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hyperref, ntheorem, amsthm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{array,xtab,ragged2e} % for symbol table +\newlength\mylengtha +\newlength\mylengthb +\newcolumntype{P}[1]{>{\RaggedRight}p{#1}} +\tabcolsep=3pt % default: 6pt +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{acronym} \usepackage{cancel} \usepackage{shortcuts} diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 3fffed7..fd6b87c 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -449,7 +449,7 @@ Die Teilraumtopologie wird auch \textit{Spurtopologie} oder \begin{bemerkung} \begin{bemenum} - \item \xindex{Homöomorphismengruppe}Für jeden topologischen Raum ist + \item \xindex{Homöomorphismengruppe}Für jeden topologischen Raum $X$ ist \[\Homoo(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}\] eine Gruppe. \item \xindex{Isometrie}Jede Isometrie $f:X \rightarrow Y$ zwischen metrischen diff --git a/documents/GeoTopo/Symbolverzeichnis.tex b/documents/GeoTopo/Symbolverzeichnis.tex index 8e238ae..f52f62c 100644 --- a/documents/GeoTopo/Symbolverzeichnis.tex +++ b/documents/GeoTopo/Symbolverzeichnis.tex @@ -7,81 +7,128 @@ % Mengenoperationen % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Mengenoperationen} -$A^C\;\;\;$ Komplement der Menge $A$\\ -$\mathcal{P}(M)\;\;\;$ Potenzmenge von $M$\\ -$\overline{M}\;\;\;$ Abschluss der Menge $M$\\ -$\partial M\;\;\;$ Rand der Menge $M$\\ -$M^\circ\;\;\;$ Inneres der Menge $M$\\ -$A \times B\;\;\;$ Kreuzprodukt zweier Mengen\\ -$A \subseteq B\;\;\;$ Teilmengenbeziehung\\ -$A \subsetneq B\;\;\;$ echte Teilmengenbeziehung\\ -$A \setminus B\;\;\;$ $A$ ohne $B$\\ -$A \cup B\;\;\;$ Vereinigung\\ -$A \dcup B\;\;\;$ Disjunkte Vereinigung\\ -$A \cap B\;\;\;$ Schnitt\\ + +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} -$AB\;\;\;$ 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 $K$\\ + +\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} -$\Homoo(X)\;\;\;$ Homöomorphismengruppe\\ -$\Iso(X)\;\;\;$ Isometriengruppe\\ -$\GL_n(K)\;\;\;$ Allgemeine lineare Gruppe\footnote{von \textit{\textbf{G}eneral \textbf{L}inear Group}}\\ -$\SL_n(K)\;\;\;$ Spezielle lineare Gruppe\\ -$\PSL_n(K)\;\;\;$ Projektive lineare Gruppe\\ -$\Perm(X)\;\;\;$ Permutationsgruppe\\ -$\Sym(X)\;\;\;$ Symmetrische Gruppe + +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} -$\gamma: I \rightarrow X\;\;\;$ Ein Weg\\ -$[\gamma]\;\;\;$ Homotopieklasse von $\gamma$\\ -$\gamma_1 * \gamma_2\;\;\;$ Zusammenhängen von Wegen\\ -$\gamma_1 \sim \gamma_2\;\;\;$ Homotopie von Wegen\\ -$\overline{\gamma}(x) = \gamma(1-x)\;\;\;$ Inverser Weg\\ -$C := \gamma([0,1])\;\;\;$ Bild eines Weges $\gamma$ +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} -$\fB\;\;\;$ Basis einer Topologie\\ -$\calS\;\;\;$ Subbasis einer Topologie\\ -$\fB_\delta(x)\;\;\;$ $\delta$-Kugel um $x$\\ -$\fT\;\;\;$ Topologie\\ -$\atlas\;\;\;$ 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 $a$\\ +\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\\ -$f \circ g\;\;\;$ 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}) +\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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -100,21 +147,32 @@ $\mdp = \Set{2, 3, 5, 7, \dots}\;\;\;$ Primzahlen\\ $\mdh = \Set{z \in \mdc | \Im{z} > 0}\;\;\;$ obere Halbebene\\ $I = [0,1] \subsetneq \mdr\;\;\;$ Einheitsintervall\\ -$f:S^1 \hookrightarrow \mdr^2\;\;\;$ 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 $f$\footnote{von \textit{\textbf{V}anishing Set}} +\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} -$D_p F: \mdr^2 \rightarrow \mdr^3\;\;\;$ 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\\ + +\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}} diff --git a/documents/GeoTopo/definitions/definitionen.pdf b/documents/GeoTopo/definitions/definitionen.pdf index 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