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Martin Thoma 2013-11-28 22:29:27 +01:00
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documents/GeoTopo/Kapitel2.tex
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@ -432,7 +432,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Mitschrieb vom 21.11.2013 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{korollar}
\begin{korollar}\label{kor:regular-surface-mannigfaltigkeit}
Jede reguläre Fläche $S \subseteq \mdr^3$ ist eine 2-dimensionale,
differenzierbare Mannigfaltigkeit.
\end{korollar}
@ -441,7 +441,13 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
\todo{Hier muss ich nochmals drüberlesen.}
\underline{z.Z.:} $F_j^{-1} \circ F_i$ ist Diffeomorphismus
\todo[inline]{Bild $F_j^{-1} \circ F_i$}
\begin{figure}[htp]
\centering
\input{figures/topology-parametric-surface-mapping.tex}
\caption{Reguläre Fläche $S$ zum Beweis von Korollar~\ref{kor:regular-surface-mannigfaltigkeit}}
\label{fig:parametric-surface-mapping}
\end{figure}
\underline{Idee:} Finde differenzierbare Funktion $\tilde{F_j^{-1}}$
in Umgebung $W$ von $s$, sodass $\tilde{F_j^{-1}}|_{S \cap W} = F_j^{-1}$.
@ -819,8 +825,6 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
\item \Obda{} sei $0 \in P$ und $P \subseteq \fB_1(0)$. Projeziere
$0P$ von $0$ aus auf $\partial \fB_1(0) = S^2$.
Erhalte Triangulierung von $S^2$.
\todo[inline]{Bild von rundem Wuerfel}
\item Sind $P_1$ und $P_2$ konvexe Polygone und $T_1, T_2$
die zugehörigen Triangulierungen von $S^2$, so gibt es
eine eine Triangulierungen $T$, die sowohl um $T_1$ als
@ -855,7 +859,9 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
Dann gilt: $d_{n-1} \circ d_n = 0$
\todo[inline]{Skizze von Dreieck}
\input{figures/topology-oriented-triangle.tex}
$a < b < c$
$d_2 \sigma = e_1 - e_2 + e_3 = c - b - (c-a) + b - a = 0$
\end{korollar}

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documents/GeoTopo/figures/topology-oriented-triangle.tex Normal file
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\begin{tikzpicture}
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
\node (a)[point,label={[label distance=0cm]210:$a$}] at (210:1cm) {};
\node (b)[point,label={[label distance=0cm]-45:$b$}] at (330:1cm) {};
\node (c)[point,label={[label distance=0cm]90:$c$}] at (90:1cm) {};
\node (sigma) at (0,0) {$\sigma$};
\draw[->, very thick] (a) edge node[label=below:$e_3$] {} (b);
\draw[->, very thick] (b) edge node[label=right:$e_1$] {} (c);
\draw[->, very thick] (c) edge node[label=left:$e_2$] {} (a);
\end{tikzpicture}

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documents/GeoTopo/figures/topology-parametric-surface-mapping.tex Normal file
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@ -0,0 +1,25 @@
\begin{tikzpicture}
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
\draw (0,0) ellipse (2cm and 1cm);
\def\ringa{(-0.3,0) circle (0.5cm)}
\def\ringb{(+0.3,0) circle (0.5cm)}
\draw \ringa;
\draw[red] \ringb;
%\node at (-1,0.3) {$U_i$};
%\node at (+1,0.3) {$U_j$};
\node at (-1.9,-2) {$U_i$};
\node[red] at (+1.9,-2) {$U_j$};
\node at (+2.0,0.7) {$S$};
\node[point,label={[label distance=-0.1cm]90:$s$}] at (0,0) {};
\path[<-] (-0.35,0) edge [bend angle=10,bend right] node[label={[label distance=0.1cm]210:$F_i$}] {} (-1,-1.5);
\path[<-,red] (+0.35,0) edge [bend angle=10,bend left] node[label={[label distance=0.1cm]-30:$F_j$}] {} (+1,-1.5);
\draw (-1,-2) circle (0.5cm);
\draw[red] (+1,-2) circle (0.5cm);
\path[->, green, thick] (-0.3,-2) edge node[label=below:$\scriptstyle F_j^{-1} \circ F_i$] {} (0.3,-2);
\end{tikzpicture}