Formula $X_{v}=e_{3}$ +\end_inset + + y +\begin_inset Formula $N(u,v)=(\cos u,\sin u,0)$ +\end_inset + +, luego +\begin_inset Formula $A_{p}=-(-\sin u,\cos u,0)=-\frac{1}{r}X_{u}$ +\end_inset + + y por tanto +\begin_inset Formula $A_{p}\equiv\text{diag}(-\frac{1}{r},0)$ +\end_inset + + con la base +\begin_inset Formula $(X_{u},X_{v})$ +\end_inset + + de +\begin_inset Formula $T_{p}S$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La silla de montar tiene curvaturas principales +\begin_inset Formula $-2$ +\end_inset + + y 2 en el origen. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $A_{p}\equiv\text{diag}(-2,2)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Una +\series bold +línea de curvatura +\series default + en una superficie regular orientada +\begin_inset Formula $S$ +\end_inset + + es una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + tal que +\begin_inset Formula $\alpha'(t)$ +\end_inset + + es una dirección principal de +\begin_inset Formula $\alpha(t)$ +\end_inset + + para todo +\begin_inset Formula $t\in I$ +\end_inset + +. + Si las curvaturas principales son distintas en todo punto de un abierto + +\begin_inset Formula $V\subseteq S$ +\end_inset + +, por cada +\begin_inset Formula $p\in V$ +\end_inset + + pasan dos únicas líneas de curvatura y estas se cortan de forma ortogonal. +\end_layout + +\begin_layout Standard + +\series bold +Fórmula de Euler: +\series default + Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $\kappa_{1}(p)\leq\kappa_{2}(p)$ +\end_inset + + las curvaturas principales de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +, +\begin_inset Formula $e_{1}$ +\end_inset + + y +\begin_inset Formula $e_{2}$ +\end_inset + + las respectivas direcciones principales, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + y +\begin_inset Formula $\theta$ +\end_inset + + tal que +\begin_inset Formula $\cos\theta=\langle e_{1},v\rangle$ +\end_inset + +, entonces +\begin_inset Formula $\kappa_{n}(v,p)=\kappa_{1}(p)\cos^{2}\theta+\kappa_{2}(p)\sin^{2}\theta$ +\end_inset + +. + En efecto, sea +\begin_inset Formula $v=:\cos\omega e_{1}+\sin\omega e_{2}$ +\end_inset + +, +\begin_inset Formula $\kappa_{n}(v,p)=\langle A_{p}v,v\rangle=\langle\kappa_{1}(p)\cos\omega e_{1}+\kappa_{2}(p)\cos\omega e_{2},\cos\omega e_{1}+\sin\omega e_{2}\rangle=\kappa_{1}(p)\cos^{2}\omega+\kappa_{2}(p)\sin^{2}\omega$ +\end_inset + +, y aunque +\begin_inset Formula $\omega=\pm\theta+2k\pi$ +\end_inset + + para algún +\begin_inset Formula $k\in\mathbb{Z}$ +\end_inset + +, el coseno y por tanto el cuadrado del seno coinciden. +\end_layout + +\begin_layout Standard +Con esto, +\begin_inset Formula $\kappa_{1}(p)=\min\{\kappa_{n}(v,p)\}_{|v|=1}$ +\end_inset + + y +\begin_inset Formula $\kappa_{2}(p)=\max\{\kappa_{n}(v,p)\}_{|v|=1}$ +\end_inset + +, pues por la fórmula, si +\begin_inset Formula $|v|=1$ +\end_inset + +, +\begin_inset Formula $\kappa_{n}(v,p)=\kappa_{1}(p)(1-\sin^{2}\theta)+\kappa_{2}(p)\sin^{2}\theta$ +\end_inset + + para algún +\begin_inset Formula $\theta$ +\end_inset + +. + Llamamos +\series bold +curvatura mínima +\series default + a +\begin_inset Formula $\kappa_{1}(p)$ +\end_inset + + y +\series bold +curvatura máxima +\series default + a +\begin_inset Formula $\kappa_{2}(p)$ +\end_inset + +. + La +\series bold +curvatura de Gauss +\series default + de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p\in S$ +\end_inset + + es +\begin_inset Formula $K(p):=\det A_{p}=\kappa_{1}(p)\kappa_{2}(p)$ +\end_inset + +, y la +\series bold +curvatura media +\series default + es +\begin_inset Formula $H(p):=\frac{1}{2}\text{tr}A_{p}=\frac{1}{2}(\kappa_{1}(p)+\kappa_{2}(p))$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Las curvaturas máxima, mínima y media cambian de signo al cambiar de orientación. + La curvatura de Gauss no, pues es el producto de dos curvaturas que cambian + de signo a la vez. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie regular, +\begin_inset Formula $p\in S$ +\end_inset + + es +\series bold +elíptico +\series default + si +\begin_inset Formula $K(p)>0$ +\end_inset + +, +\series bold +hiperbólico +\series default + si +\begin_inset Formula $K(p)<0$ +\end_inset + +, +\series bold +parabólico +\series default + si +\begin_inset Formula $K(p)=0$ +\end_inset + + pero +\begin_inset Formula $A_{p}\not\equiv0$ +\end_inset + + y +\series bold +llano +\series default + o +\series bold +plano +\series default + si +\begin_inset Formula $A_{p}\equiv0$ +\end_inset + +. + Ejemplos: +\end_layout + +\begin_layout Enumerate +Los puntos de un plano son planos. +\end_layout + +\begin_layout Enumerate +Los puntos de una esfera son elípticos. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $r$ +\end_inset + + es el radio, +\begin_inset Formula $K(p)=(-\frac{1}{r})^{2}=\frac{1}{r^{2}}>0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +El origen en la silla de montar es hiperbólico. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $A_{p}\equiv\text{diag}(-2,2)$ +\end_inset + + respecto de cierta base, luego +\begin_inset Formula $K(p)=-4$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Los puntos de un cilindro son parabólicos. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $r$ +\end_inset + + es el radio, +\begin_inset Formula $A_{p}\equiv\text{diag}(-\frac{1}{r},0)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +En +\begin_inset Formula $\{z=(x^{2}+y^{2})^{2}\}$ +\end_inset + +, el origen es un punto plano. +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +La superficie es el grafo +\begin_inset Formula $S:=\{X(u,v):=(u,v,(u^{2}+v^{2})^{2}\}_{u,v\in\mathbb{R}}$ +\end_inset + +, de modo que +\begin_inset Formula $X_{u}=(1,0,2(u^{2}+v^{2})u)$ +\end_inset + +, +\begin_inset Formula $X_{v}=(0,1,2(u^{2}+v^{2})v)$ +\end_inset + +, +\begin_inset Formula $N=\frac{(-4(u^{2}+v^{2})u,-4(u^{2}+v^{2})v,1)}{\sqrt{16(u^{2}+v^{2})^{2}+1}}$ +\end_inset + +, +\begin_inset Formula $N_{u}=\frac{(-4(3u^{2}+v^{2})(16(u^{2}+v^{2})^{2}+1)+256(u^{2}+v^{2})^{2}u^{2},-8uv(16(u^{2}+v^{2})^{2}+1)+256(u^{2}+v^{2})^{2}uv,64(u^{2}+v^{2})u)}{(16(u^{2}+v^{2})^{2}+1)^{3/2}}$ +\end_inset + + y entonces +\begin_inset Formula $N_{u}(0,0)=(0,0,0)$ +\end_inset + + y, por simetría, +\begin_inset Formula $N_{v}(0,0)=(0,0,0)$ +\end_inset + +, por lo que +\begin_inset Formula $A_{p}\equiv0$ +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +En una superficie regular +\begin_inset Formula $S$ +\end_inset + + orientada, +\begin_inset Formula $p\in S$ +\end_inset + + es un +\series bold +punto umbilical +\series default + si +\begin_inset Formula $\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +. + +\begin_inset Formula $S$ +\end_inset + + es +\series bold +totalmente umbilical +\series default + si todos sus puntos son umbilicales. + Así, el plano y la esfera son totalmente umbilicales. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, toda superficie regular, orientable con orientación +\begin_inset Formula ${\cal C}^{2}$ +\end_inset + +, conexa y totalmente umbilical es un trozo de esfera o plano. +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula $S$ +\end_inset + + la superficie y +\begin_inset Formula $N$ +\end_inset + + una orientación de +\begin_inset Formula $S$ +\end_inset + +, para +\begin_inset Formula $p\in S$ +\end_inset + + es +\begin_inset Formula $H(p)=\kappa_{1}(p)=\kappa_{2}(p)$ +\end_inset + +, luego +\begin_inset Formula $A_{p}\equiv\text{diag}(H(p),H(p))$ +\end_inset + + y +\begin_inset Formula $A_{p}=H(p)1_{T_{p}S}$ +\end_inset + +. + +\begin_inset Formula $H:S\to\mathbb{R}$ +\end_inset + + es diferenciable, y queremos ver que es constante. + Sean +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +, +\begin_inset Formula $q:=(u_{0},v_{0}):=X^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $\alpha(u):=X(u_{0}+u,v_{0})$ +\end_inset + +, como +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=q$ +\end_inset + +, +\begin_inset Formula $dH_{p}(X_{u}(q))=\frac{d(H\circ\alpha)}{dt}(0)=\frac{d}{dt}(H(X(u_{0}+u,v_{0})))(0)=(H\circ X)_{u}(q)$ +\end_inset + +, y por simetría +\begin_inset Formula $dH_{p}(X_{v}(q))=\frac{\partial(H\circ X)}{\partial v}(q)$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +Como +\begin_inset Formula $A_{p}=H(p)1_{T_{p}S}$ +\end_inset + +, +\begin_inset Formula $(H\circ X)(q)X_{u}(q)=H(p)X_{u}(q)=A_{p}(X_{u}(q))=-dN_{p}(X_{u}(q))=-(N\circ X)(q)$ +\end_inset + +, y como esto es cierto para todo +\begin_inset Formula $q\in U$ +\end_inset + +, +\begin_inset Formula $(N\circ X)_{u}=-(H\circ X)X_{u}$ +\end_inset + +, y por simetría +\begin_inset Formula $(N\circ X)_{v}=-(H\circ X)X_{v}$ +\end_inset + +. + Derivando, +\begin_inset Formula $(N\circ X)_{uv}=-(H\circ X)_{v}X_{u}-(H\circ X)X_{uv}$ +\end_inset + + y +\begin_inset Formula $(N\circ X)_{vu}=-(H\circ X)_{u}X_{v}-(H\circ X)X_{vu}$ +\end_inset + +, y como las derivadas cruzadas coinciden, +\begin_inset Formula $(H\circ X)_{v}X_{u}=(H\circ X)_{u}X_{v}$ +\end_inset + +. + Como +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ +\end_inset + + es una base en cada +\begin_inset Formula $q\in U$ +\end_inset + +, necesariamente +\begin_inset Formula $(H\circ X)_{u},(H\circ X)_{v}\equiv0$ +\end_inset + +, luego +\begin_inset Formula $dH_{p}(X_{u}(q)),dH_{p}(X_{v}(q))=0$ +\end_inset + + y al ser +\begin_inset Formula $S$ +\end_inset + + conexa, +\begin_inset Formula $H\equiv c$ +\end_inset + + para algún +\begin_inset Formula $c\in\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $c=0$ +\end_inset + +, +\begin_inset Formula $H\equiv0$ +\end_inset + + y +\begin_inset Formula $dN_{p}=-A_{p}\equiv0$ +\end_inset + +, luego +\begin_inset Formula $N$ +\end_inset + + es constante en algún +\begin_inset Formula $a\in\mathbb{R}^{3}$ +\end_inset + +. + Sean ahora +\begin_inset Formula $\phi(p):=\langle p,a\rangle$ +\end_inset + +, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + y +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +, entonces +\begin_inset Formula $d\phi_{p}(v)=\frac{d(\phi\circ\alpha)}{dt}(0)=\frac{d}{dt}(\langle\alpha(t),a\rangle)(0)=\langle\alpha'(0),a\rangle=\langle v,a\rangle\overset{a=N(p)}{=}0$ +\end_inset + +, luego +\begin_inset Formula $\phi$ +\end_inset + + es constante en algún +\begin_inset Formula $d\in\mathbb{R}$ +\end_inset + + y +\begin_inset Formula $S\subseteq\{\langle p,a\rangle=d\}=\{\langle p-p',a\rangle=0\}$ +\end_inset + + para algún +\begin_inset Formula $p'$ +\end_inset + + con +\begin_inset Formula $\langle p',a\rangle=d$ +\end_inset + +, pero +\begin_inset Formula $\{\langle p-p',a\rangle=0\}=p'+\langle a\rangle^{\bot}$ +\end_inset + +, luego +\begin_inset Formula $S$ +\end_inset + + esta contenido en un plano. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $c\neq0$ +\end_inset + +, sea +\begin_inset Formula $\phi:S\to\mathbb{R}^{3}$ +\end_inset + + la función diferenciable dada por +\begin_inset Formula $\phi(p):=p+\frac{1}{c}N(p)$ +\end_inset + +, para +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $v\in T_{p}S$ +\end_inset + + y una curva +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +, entonces +\begin_inset Formula +\begin{align*} +d\phi_{p}(v) & =\frac{d(\phi\circ\alpha)}{dt}(0)=\frac{d}{dt}\left(\alpha(t)+\frac{1}{c}N(\alpha(t))\right)(0)=\alpha'(0)+\frac{1}{c}(N\circ\alpha)'(0)\\ + & =v+\frac{1}{c}dN_{p}(v)=v-\frac{1}{c}A_{p}v=v-\frac{1}{c}cv=0, +\end{align*} + +\end_inset + +luego +\begin_inset Formula $\phi$ +\end_inset + + es constante en algún +\begin_inset Formula $a\in\mathbb{R}^{3}$ +\end_inset + +. + Pero para +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $p-a=-\frac{1}{c}N(p)$ +\end_inset + +, luego +\begin_inset Formula $\Vert p-a\Vert^{2}=\frac{1}{c^{2}}$ +\end_inset + + y todos los puntos de +\begin_inset Formula $S$ +\end_inset + + están en la esfera +\begin_inset Formula $a+\mathbb{S}^{2}(\frac{1}{c^{2}})$ +\end_inset + +. +\end_layout + \end_body \end_document |