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a/documents/GeoTopo/Fragen/Fragen.tex +++ b/documents/GeoTopo/Fragen/Fragen.tex @@ -159,4 +159,13 @@ hinzufügen, dass der Atlas $n$-dimensional sein soll? \end{definition} Gibt es ein Beispiel, das zegit, dass nicht $\fB = \fT$ gilt? +\section{12.) $\Delta^2$ explizit} +Wie sieht der Standard-Simplex der dim. 2, also $\Delta^2$, explizit +notiert aus? Praktisch ist das ja die konvexe Hülle der Standard-Basisvektoren +$e_0, e_1, e_2$ (also $\begin{pmatrix}0\\0\\1\end{pmatrix},\begin{pmatrix}0\\1\\0\end{pmatrix},\begin{pmatrix}1\\0\\0\end{pmatrix}$), +also ein Polyeder mit vier Flächen im $\mdr^3$ (jedoch kein regelmäßiges Tetraeder, oder?) + +Das ist dann nur das Gitter dieses Polyeders, aber nicht die Flächen +oder sogar etwas innerhalb vom Polyeder, oder? + \end{document} diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 112b7775bc5f6c0be5233f8e868843f5a6800bef..b592aa63cb05f6676de0a0e98b3c3890e95cc247 100644 GIT binary patch delta 114533 zcma%>LvSWsus~zmwr$&**mg3pF~Jvnv2EM7ZQIUFZ0r4R`F5|W7rp3P)r+p`Tc=OK zR6E$~bURo85EKXo1P4L@A%RdpXdny_76=D~2O3pZyemF8TJMFHlM+>x0>n+28t=L-?}gVfNOkDbYiMrs~=36g)qBIuJ%fvSuN9+Upxzv&HjX_$c1 zjce1EDJ=QYiO)3+mE)|bl_%0YUqcqYx%(j;D+L*#Wb=So{@Cm1R?xY(CPuE5+oz4- zzffDETH+MwF@*1=gGvTwwm#?k0KjdJ@cyEzBgHUAp{Ac0Ijy{E3If9q3Anbx-wQj@ 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zL3(|5CJA$A)zNCD=0K?oafp_=rNX4KMHl5Iwf{dPt46=CE{cm67&$h8rc>_c3AqsM zgSwa!-K#Qlnq*PXe_o; s^lAYGV%)!paAYsnZTqXR9ZAcOFmiV`3u}eltcS37(A&1n6X&J(PnEIUasU7T diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index e664289..810e199 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -297,7 +297,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \begin{definition}% Sei $X$ eine differenzierbare Mannigfaltigkeit der Klasse $C^k$ - ($k \in \mdn \cup \Set{\infty}$) mit Atlas $(U_i, \varphi_i)_{i \in I}$. + ($k \in \mdn \cup \Set{\infty}$) mit Atlas $\atlas = (U_i, \varphi_i)_{i \in I}$. \begin{defenum} \item Eine Karte $(U, \varphi)$ auf $X$ heißt \textbf{verträglich}\xindex{verträglich} @@ -315,7 +315,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \begin{bemerkung} Für $n \geq 4$ gibt es auf $S^n$ mehrere verschiedene differenzierbare - Strukturen, die sog. \enquote{exotische Sphären}\xindex{Sphäre!exotische}. + Strukturen, die sogenannten \enquote{exotische Sphären}\xindex{Sphäre!exotische}. \end{bemerkung} \begin{definition} @@ -501,15 +501,13 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \end{beweis} \begin{definition}% - Sei $G$ eine Mannigfaltigkeit, $\circ: G \times G \rightarrow G$ - eine Abbildung, $(g,h) \mapsto g \cdot h$, sodass $(G, \circ)$ - eine Gruppe ist. + Sei $G$ eine Mannigfaltigkeit und $(G, \circ)$ eine Gruppe. \begin{defenum} \item $G$ heißt \textbf{topologische Gruppe}\xindex{Gruppe!topologische}, wenn die Abbildungen $\circ: G \times G \rightarrow G$ - und $\iota: G \rightarrow G$. - \[(g, h) \mapsto g \cdot h\;\;\; g \mapsto g^{-1}\] + und $\iota: G \rightarrow G$ definiert durch + \[g \circ h := g \cdot h \text{ und } \iota(g) := g^{-1}\] stetig sind. \item Ist $G$ eine differenzierbare Mannigfaltigkeit, so heißt $G$ \textbf{Lie-Gruppe}\xindex{Lie-Gruppe}, wenn @@ -538,7 +536,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch $\det A_{ij}$ kann $0$ werden, da: \[\begin{pmatrix}1 & 1\\-1&0\end{pmatrix}\] - \item $\SL_n(\mdr) = \Set{A \in \GL_n(\mdr) | \det(A) = 1} $ \todo{Besser strukturieren} + \item $\SL_n(\mdr) = \Set{A \in \GL_n(\mdr) | \det(A) = 1} $ \todo{Was soll das '?'} $\grad(\det-1)(A) = 0$? @@ -560,7 +558,7 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \section{Simplizialkomplex} \begin{definition}\xindex{Lage!allgemeine}% - Seien $v_0, \dots, v_k \in \mdr^n$ Punkte. + Seien $v_0, \dots, v_k \in \mdr^n$ Punkte.\xindex{Punkt} \begin{defenum} \item $v_0, \dots, v_k$ sind \textbf{in allgemeiner Lage} $\gdw$ es gibt keinen $(k-1)$-dimensionalen affinen Untervektorraum, der $v_0, \dots, v_k$ enthält @@ -687,14 +685,14 @@ Differenzierbare Mannigfaltigkeiten der Klasse $C^\infty$ werden auch \end{defenum} \end{definition} -\begin{beispiel} +\begin{beispiel}[Simpliziale Abbildungen] \begin{bspenum} \item $\varphi(e_1) := b_1$, $\varphi(e_2) := b_2$\\ $\varphi$ ist eine eindeutig bestimmte lineare Abbildung \input{figures/topology-linear-mapping.tex} - \item Folgende Abbildung $\Delta^n \rightarrow \Delta^{n-1}$ + \item Folgende Abbildung $\varphi: \Delta^n \rightarrow \Delta^{n-1}$ ist simplizial: \input{figures/topology-triangle-to-line.tex} diff --git a/documents/GeoTopo/Symbolverzeichnis.tex b/documents/GeoTopo/Symbolverzeichnis.tex index 47df39c..a00d213 100644 --- a/documents/GeoTopo/Symbolverzeichnis.tex +++ b/documents/GeoTopo/Symbolverzeichnis.tex @@ -43,6 +43,7 @@ $\calS\;\;\;$ 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 $\sim$\\ @@ -63,7 +64,9 @@ $\chi(K)\;\;\;$ Euler-Charakteristik von $K$\\ $\Delta^k\;\;\;$ Standard-Simplex\\ $X \# Y\;\;\;$ Verklebung von $X$ und $Y$\\ $\gamma_1 * \gamma_2\;\;\;$ Zusammenhängen von Wegen\\ -$d_n\;\;\;$ Lineare Abbildung aus \cref{kor:9.11} +$\gamma_1 \sim \gamma_2\;\;\;$ Homotopie von Wegen\\ +$d_n\;\;\;$ Lineare Abbildung aus \cref{kor:9.11}\\ +$A \cong B\;\;\;$ $A$ ist isometrisch zu $B$ \onecolumn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%