%!TEX root = GeoTopo.tex \markboth{Symbolverzeichnis}{Symbolverzeichnis} \twocolumn \chapter*{Symbolverzeichnis} \addcontentsline{toc}{chapter}{Symbolverzeichnis} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mengenoperationen % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Mengenoperationen} Seien $A, B$ und $M$ Mengen. % Set \mylengtha to widest element in first column; adjust % \mylengthb so that the width of the table is \columnwidth \settowidth\mylengtha{$A \subsetneq B$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $A^C $ & Komplement von $A$\\ $\mathcal{P}(M)$ & Potenzmenge von $M$\\ $\overline{M}$ & Abschluss von $M$\\ $\partial M$ & Rand der Menge $M$\\ $M^\circ$ & Inneres der Menge $M$\\ $A \times B$ & Kreuzprodukt\\ $A \subseteq B$ & Teilmengenbeziehung\\ $A \subsetneq B$ & echte Teilmengenbeziehung\\ $A \setminus B$ & Differenzmenge\\ $A \cup B$ & Vereinigung\\ $A \dcup B$ & Disjunkte Vereinigung\\ $A \cap B$ & Schnitt\\ \end{xtabular} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Geometrie % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Geometrie} \settowidth\mylengtha{$\overline{AB} \cong \overline{CD}$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $AB$ & Gerade durch die Punkte $A$ und $B$\\ $\overline{AB}$ & Strecke mit Endpunkten $A$ und $B$\\ $\triangle ABC$ & Dreieck mit Eckpunkten $A, B, C$\\ $\overline{AB} \cong \overline{CD}$& Die Strecken $\overline{AB}$ und $\overline{CD}$ sind isometrisch\\ $|K|$ & Geometrische Realisierung des Simplizialkomplexes~$K$\\ \end{xtabular} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Gruppen % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Gruppen} Sei $X$ ein topologischer Raum und $K$ ein Körper. \settowidth\mylengtha{$\Homoo(X)$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $\Homoo(X)$ & Homöomorphis\-men\-gruppe\\ $\Iso(X)$ & Isometrien\-gruppe\\ $\GL_n(K)$ & Allgemeine lineare Gruppe (von \textit{\textbf{G}eneral \textbf{L}inear Group})\\ $\SL_n(K)$ & Spezielle lineare Gruppe\\ $\PSL_n(K)$ & Projektive lineare Gruppe\\ $\Perm(X)$ & Permutations\-gruppe\\ $\Sym(X)$ & Symmetrische Gruppe\\ \end{xtabular} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Wege % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Wege} Sei $\gamma: I \rightarrow X$ ein Weg. \settowidth\mylengtha{$\gamma_1 \sim \gamma_2$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $[\gamma]$ & Homotopieklasse von $\gamma$\\ $\gamma_1 * \gamma_2$ & Zusammenhängen von Wegen\\ $\gamma_1 \sim \gamma_2$ & Homotopie von Wegen\\ $\overline{\gamma}(x)$ & Inverser Weg, also $\overline{\gamma}(x) := \gamma(1-x)$\\ $C$ & Bild eines Weges $\gamma$, also $C := \gamma([0,1])$ \end{xtabular} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Weiteres % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Weiteres} \settowidth\mylengtha{$\fB_\delta(x)$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $\fB$ & Basis einer Topologie\\ $\fB_\delta(x)$& $\delta$-Kugel um $x$\\ $\calS$ & Subbasis einer Topologie\\ $\fT$ & Topologie\\ \end{xtabular} \settowidth\mylengtha{$X /_\sim$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $\atlas$ & Atlas\\ $\praum$ & Projektiver Raum\\ $\langle \cdot , \cdot \rangle$ & Skalarprodukt\\ $X /_\sim$ & $X$ modulo $\sim$\\ $[x]_\sim$ & Äquivalenzklassen von $x$ bzgl. $\sim$\\ $\| x \|$ & Norm von $x$\\ $| x |$ & Betrag von $x$\\ $\langle a \rangle$ & Erzeugnis von $a$\\ \end{xtabular} $S^n\;\;\;$ Sphäre\\ $T^n\;\;\;$ Torus\\ \settowidth\mylengtha{$f^{-1}(M)$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $f \circ g$&Verkettung von $f$ und $g$\\ $\pi_X$ &Projektion auf $X$\\ $f|_U$ $f$ &eingeschränkt auf $U$\\ $f^{-1}(M)$&Urbild von $M$\\ $\rang(M)$ & Rang von $M$\\ $\chi(K)$ & Euler-Charakteristik von $K$\\ $\Delta^k$ & Standard-Simplex\\ $X \# Y$ & Verklebung von $X$ und $Y$\\ $d_n$ & Lineare Abbildung aus \cref{kor:9.11}\\ $A \cong B$& $A$ ist isometrisch zu $B$\\ $f_*$ & Abbildung zwischen Fundamentalgruppen (vgl. \cpageref{korr:11.5}) \end{xtabular} \onecolumn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Zahlenmengen % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Zahlenmengen} $\mdn = \Set{1, 2, 3, \dots} \;\;\;$ Natürliche Zahlen\\ $\mdz = \mdn \cup \Set{0, -1, -2, \dots} \;\;\;$ Ganze Zahlen\\ $\mdq = \mdz \cup \Set{\frac{1}{2}, \frac{1}{3}, \frac{2}{3}} = \Set{\frac{z}{n} \text{ mit } z \in \mdz \text{ und } n \in \mdz \setminus \Set{0}} \;\;\;$ Rationale Zahlen\\ $\mdr = \mdq \cup \Set{\sqrt{2}, -\sqrt[3]{3}, \dots}\;\;\;$ Reele Zahlen\\ $\mdr_+\;$ Echt positive reele Zahlen\\ $\mdr_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_n \geq 0}\;\;\;$ Halbraum\\ $\mdr^\times = \mdr \setminus \Set{0} \;$ Einheitengruppe von $\mdr$\\ $\mdc = \Set{a+ib|a,b \in \mdr}\;\;\;$ Komplexe Zahlen\\ $\mdp = \Set{2, 3, 5, 7, \dots}\;\;\;$ Primzahlen\\ $\mdh = \Set{z \in \mdc | \Im{z} > 0}\;\;\;$ obere Halbebene\\ $I = [0,1] \subsetneq \mdr\;\;\;$ Einheitsintervall\\ \settowidth\mylengtha{$f:S^1 \hookrightarrow \mdr^2$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $f:S^1 \hookrightarrow \mdr^2$& Einbettung der Kreislinie in die Ebene\\ $\pi_1(X,x)$ & Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\ $\Fix(f)$ & Menge der Fixpunkte der Abbildung $f$\\ $\|\cdot\|_2$ & 2-Norm; Euklidische Norm\\ $\kappa$ & Krümmung\\ $\kappa_{\ts{Nor}}$ & Normalenkrümmung\\ $V(f)$ & Nullstellenmenge von $f$\footnotemark \end{xtabular} \footnotetext{von \textit{\textbf{V}anishing Set}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Krümmung % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Krümmung} \settowidth\mylengtha{$D_p F: \mdr^2 \rightarrow \mdr^3$} \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} $D_p F: \mdr^2 \rightarrow \mdr^3$& Lineare Abbildung mit Jacobi-Matrix in $p$ (siehe \cpageref{def:Tangentialebene})\\ $T_s S$ & Tangentialebene an $S \subseteq \mdr^3$ durch $s \in S$\\ $d_s n(x)$ & Weingarten-Abbildung\\ \end{xtabular} \index{Faser|see{Urbild}} \index{kongruent|see{isometrisch}} \index{Kongruenz|see{Isometrie}}