GGS tema 1: Campos paralelos

2 files changed, 1811 insertions, 0 deletions

diff --git a/ggs/n.lyx b/ggs/n.lyx
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index 0000000..31e8f95
--- /dev/null
+++ b/ggs/n.lyx
@@ -0,0 +1,188 @@
+#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
+\begin_modules
+algorithm2e
+\end_modules
+\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"
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+\index_command default
+\paperfontsize 10
+\spacing single
+\use_hyperref false
+\papersize a5paper
+\use_geometry true
+\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
+\leftmargin 0.2cm
+\topmargin 0.7cm
+\rightmargin 0.2cm
+\bottommargin 0.7cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
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+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Geometría Global de Superficies
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2021}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Luis Alías.
+ 
+\emph on
+Bloque 1.
+ Geodésicas en superficies
+\emph default
+.
+\end_layout
+
+\begin_layout Chapter
+Campos paralelos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Chapter
+Geodésicas
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset CommandInset include
+LatexCommand input
+filename "n2.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/ggs/n1.lyx b/ggs/n1.lyx
new file mode 100644
index 0000000..dff032c
--- /dev/null
+++ b/ggs/n1.lyx
@@ -0,0 +1,1623 @@
+#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
+
+.
+\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 $\mathbf{t}(s):=\alpha'(s)$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbf{n}(s):=J\mathbf{t}(s)$
+\end_inset
+
+ [...], [...] 
+\begin_inset Formula $\kappa_{\alpha}(s):=\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle$
+\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 $\kappa(s):=|\mathbf{t}'(s)|$
+\end_inset
+
+.
+ [...] 
+\begin_inset Formula $\mathbf{n}(s):=\frac{\mathbf{t}'(s)}{\kappa(s)}[...],$
+\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
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GCS}
+\end_layout
+
+\end_inset
+
+
+\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 $V(t)^{\top}:=\pi_{T_{\alpha(t)}S}V(t)$
+\end_inset
+
+ y 
+\begin_inset Formula $V(t)^{\bot}:=\pi_{(T_{\alpha(t)}S)^{\bot}}V(t)$
+\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
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\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
+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 $\pi:=\pi_{T_{\alpha(t)}S}$
+\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 $\tilde{\alpha}:=(u,v):=X^{-1}\circ\alpha:I\to U$
+\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 $p:=\alpha(t)$
+\end_inset
+
+, 
+\begin_inset Formula $q:=X^{-1}(p)$
+\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 $(u,v):=X^{-1}\circ\alpha:I\to U$
+\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 $p:=\alpha(a)$
+\end_inset
+
+, 
+\begin_inset Formula $q:=\alpha(b)$
+\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 $P_{\alpha}:=P_{a}^{b}(\alpha):T_{p}S\to T_{q}S$
+\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