#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass book \begin_preamble \input{../defs} \end_preamble \use_default_options true \maintain_unincluded_children false \language spanish \language_package default \inputencoding auto \fontencoding global \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style french \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard Una función real es \series bold diferenciable \series default si es de clase \begin_inset Formula ${\cal C}^{\infty}$ \end_inset . Una función \begin_inset Formula $F:A\to B$ \end_inset diferenciable entre abiertos de superficies o de \begin_inset Formula $\mathbb{R}^{n}$ \end_inset es un difeomorfismo local en \begin_inset Formula $p\in A$ \end_inset si y sólo si \begin_inset Formula $dF_{p}$ \end_inset es un isomorfismo lineal. \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash sremember{GCS} \end_layout \end_inset \begin_inset Formula \[ J:=\begin{pmatrix}0 & -1\\ 1 & 0 \end{pmatrix}. \] \end_inset Entonces, dada una curva \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset p.p.a., si \begin_inset Formula $\mathbf{t}(s)\coloneqq \alpha'(s)$ \end_inset y \begin_inset Formula $\mathbf{n}(s)\coloneqq J\mathbf{t}(s)$ \end_inset [...], [...] \begin_inset Formula $\kappa_{\alpha}(s)\coloneqq \langle\mathbf{t}'(s),\mathbf{n}(s)\rangle$ \end_inset [...]. \end_layout \begin_layout Standard Las \series bold fórmulas de Frenet \series default son \begin_inset Formula \[ \left\{ \begin{aligned}\mathbf{t}'(s) & =\kappa(s)\mathbf{n}(s),\\ \mathbf{n}'(s) & =-\kappa(s)\mathbf{t}(s). \end{aligned} \right. \] \end_inset [...] Una curva regular \begin_inset Formula $\alpha:I\to\mathbb{R}^{2}$ \end_inset [...], [...] la curvatura [...] es \begin_inset Formula \[ \kappa_{\alpha}(t)=\frac{\langle\alpha''(t),J\alpha'(t)\rangle}{|\alpha'(t)|^{3}}. \] \end_inset \end_layout \begin_layout Standard [...] Sea \begin_inset Formula $\alpha:I\to\mathbb{R}^{3}$ \end_inset una curva regular p.p.a., si \begin_inset Formula $\mathbf{t}(s)$ \end_inset es su vector tangente, [...] \begin_inset Formula $\kappa(s)\coloneqq |\mathbf{t}'(s)|$ \end_inset . [...] \begin_inset Formula $\mathbf{n}(s)\coloneqq \frac{\mathbf{t}'(s)}{\kappa(s)}[...],$ \end_inset [...] \begin_inset Formula $\mathbf{b}(s)=\mathbf{t}(s)\land\mathbf{n}(s)$ \end_inset [...]. [...] \begin_inset Formula $\tau(s)[...]=\langle\mathbf{b}'(s),\mathbf{n}(s)\rangle$ \end_inset . [...] \end_layout \begin_layout Standard \begin_inset Formula \[ \begin{pmatrix}\mathbf{t}\\ \mathbf{n}\\ \mathbf{b} \end{pmatrix}^{\prime}=\begin{pmatrix}\kappa\mathbf{n}\\ -\kappa\mathbf{t}-\tau\mathbf{b}\\ \tau\mathbf{n} \end{pmatrix}=\begin{pmatrix} & \kappa\\ -\kappa & & -\tau\\ & \tau \end{pmatrix}\begin{pmatrix}\mathbf{t}\\ \mathbf{n}\\ \mathbf{b} \end{pmatrix}. \] \end_inset \end_layout \begin_layout Standard [...] \begin_inset ERT status open \begin_layout Plain Layout { \end_layout \end_inset \begin_inset Formula \begin{align*} \kappa_{\alpha}(t) & :=\frac{|\alpha'(t)\land\alpha''(t)|}{|\alpha'(t)|^{3}}, & \tau_{\alpha}(t) & =-\frac{\det(\alpha'(t),\alpha''(t),\alpha'''(t))}{|\alpha'(t)\land\alpha''(t)|^{2}}. \end{align*} \end_inset \begin_inset ERT status open \begin_layout Plain Layout } \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout { \end_layout \end_inset \begin_inset Formula \begin{align*} e & :=\langle N,X_{uu}\rangle=-\langle N_{u},X_{u}\rangle,\\ f & :=\langle N,X_{uv}\rangle=-\langle N_{v},X_{u}\rangle=-\langle N_{u},X_{v}\rangle,\\ g & :=\langle N,X_{vv}\rangle=-\langle N_{v},X_{v}\rangle \end{align*} \end_inset \begin_inset ERT status open \begin_layout Plain Layout } \end_layout \end_inset \end_layout \begin_layout Standard [...]. [...] Si \begin_inset Formula \[ dN_{p}\equiv\begin{pmatrix}a_{11} & a_{12}\\ a_{21} & a_{22} \end{pmatrix} \] \end_inset respecto de la base \begin_inset Formula $(X_{u},X_{v})$ \end_inset , entonces \begin_inset Formula \[ \begin{pmatrix}-e & -f\\ -f & -g \end{pmatrix}=\begin{pmatrix}a_{11} & a_{21}\\ a_{12} & a_{22} \end{pmatrix}\begin{pmatrix}E & F\\ F & G \end{pmatrix} \] \end_inset y tenemos las \series bold fórmulas de Weingarten: \series default \begin_inset ERT status open \begin_layout Plain Layout { \end_layout \end_inset \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 \begin_inset ERT status open \begin_layout Plain Layout } \end_layout \end_inset [...] \begin_inset ERT status open \begin_layout Plain Layout { \end_layout \end_inset \begin_inset Formula \begin{align*} K(p) & =\frac{eg-f^{2}}{EG-F^{2}}, & H(p) & =\frac{1}{2}\frac{eG+gE-2fF}{EG-F^{2}} \end{align*} \end_inset \begin_inset ERT status open \begin_layout Plain Layout } \end_layout \end_inset [...]. Las \series bold fórmulas de Gauss \series default son \begin_inset Formula \[ \left\{ \begin{aligned}X_{uu} & =\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN,\\ X_{uv} & =\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN,\\ X_{vu} & =\Gamma_{21}^{1}X_{u}+\Gamma_{21}^{2}X_{v}+fN,\\ X_{vv} & =\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN \end{aligned} \right. \] \end_inset donde los \begin_inset Formula $\Gamma_{ij}^{k}$ \end_inset son los \series bold símbolos de Christoffel \series default [...]. \begin_inset Formula $\Gamma_{12}^{1}=\Gamma_{21}^{1}$ \end_inset y \begin_inset Formula $\Gamma_{12}^{2}=\Gamma_{21}^{2}$ \end_inset [...]. Además, \begin_inset Formula \[ \begin{pmatrix}\Gamma_{11}^{1} & \Gamma_{12}^{1} & \Gamma_{22}^{1}\\ \Gamma_{11}^{2} & \Gamma_{12}^{2} & \Gamma_{22}^{2} \end{pmatrix}=\frac{1}{EG-F^{2}}\begin{pmatrix}G & -F\\ -F & E \end{pmatrix}\begin{pmatrix}\frac{E_{u}}{2} & \frac{E_{v}}{2} & F_{v}-\frac{G_{u}}{2}\\ F_{u}-\frac{E_{v}}{2} & \frac{G_{u}}{2} & \frac{G_{v}}{2} \end{pmatrix}. \] \end_inset \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 Si \begin_inset Formula $F=0$ \end_inset , la curvatura de Gauss es \begin_inset Formula \[ K=\frac{-1}{2\sqrt{EG}}\left[\left(\frac{E_{v}}{\sqrt{EG}}\right)_{v}+\left(\frac{G_{u}}{\sqrt{EG}}\right)_{u}\right]. \] \end_inset \series bold Demostración: \series default \begin_inset Formula \[ \begin{pmatrix}\Gamma_{11}^{1} & \Gamma_{12}^{1} & \Gamma_{22}^{1}\\ \Gamma_{11}^{2} & \Gamma_{12}^{2} & \Gamma_{22}^{2} \end{pmatrix}=\begin{pmatrix}\frac{E_{u}}{2E} & \frac{E_{v}}{2E} & -\frac{G_{v}}{2E}\\ -\frac{E_{v}}{2G} & \frac{G_{u}}{2G} & \frac{G_{v}}{2G} \end{pmatrix}, \] \end_inset y por la ecuación de Gauss, \begin_inset Formula \begin{align*} K & =\frac{1}{E}\left(\frac{E_{u}G_{u}}{4EG}-\frac{E_{vv}}{2G}+\frac{E_{v}G_{u}}{2G^{2}}-\frac{E_{v}G_{v}}{4G^{2}}+\frac{E_{v}^{2}}{4EG}-\frac{G_{uu}}{2G}+\frac{G_{u}^{2}}{2G^{2}}-\frac{G_{u}^{2}}{4G^{2}}\right)\\ & =\left(\frac{E_{u}G_{u}}{4E^{2}G}-\frac{E_{vv}}{4EG}+\frac{E_{v}G_{u}}{2EG^{2}}-\frac{E_{v}G_{v}}{4EG^{2}}-\frac{G_{uu}}{2EG}+\frac{G_{u}^{2}}{4EG^{2}}\right), \end{align*} \end_inset pero \begin_inset Formula \begin{align*} \left(\frac{E_{v}}{\sqrt{EG}}\right)_{v} & =\frac{E_{vv}}{\sqrt{EG}}-\frac{E_{v}(E_{v}G+EG_{v})}{2(EG)^{3/2}}=\sqrt{EG}\left(\frac{E_{vv}}{EG}-\frac{E_{v}^{2}}{2E^{2}G}-\frac{E_{v}G_{v}}{2EG^{2}}\right),\\ \left(\frac{G_{u}}{\sqrt{EG}}\right)_{u} & =\frac{G_{uu}}{\sqrt{EG}}-\frac{G_{u}(E_{u}G+EG_{u})}{2(EG)^{3/2}}=\sqrt{EG}\left(\frac{G_{uu}}{EG}-\frac{E_{u}G_{u}}{2E^{2}G}-\frac{G_{u}^{2}}{2EG^{2}}\right), \end{align*} \end_inset de modo que \begin_inset Formula \[ -\frac{1}{2\sqrt{EG}}\left[\left(\frac{E_{v}}{\sqrt{EG}}\right)_{v}+\left(\frac{G_{u}}{\sqrt{EG}}\right)_{u}\right]=K. \] \end_inset \end_layout \begin_layout Section La derivada covariante \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash sremember{GCS} \end_layout \end_inset \end_layout \begin_layout Standard Sean \begin_inset Formula $S$ \end_inset una superficie regular y \begin_inset Formula $\alpha:I\to S$ \end_inset una curva regular, un \series bold campo de vectores a lo largo de \begin_inset Formula $\alpha$ \end_inset \series default es una función \begin_inset Formula $V:I\to\mathbb{R}^{3}$ \end_inset , y es \series bold tangente \series default a \begin_inset Formula $S$ \end_inset (a lo largo de \begin_inset Formula $\alpha$ \end_inset ) si para \begin_inset Formula $t\in S$ \end_inset es \begin_inset Formula $V(t)\in T_{\alpha(t)}S$ \end_inset . \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 Para un \begin_inset Formula $t\in I$ \end_inset , \begin_inset Formula $V(t)^{\top}\coloneqq \pi_{T_{\alpha(t)}S}V(t)$ \end_inset y \begin_inset Formula $V(t)^{\bot}\coloneqq \pi_{(T_{\alpha(t)}S)^{\bot}}V(t)$ \end_inset . Llamamos \begin_inset Formula $\mathfrak{X}(\alpha)$ \end_inset al conjunto de campos de vectores a lo largo de \begin_inset Formula $\alpha$ \end_inset diferenciables y tangentes. Así: \end_layout \begin_layout Enumerate La velocidad \begin_inset Formula $\alpha'\in\mathfrak{X}(\alpha)$ \end_inset . \end_layout \begin_layout Enumerate La rotación de la velocidad \begin_inset Formula $N\wedge\alpha'\in\mathfrak{X}(\alpha)$ \end_inset . \end_layout \begin_layout Enumerate La aceleración \begin_inset Formula $\alpha''(t)$ \end_inset es un campo de vectores diferenciable. \end_layout \begin_layout Enumerate Dado un campo de vectores diferenciable \begin_inset Formula $V:I\to\mathbb{R}^{3}$ \end_inset , \begin_inset Formula $V'$ \end_inset es otro campo de vectores, pero \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset no implica \begin_inset Formula $V'\in\mathfrak{X}(\alpha)$ \end_inset . \end_layout \begin_layout Standard Un \series bold campo normal unitario \series default a lo largo de \begin_inset Formula $\alpha$ \end_inset es un campo \begin_inset Formula $N:I\to\mathbb{R}^{3}$ \end_inset diferenciable y unitario tal que todo \begin_inset Formula $N(t)$ \end_inset es normal a \begin_inset Formula $S$ \end_inset en \begin_inset Formula $\alpha(t)$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash sremember{GCS} \end_layout \end_inset \end_layout \begin_layout Standard Sea \begin_inset Formula $V:I\to\mathbb{R}^{3}$ \end_inset un campo de vectores tangente y diferenciable, llamamos \series bold derivada covariante \series default [o \series bold intrínseca \series default ] a \begin_inset Formula \[ \frac{DV}{dt}(t):=\pi_{T_{\alpha(t)}S}V'(t) \] \end_inset \end_layout \begin_layout Standard Propiedades: Sean \begin_inset Formula $V,W:I\to T_{p}S$ \end_inset y \begin_inset Formula $f:I\to\mathbb{R}$ \end_inset diferenciables, siendo \begin_inset Formula $I$ \end_inset un intervalo: \end_layout \begin_layout Enumerate \begin_inset Formula $\frac{D(fV)}{dt}=f'V+f\frac{DV}{dt}$ \end_inset . \end_layout \begin_deeper \begin_layout Standard Si \begin_inset Formula $\pi\coloneqq \pi_{T_{\alpha(t)}S}$ \end_inset , \begin_inset Formula $\frac{D(fV)}{dt}=\pi((fV)')=\pi(fV'+f'V)=f\pi(V')+f'\pi(V)=f\frac{DV}{dt}+f'V$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate \begin_inset Formula $\frac{D(V+W)}{dt}=\frac{DV}{dt}+\frac{DW}{dt}$ \end_inset . \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula $\frac{D(V+W)}{dt}=\pi((V+W)')=\pi(V')+\pi(W')=\frac{DV}{dt}+\frac{DW}{dt}$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate \begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{DV}{dt}W\rangle+\langle V,\frac{DW}{dt}\rangle$ \end_inset . \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{dV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle$ \end_inset , pero dada una base ortonormal \begin_inset Formula $(v_{1},v_{2},v_{3})$ \end_inset con \begin_inset Formula $T_{p}S=\text{span}\{v_{1},v_{2}\}$ \end_inset , si \begin_inset Formula $\frac{dV}{dt}(t)=\sum_{i}x_{i}v_{i}$ \end_inset y \begin_inset Formula $W(t)=\sum_{i}y_{i}v_{i}$ \end_inset , \begin_inset Formula $\langle\frac{dV}{dt}(t),W(t)\rangle=\sum_{i=1}^{3}x_{i}y_{i}\overset{y_{3}=0}{=}x_{1}y_{1}+x_{2}y_{2}=\langle\pi(\frac{dV}{dt}(t)),W(t)\rangle=\langle\frac{DV}{dt}(t),W(t)\rangle$ \end_inset , y análogamente para \begin_inset Formula $\langle V,\frac{dW}{dt}\rangle$ \end_inset , luego \begin_inset Formula $\frac{d}{dt}\langle V,W\rangle=\langle\frac{dV}{dt},W\rangle+\langle V,\frac{dW}{dt}\rangle=\langle\frac{DV}{dt},W\rangle+\langle V,\frac{DW}{dt}\rangle$ \end_inset . \end_layout \end_deeper \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash eremember \end_layout \end_inset \end_layout \begin_layout Standard Sean \begin_inset Formula $(U,X)$ \end_inset una carta local de \begin_inset Formula $S$ \end_inset , \begin_inset Formula $\alpha:I\to X(U)$ \end_inset una curva sobre \begin_inset Formula $S$ \end_inset , \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset , \begin_inset Formula $\tilde{\alpha}\coloneqq (u,v)\coloneqq X^{-1}\circ\alpha:I\to U$ \end_inset y \begin_inset Formula $(a,b):I\to U$ \end_inset con \begin_inset Formula $V(t)=a(t)X_{u}(\tilde{\alpha}(t))+b(t)X_{v}(\tilde{\alpha}(t))$ \end_inset , entonces, para \begin_inset Formula $t\in I$ \end_inset , \begin_inset Formula \begin{align*} \frac{DV}{dt} & =\left(a'+au'\Gamma_{11}^{1}+(av'+bu')\Gamma_{12}^{1}+bv'\Gamma_{22}^{1}\right)X_{u}(\tilde{\alpha})\\ & +\left(b'+au'\Gamma_{11}^{2}+(av'+bu')\Gamma_{12}^{2}+bv'\Gamma_{22}^{2}\right)X_{v}(\tilde{\alpha}). \end{align*} \end_inset \series bold Demostración: \series default Sean \begin_inset Formula $t\in I$ \end_inset , \begin_inset Formula $p\coloneqq \alpha(t)$ \end_inset , \begin_inset Formula $q\coloneqq X^{-1}(p)$ \end_inset y \begin_inset Formula $N:X(U)\to\mathbb{R}^{3}$ \end_inset un campo normal tal que la base \begin_inset Formula $(X_{u}(q),X_{v}(q),N(p))$ \end_inset está orientada positivamente, derivando en \begin_inset Formula $V=aX_{u}(u,v)+bX_{v}(u,v)$ \end_inset , \begin_inset Formula \begin{align*} V'(t) & =a'X_{u}(u,v)+a\left(X_{uu}(u,v)u'+X_{uv}(u,v)v'\right)+b'X_{v}(u,v)+b\left(X_{vu}(u,v)u'+X_{vv}(u,v)v'\right)\\ & =a'X_{u}(u,v)+a\left[(\Gamma_{11}^{1}X_{u}(u,v)+\Gamma_{11}^{2}X_{v}(u,v)+eN)u'+(\Gamma_{12}^{1}X_{u}(u,v)+\Gamma_{12}^{2}X_{v}(u,v)+fN)v'\right]\\ & +b'X_{v}(u,v)+b\left[(\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN)u'+(\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN)v'\right], \end{align*} \end_inset y entonces \begin_inset Formula $\frac{DV}{dt}$ \end_inset es la parte tangente de esto último, \begin_inset Formula \begin{align*} \frac{DV}{dt} & =\left(a'+a\Gamma_{11}^{1}u'+a\Gamma_{12}^{1}v'+b\Gamma_{12}^{1}u'+b\Gamma_{22}^{1}v'\right)X_{u}(u,v)\\ & +\left(a\Gamma_{11}^{2}u'+a\Gamma_{12}^{2}v'+b'+b\Gamma_{12}^{2}u'+b\Gamma_{22}^{2}v'\right)X_{v}(u,v). \end{align*} \end_inset \end_layout \begin_layout Section Campos paralelos \end_layout \begin_layout Standard Sean \begin_inset Formula $S$ \end_inset una superficie regular y \begin_inset Formula $\alpha:I\to S$ \end_inset una curva regular, \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset es \series bold paralelo \series default ( \series bold a lo largo de \begin_inset Formula $\alpha$ \end_inset \series default ) si \begin_inset Formula $\frac{DV}{dt}=0$ \end_inset . Si \begin_inset Formula $V,W\in\mathfrak{X}(\alpha)$ \end_inset son paralelos: \end_layout \begin_layout Enumerate Para \begin_inset Formula $a,b\in\mathbb{R}$ \end_inset , \begin_inset Formula $aV+bW$ \end_inset es paralelo. \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula $\frac{D(aV+bW)}{dt}=a\frac{DV}{dt}+b\frac{DW}{dt}=0$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate \begin_inset Formula $\langle V(t),W(t)\rangle$ \end_inset es constante, por lo que también lo son \begin_inset Formula $\Vert V(t)\Vert$ \end_inset y \begin_inset Formula $\angle(V,W)$ \end_inset . \end_layout \begin_deeper \begin_layout Standard \begin_inset Formula $\langle V,W\rangle'=\langle\frac{DV}{dt},W\rangle+\langle V,\frac{DW}{dt}\rangle=0+0=0$ \end_inset . \end_layout \end_deeper \begin_layout Standard \series bold E.d.o extrínseca de los campos paralelos: \series default \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset es paralelo a lo largo de \begin_inset Formula $\alpha$ \end_inset si y sólo si \begin_inset Formula \[ V'(t)+\langle V(t),N'(t)\rangle N(t)=0, \] \end_inset donde \begin_inset Formula $N:I\to\mathbb{R}^{3}$ \end_inset es un campo normal unitario de \begin_inset Formula $S$ \end_inset a lo largo de \begin_inset Formula $\alpha$ \end_inset . \series bold Demostración: \series default \begin_inset Formula $V$ \end_inset es paralelo si y sólo si \begin_inset Formula $V'(t)$ \end_inset es proporcional a \begin_inset Formula $N(t)$ \end_inset en todo \begin_inset Formula $t\in I$ \end_inset , si y sólo si \begin_inset Formula $V'(t)=\langle V'(t),N(t)\rangle$ \end_inset , pero como \begin_inset Formula $\langle V(t),N(t)\rangle=0$ \end_inset en todo punto, derivando es \begin_inset Formula $\langle V'(t),N(t)\rangle+\langle V(t),N'(t)\rangle=0$ \end_inset , luego \begin_inset Formula $V'(t)=\langle V'(t),N(t)\rangle$ \end_inset si y sólo si \begin_inset Formula $V'(t)-\langle V'(t),N(t)\rangle=V'(t)+\langle V(t),N'(t)\rangle=0$ \end_inset . \end_layout \begin_layout Standard \series bold E.d.o intrínseca de los campos paralelos: \series default Sean \begin_inset Formula $(U,X)$ \end_inset una carta local de la superficie regular \begin_inset Formula $S$ \end_inset , \begin_inset Formula $\alpha:I\to X(U)$ \end_inset una curva sobre \begin_inset Formula $S$ \end_inset , \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset , \begin_inset Formula $(u,v)\coloneqq X^{-1}\circ\alpha:I\to U$ \end_inset y \begin_inset Formula $(a,b):I\to U$ \end_inset tal que \begin_inset Formula $V=aX_{u}(\tilde{\alpha})+bX_{v}(\tilde{\alpha})$ \end_inset , entonces \begin_inset Formula $V$ \end_inset es paralelo a lo largo de \begin_inset Formula $\alpha$ \end_inset si y sólo si satisface \begin_inset Formula \[ \left\{ \begin{aligned}a'+au'\Gamma_{11}^{1}(u,v)+(av'+bu')\Gamma_{12}^{1}(u,v)+bv'\Gamma_{22}^{1}(u,v) & =0,\\ b'+au'\Gamma_{11}^{2}(u,v)+(av'+bu')\Gamma_{12}^{2}(u,v)+bv'\Gamma_{22}^{2}(u,v) & =0, \end{aligned} \right. \] \end_inset ecuaciones que resultan de sustituir la fórmula intrínseca de la derivada covariante en \begin_inset Formula $\frac{DV}{dt}=0$ \end_inset y usar que \begin_inset Formula $X_{u}(\tilde{\alpha})$ \end_inset y \begin_inset Formula $X_{v}(\tilde{\alpha})$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash sremember{EDO} \end_layout \end_inset Una e.d.o. es \series bold lineal \series default si es de la forma \begin_inset Formula $\dot{x}=A(t)x+b(t)$ \end_inset , con \begin_inset Formula $A:I\subseteq\mathbb{R}\to{\cal L}(\mathbb{R}^{n})$ \end_inset y \begin_inset Formula $b:I\subseteq\mathbb{R}\to\mathbb{R}^{n}$ \end_inset [...]. [...] Si \begin_inset Formula $A$ \end_inset y \begin_inset Formula $b$ \end_inset son continuas, para \begin_inset Formula $(t_{0},x_{0})\in I\times\mathbb{R}^{n}$ \end_inset [...] \begin_inset Formula \[ \left\{ \begin{aligned}\dot{x} & =A(t)x+b(t)\\ x(t_{0}) & =x_{0} \end{aligned} \right. \] \end_inset tiene solución única definida en todo \begin_inset Formula $I$ \end_inset [...]. \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash eremember \end_layout \end_inset \end_layout \begin_layout Section Transporte paralelo \end_layout \begin_layout Standard Como \series bold teorema \series default , sean \begin_inset Formula $S$ \end_inset una superficie regular, \begin_inset Formula $\alpha:I\to S$ \end_inset una curva, \begin_inset Formula $t_{0}\in I$ \end_inset y \begin_inset Formula $v\in T_{\alpha(t_{0})}S$ \end_inset , existe un único \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset paralelo tal que \begin_inset Formula $V(t_{0})=v$ \end_inset . \series bold Demostración: \series default Sea \begin_inset Formula $N$ \end_inset un campo normal unitario de \begin_inset Formula $S$ \end_inset a lo largo de \begin_inset Formula $\alpha$ \end_inset , \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset es paralelo si y sólo si \begin_inset Formula \[ 0=V'+\langle V,N'\rangle N=\begin{pmatrix}V_{1}'\\ V_{2}'\\ V_{3}' \end{pmatrix}+\sum_{j=1}^{3}V_{j}N_{j}'\begin{pmatrix}N_{1}\\ N_{2}\\ N_{3} \end{pmatrix}=\begin{pmatrix}V'_{1}\\ V'_{2}\\ V'_{3} \end{pmatrix}+\begin{pmatrix}N_{1}N'_{1} & N_{1}N'_{2} & N_{1}N'_{3}\\ N_{2}N'_{1} & N_{2}N'_{2} & N_{2}N'_{3}\\ N_{3}N'_{1} & N_{3}N'_{2} & N_{3}N'_{3} \end{pmatrix}\begin{pmatrix}V_{1}\\ V_{2}\\ V_{3} \end{pmatrix}, \] \end_inset lo que nos da una e.d.o. lineal que, añadiendo la condición inicial \begin_inset Formula $V(t_{0})=v$ \end_inset , tiene solución única definida en todo \begin_inset Formula $I$ \end_inset . Para ver que realmente la solución es tangente, sabemos que \begin_inset Formula $\langle V,N\rangle(t_{0})=\langle v,N(t_{0})\rangle=0$ \end_inset , y como por la ecuación es \begin_inset Formula $V'=-\langle V,N'\rangle N$ \end_inset , \begin_inset Formula \[ \langle V,N\rangle'=\langle V',N\rangle+\langle V,N'\rangle=-\langle V,N'\rangle\langle N,N\rangle+\langle V,N'\rangle\overset{\langle N,N\rangle=1}{=}0. \] \end_inset \end_layout \begin_layout Standard Sean \begin_inset Formula $S$ \end_inset una superficie regular, \begin_inset Formula $\alpha:I\to S$ \end_inset una curva regular, \begin_inset Formula $a,b\in I$ \end_inset , \begin_inset Formula $p\coloneqq \alpha(a)$ \end_inset , \begin_inset Formula $q\coloneqq \alpha(b)$ \end_inset y \begin_inset Formula $v\in T_{p}S$ \end_inset y \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset el único campo paralelo con \begin_inset Formula $V(a)=v$ \end_inset , el \series bold transporte paralelo \series default de \begin_inset Formula $v$ \end_inset a lo largo de \begin_inset Formula $\alpha$ \end_inset en el punto \begin_inset Formula $q$ \end_inset es \begin_inset Formula $V(b)$ \end_inset . \end_layout \begin_layout Standard La \series bold aplicación transporte paralelo \series default es la \begin_inset Formula $P_{\alpha}\coloneqq P_{a}^{b}(\alpha):T_{p}S\to T_{q}S$ \end_inset que a cada \begin_inset Formula $v\in T_{p}S$ \end_inset le asigna su transporte paralelo a lo largo de \begin_inset Formula $\alpha$ \end_inset en \begin_inset Formula $q$ \end_inset . Como \series bold teorema \series default , \begin_inset Formula $P_{\alpha}$ \end_inset es una isometría lineal. \series bold Demostración: \series default Para \begin_inset Formula $v\in T_{p}S$ \end_inset , sea \begin_inset Formula $V\in\mathfrak{X}(\alpha)$ \end_inset el único campo paralelo con \begin_inset Formula $V(a)=v$ \end_inset , este también es el único campo paralelo con \begin_inset Formula $V(b)=P_{a}^{b}(\alpha)(v)$ \end_inset , por lo que \begin_inset Formula $v=P_{b}^{a}(\alpha)(P_{a}^{b}(\alpha)(v))$ \end_inset y, por simetría, para \begin_inset Formula $w\in T_{q}S$ \end_inset , \begin_inset Formula $w=P_{a}^{b}(\alpha)(P_{b}^{a}(\alpha)(v))$ \end_inset , de modo que \begin_inset Formula $P_{\alpha}$ \end_inset es invertible. Sean ahora \begin_inset Formula $v,w\in T_{p}S$ \end_inset , \begin_inset Formula $V$ \end_inset el único campo paralelo con \begin_inset Formula $V(a)=v$ \end_inset y \begin_inset Formula $W$ \end_inset el único con \begin_inset Formula $W(a)=w$ \end_inset , entonces \begin_inset Formula $V+W$ \end_inset es otro campo paralelo con \begin_inset Formula $(V+W)(a)=v+w$ \end_inset y por tanto el único, luego \begin_inset Formula $P_{\alpha}(v+w)=(V+W)(b)=V(b)+W(b)=P_{\alpha}(v)+P_{\alpha}(w)$ \end_inset . Del mismo modo, si \begin_inset Formula $\lambda\in\mathbb{R}$ \end_inset , \begin_inset Formula $\lambda V$ \end_inset es un campo paralelo con \begin_inset Formula $(\lambda V)(a)=\lambda v$ \end_inset , luego \begin_inset Formula $P_{\alpha}(\lambda v)=\lambda V(a)=\lambda P_{\alpha}(v)$ \end_inset , y con esto \begin_inset Formula $P_{\alpha}$ \end_inset es lineal. Finalmente, como \begin_inset Formula $\langle V(t),W(t)\rangle$ \end_inset es constante en \begin_inset Formula $t$ \end_inset , \begin_inset Formula $\langle v,w\rangle=\langle V(a),W(a)\rangle=\langle V(b),W(b)\rangle=\langle P_{\alpha}(v),P_{\alpha}(v)\rangle$ \end_inset y \begin_inset Formula $P_{\alpha}$ \end_inset es una isometría. \end_layout \end_body \end_document