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z#zoS6IB!h$bJKaw!OXR2{}NVr&Lc>{qKLYWRd)J^?FaO4#Q6DCVqaKtv`>P3+&n0B KbTZ1aDE|eQ0SamW diff --git a/documents/GeoTopo/Fragen/Fragen.tex b/documents/GeoTopo/Fragen/Fragen.tex index a6492f2..0456548 100644 --- a/documents/GeoTopo/Fragen/Fragen.tex +++ b/documents/GeoTopo/Fragen/Fragen.tex @@ -51,95 +51,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \chapter{Fragen zu Definitionen} -\section*{6.) Basisbeispiele} -\todo[inline]{Kennst du ein Beispiel für eine Subbasis in einem Topologischen Raum, -die keine Basis ist?} - -Wie ist es mit folgendem? - -Sei $(X, \fT)$ ein topologischer Raum mit - $X = \Set{0,1,2}$ und $\fT = \Set{\emptyset, \Set{0}, \Set{0,1}, X}$.\\ - Dann ist $\calS = \Set{\emptyset, \Set{0,1}, \Set{0,2}}$ eine Subbasis von - $\fT$, da gilt: - \begin{itemize} - \item $\emptyset \in \calS$ - \item $\Set{0} = \Set{0, 1} \cap \Set{0,2}$ - \item $\Set{0,1} \in \calS$ - \item $X = \Set{0,1} \cup \Set{0,2}$ - \end{itemize} - Allerings ist $\calS$ keine Basis von $(X, \fT)$, da - $\Set{0}$ nicht als Vereinigung von Elementen aus $\calS$ - erzeugt werden kann. - - -\section*{9.) Mannigfaltigkeit mit Rand} -\begin{definition}% - Sei $X$ ein topologischer Raum und $n \in \mdn$. - \begin{defenum} - \item Eine $n$-dimensionale \textbf{Karte}\xindex{Karte} auf - $X$ ist ein Paar $(U, \varphi)$, wobei $U \subseteq X$ - offen und $\varphi: U \rightarrow V$ Homöomorphismus - von $U$ auf eine offene Teilmenge $V \subseteq \mdr^n$. - \item Ein $n$-dimensionaler \textbf{Atlas}\xindex{Atlas} $\atlas$ auf $X$ ist eine - Familie $(U_i, \varphi_i)_{i \in I}$ von Karten auf $X$, - sodass $\bigcup_{i \in I} U_i = X$. - \item $X$ heißt (topologische) $n$-dimensionale \textbf{Mannigfaltigkeit}\xindex{Mannigfaltigkeit}, - wenn $X$ hausdorffsch ist, eine abzählbare Basis der - Topologie hat und ein $n$-dimensionalen Atlas besitzt. - \end{defenum} -\end{definition} -\begin{definition}\xindex{Mannigfaltigkeit!mit Rand}% - Sei $X$ ein Hausdorffraum mit abzählbarer Basis der Topologie. - $X$ heißt $n$-dimensionale \textbf{Mannigfaltigkeit mit Rand}, - wenn es einen Atlas $(U_i, \varphi_i)$ gibt, wobei $U_i \subseteq X_i$ - offen und $\varphi_i$ ein Homöomorphismus auf eine offene - Teilmenge von - \[R_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_{\color{red}m} \geq 0}\] - ist. -\end{definition} - -\todo[inline]{Sind Mannigfaltigkeiten mit Rand auch Mannigfaltigkeiten? Sollte das rote $m$ eventuell $n$ sein? Oder sollte es ein $i$ sein, mit $i = 1..n$?} - -Laut \url{https://de.wikipedia.org/wiki/Mannigfaltigkeit_mit_Rand}: - -\enquote{Eine Mannigfaltigkeit mit Rand ist mathematisches Objekt aus der Differentialgeometrie. Es handelt sich hierbei nicht um einen Spezialfall einer Mannigfaltigkeit, sondern ganz im Gegenteil um eine Verallgemeinerung.} - -\todo[inline]{Ist die Aussage auf Wikipedia korrekt? Für mich sieht das so aus, also ob folgende Definition auch richtig wäre:} - -\begin{definition}\xindex{Mannigfaltigkeit!mit Rand}% - Sei $X$ eine Mannigfaltigkeit mit Atlas $\atlas$ und - \[\mdr_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_m \geq 0}\] - - $X$ heißt \textbf{Mannigfaltigkeit mit Rand}, wenn gilt: - \[\forall (U, \varphi) \in \atlas: \varphi(U) \subseteq \mdr_{+,0}^n\] -\end{definition} - -\section*{11.) Produkttopologie} -\begin{definition}\xindex{Produkttopologie}% - Seien $X_1, X_2$ topologische Räume.\\ - $U \subseteq X_1 \times X_2$ sei offen, wenn es zu jedem $x = (x_1, x_2) \in U$ - Umgebungen $U_i$ um $x_i$ mit $i=1,2$ gibt, sodass $U_1 \times U_2 \subseteq U$ - gilt. - - $\fT = \Set{U \subseteq X_1 \times X_2 | U \text{ offen}}$ - ist eine Topologie auf $X_1 \times X_2$. Sie heißt \textbf{Produkttopologie}. - $\fB = \Set{U_1 \times U_2 | U_i \text{ offen in } X_i, i=1,2}$ - ist eine Basis von $\fT$. -\end{definition} - -\todo[inline]{Gibt es ein Beispiel, das zegit, dass nicht $\fB = \fT$ gilt?} - -\section*{15.) Existenz der Parallelen} -\begin{definition}% - \begin{enumerate}[label=§\arabic*),ref=§\arabic*,start=5] - \item \label{axiom:5}\textbf{Parallelenaxiom}\xindex{Parallele}: - Für jedes $g \in G$ und jedes - $P \in X \setminus g$ gibt es höchstens ein $h \in G$ mit - $h \cap g = \emptyset$. $h$ heißt \textbf{Parallele zu $g$ durch $P$}. - \end{enumerate} -\end{definition} - -\todo[inline]{Wie beweist man, dass es genau eine gibt? (Verschiebung der Geraden in den entsprechenden Punkt mit der Isometrie, die die Halbebenen gleich lässt)} \section*{17.) Simpliziale Abbildungen} Wenn man Simpliziale Abbildungen wie folgt definiert @@ -219,51 +130,6 @@ Da $x \in U$ beliebig gewählt war gilt: $\fT|_A \subseteq \fT'$ \todo[inline]{Wenn $G$ eine topologische Gruppe ist, dann ist $\circ$ doch auf jeden Fall stetig! Was soll die Definition? Des Weiteren verstehe ich $g \circ h := g \cdot h$ nicht. Was ist $\cdot$?} -\section*{20.) Hyperbolische Metrik und Geraden} -\begin{definition}\xindex{Gerade!hyperbolische}% - Sei - \[\mdh:= \Set{z \in \mdc | \Im(z) > 0} = \Set{(x,y) \in \mdr^2 | y > 0}\] - die obere Halbebene bzw. Poincaré-Halbebene und $G = G_1 \cup G_2$ - mit - \begin{align*} - G_1 &= \Set{g_1 \subseteq \mdh | \exists m \in \mdr, r \in \mdr_{>0}: g_1 = \Set{z \in \mdh : |z-m|=r}}\\ - G_2 &= \Set{g_2 \subseteq \mdh | \exists x \in \mdr: g_2 = \Set{z \in \mdh: \Re(z) = x}} - \end{align*} - - Die Elemente von $\mdh$ heißen \textbf{hyperbolische Geraden}. -\end{definition} -\begin{definition}\xindex{Metrik!hyperbolische}% - Für $z_1, z_2 \in \mdh$ sei $g_{z_1, z_2}$ die eindeutige hyperbolische - Gerade durch $z_1$ und $z_2$ und $a_1, a_2$ die - \enquote{Schnittpunkte} von $g_{z_1, z_2}$ mit $\mdr \cup \Set{\infty}$. - - Dann sei $d(z_1, z_2) := \frac{1}{2} \ln |\DV(a_1, z_4, a_2, z_2) |$ - und heiße \textbf{hyperbolische Metrik}. -\end{definition} - -\todo[inline]{Wir haben hyperbolische Geraden mit der euklidischen Metrik beschrieben. Kann man hyperbolische Geraden auch mit der hyperbolischen Metrik beschreiben? Wie?} -vgl. Beweis von Bemerkung 68 b) - -\section*{21.) Defintion Normalenvektor} -\begin{definition}%In Vorlesung: Definition 16.2 - Sei $\gamma: I \rightarrow \mdr^2$ eine durch Bogenlänge - parametrisierte Kurve. - - \begin{defenum} - \item Für $t \in I$ sei $n(t)$ \textbf{Normalenvektor}\xindex{Normalenvektor} - an $\gamma$ in $t$, d.~h. - \[\langle n(t), \gamma'(t) \rangle = 0, \;\;\; \|n(t)\|=1 \] - und $\det((\gamma_1(t), n(t))) = +1$. - \item Nach \cref{bem:16.1d} sind $n(t)$ und $\gamma''(t)$ linear - abhängig, d.~h. es gibt $\kappa(t) \in \mdr$ mit - \[\gamma''(t) = \kappa(t) \cdot n(t)\] - $\kappa(t)$ heißt \textbf{Krümmung}\xindex{Krümmung} - von $\gamma$ in $t$. - \end{defenum} -\end{definition} - -\todo[inline]{Sollte es in a) $\det((\gamma_1{\color{red}'}(t), n(t))) = +1$ sein?} - \section*{22.) MF-Beispiel} $\praum^n(\mdr) = (\mdr^{n+1} \setminus \Set{0})/_\sim = S^n /_\sim$ und $\praum^n(\mdc)$ sind Mannigfaltigkeiten der Dimension $n$ bzw. $2n$, da gilt: @@ -331,14 +197,6 @@ $\Rightarrow$ Widerspruch \todo[inline]{TODO} \end{beweis} - -\section*{24) Tangentialebene} -Erinnerung Sie sich an \cref{def:8.5} \enquote{reguläre Fläche}. - -Äquivalent dazu ist: $S$ ist lokal von der Form -\[V(f) = \Set{x \in \mdr^3 | f(x) = 0 }\] -für eine $C^\infty$-Funktion $f: \mdr^\infty \rightarrow \mdr$.\todo{Wirklich $\mdr^\infty$?} - \section*{25.) Fragen} \begin{enumerate} \item Kapitel II: diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index c01efcfb851fd5dfc233377c7e686af8318f3bdd..1a2ce3005a303608b62140ab4cb7794fb5772d36 100644 GIT binary patch delta 83404 zcmY(LcRZEv|HrfU-r0L(onxzBSO|~lo=|CA`+67WS3DSGBdI& zBkOnFeR9tC_s8{locH~{uGjUvuj{(sulu0S3#b9VKBED@>Om&7!HpEH?#-Mdz*raz zh9Wvwdb(fSd-2DD;fsgskBA!z4oail9`lXYglOJ=Kke-KX4m`4^Q-eLarX$EDs5Q{W zCUS>%wrq!?wnSs!!_G;xR)(YYTqZ|*{=}0!1-EPS{SKkui`g?fa{uHc?PTT8JF-&u zDe~yZquEZ78E0l1XV|M+ea@>ZepPtKh*f<2S@E5=SJ`cyJC31$q0T*GyuKj9>9xL(O1Gre)#6Q7SoF27Db7Ci zT};IFZd_`wWN7#2z0LkflUpY&Q!|v$cu!SMb#-YyzcuZ0$B6%_Z>9^GDEDYD-o<4- z33C*aMCQfQcuw1|CeZFms;xAX(`r>wSvEmA-K6Y_9WTg|+P>fg?9WvuG(MkdAj9-c zWWQ>PCCuY|nVID1>6SdLwERvil7vaGD(Oj&w)4tb_vxVki>JPH{^FyRb|e`OjI?zBMAR

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z*kX@+#a$&3Bj<`y7HD~Le6NJwV8>#2F*kJhco_W1qJeMut_i}#tlKS=C8<1}g{*oda{{hZcGX0)8{j)e$gX4EA9S$FL-Y80 zy3BG^RGjR1`k>(RzMJWT_s3wz_xa9Cq9ee9kQ1D_vGU!51?g7FrJ-r5bb9Rb>%SeJ z9o3q%xP*KlRuXEZntA?)={+NhkGz)#Q-(x_E;JqJ<%|W{=H*SpKe}=&WVC+^dA-jVKau(bllIC=lucb#Qyn5LeahMd@?S$jc9{SG diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 5117dc8..b99c30e 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -243,12 +243,14 @@ Anschaulich ist also ein $n$-dimensionale Mannigfaltigkeit lokal dem $\mdr^n$ ä wenn es einen Atlas $(U_i, \varphi_i)$ gibt, wobei $U_i \subseteq X_i$ offen und $\varphi_i$ ein Homöomorphismus auf eine offene Teilmenge von - \[R_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_m \geq 0}\] + \[R_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_n \geq 0}\] ist. \end{definition} $R_{+,0}^n$ ist ein \enquote{Halbraum}\xindex{Halbraum}. +\underline{Hinweis:} Mannigfaltigkeiten mit Rand sind keine Mannigfaltigkeiten. + \begin{figure}[ht] \centering \subfloat[Halbraum]{ diff --git a/documents/GeoTopo/Kapitel5.tex b/documents/GeoTopo/Kapitel5.tex index 7a654dc..d713d0d 100644 --- a/documents/GeoTopo/Kapitel5.tex +++ b/documents/GeoTopo/Kapitel5.tex @@ -46,7 +46,7 @@ \item Für $t \in I$ sei $n(t)$ \textbf{Normalenvektor}\xindex{Normalenvektor} an $\gamma$ in $t$, d.~h. \[\langle n(t), \gamma'(t) \rangle = 0, \;\;\; \|n(t)\|=1 \] - und $\det((\gamma_1(t), n(t))) = +1$. + und $\det((\gamma_1'(t), n(t))) = +1$. \item Nach \cref{bem:16.1d} sind $n(t)$ und $\gamma''(t)$ linear abhängig, d.~h. es gibt $\kappa(t) \in \mdr$ mit \[\gamma''(t) = \kappa(t) \cdot n(t)\] @@ -108,7 +108,7 @@ Erinnerung Sie sich an \cref{def:8.5} \enquote{reguläre Fläche}. Äquivalent dazu ist: $S$ ist lokal von der Form \[V(f) = \Set{x \in \mdr^3 | f(x) = 0 }\] -für eine $C^\infty$-Funktion $f: \mdr^\infty \rightarrow \mdr$.\todo{Wirklich $\mdr^\infty$?} +für eine $C^\infty$-Funktion $f: \mdr^3 \rightarrow \mdr$. \begin{definition}%In Vorlesung: 17.1 Sei $S \subseteq \mdr^3$ eine reguläre Fläche, $s \in S$, diff --git a/documents/GeoTopo/definitions/definitionen.pdf b/documents/GeoTopo/definitions/definitionen.pdf index a3458def792aa9bd9095020f11dbb327944c3bc8..2d1612d48dac73d971dc935766f18e774db88cc8 100644 GIT binary patch delta 28796 zcmV(vKXDHFbO|(>n=3?^fYVu$C6R zNt!%KRl#B8Bs}wT`9#_lWrd=;dSbEKT+Fy4)4$CRyQ|ynOTzhfez<`zrJdfa_Tkt0 z<=smz#Ow9te7o3&AOAext#*Im=d0?wXBUP~Z{J+LZ)dLm34ACllam>xg_SaN zyj(2`Z2Sf!PY1@Yy};;?tJ}?dy$H|p zv^B*G!dUh6-{+gnYI!|h?^nyW7o33{K~jQ?ATsynhxz6T8$Oo>p{jps=us&lMG7ND zE}{&d+&k{}Y|h0&sV=BM%j zG67xG-1tcshHw~EA#6FO71|7k2_+9bzL-&^rw@@9DeMX%u_jHUMV+_WChc7=Of@6QMV|i{-Htl)ILNMO3{Y9>cqBhCpwu4_1D_96pdjMP~DD=_RiL4s~ 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