Merge remote-tracking branch 'upstream/master'

1 files changed, 99 insertions, 0 deletions

diff --git a/vorlesungen/slides/7/zusammenhang.tex b/vorlesungen/slides/7/zusammenhang.tex
new file mode 100644
index 0000000..6a43cd8
--- /dev/null
+++ b/vorlesungen/slides/7/zusammenhang.tex
@@ -0,0 +1,99 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zusammenhang}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zusammenhängend --- oder nicht}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\ds{2.4}
+\coordinate (A) at (0,0);
+\coordinate (B) at (\ds,0);
+\coordinate (C) at ({2*\ds},0);
+
+\node at (A) {$\operatorname{SO}(n)$};
+\node at (B) {$\operatorname{O}(n)$};
+\node at (C) {$\{\pm 1\}$};
+
+\draw[->,shorten <= 0.6cm,shorten >= 0.5cm] (A) -- (B);
+\draw[->,shorten <= 0.5cm,shorten >= 0.5cm] (B) -- (C);
+\node at ($0.5*(B)+0.5*(C)$) [above] {$\det$};
+
+\coordinate (A2) at (0,-1.0);
+\coordinate (B2) at (\ds,-1.0);
+\coordinate (C2) at ({2*\ds},-1.0);
+
+\draw[color=blue] (A2) ellipse (1cm and 0.3cm);
+\draw[color=blue] (B2) ellipse (1cm and 0.3cm);
+\node[color=blue] at (C2) {$+1$};
+
+\coordinate (A3) at (0,-1.7);
+\coordinate (B3) at (\ds,-1.7);
+\coordinate (C3) at ({2*\ds},-1.7);
+
+\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B2) -- (C2);
+\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B3) -- (C3);
+
+\draw[color=red] (B3) ellipse (1cm and 0.3cm);
+\node[color=red] at (C3) {$-1$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\begin{block}{Zusammenhangskomponente von $e$}
+$G_e\subset G$ grösste zusammenhängende Menge, die $e$ enthält:
+\begin{align*}
+\operatorname{SO}(n)&\subset \operatorname{O}(n)
+\\
+\{A\in\operatorname{GL}_n(\mathbb{R})\,|\, \det A > 0\}
+	&\subset \operatorname{GL}_n(\mathbb{R})
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenschaften}
+\begin{itemize}
+\item
+{\bf Untergruppe}: $\gamma_i(t)$ Weg von $e$ nach $g_i$,
+dann ist
+\begin{itemize}
+\item
+$\gamma_1(t)\gamma_2(t)$ ein Weg von $e$ nach $g_1g_2$
+\item
+$\gamma_1(t)^{-1}$ Weg von $e$ nach $g_1^{-1}$
+\end{itemize}
+\item
+{\bf Normalteiler}: $\gamma(t)$ ein Weg von $e$ nach $g$, dann 
+ist $h\gamma(t)h^{-1}$ ein Weg von $h$ nach $hgh^{-1}$
+$\Rightarrow hG_eh^{-1}\subset G_e$
+\end{itemize}
+\end{block}
+\begin{block}{Quotient}
+$G/G_e$ ist eine diskrete Gruppe
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (0,0);
+\coordinate (B) at (2,0);
+\coordinate (C) at (4,0);
+\node at (A) {$G_e$};
+\node at (B) {$G$};
+\node at (C) {$G/G_e$};
+\draw [->,shorten <= 0.3cm,shorten >= 0.3cm] (A) -- (B);
+\draw [->,shorten <= 0.3cm,shorten >= 0.5cm] (B) -- (C);
+\end{tikzpicture}
+\end{center}
+\vspace{-7pt}
+$\Rightarrow$ $G_e$ und $G/G_e$ separat studieren
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup