Commit fe4faa19 authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

Minor corrections.

parent c64bf571
\tableofcontents \tableofcontents
\chapter{Tosion Invariants [Roman Sauer]} \chapter{Torsion Invariants [Roman Sauer]}
Torsion invariants fall into a class of so-called ``secondary invariants'' of topological spaces in the sense that they are only defined if a certain class of ``primary invariants'' (e.g. Betti numbers) vanish. Torsion invariants fall into a class of so-called ``secondary invariants'' of topological spaces in the sense that they are only defined if a certain class of ``primary invariants'' (e.g. Betti numbers) vanish.
Often they reveal more subtle geometric information. Often they reveal more subtle geometric information.
...@@ -865,7 +865,7 @@ where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. ...@@ -865,7 +865,7 @@ where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$.
\begin{align*} \begin{align*}
\Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij}) \Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij})
\end{align*} \end{align*}
($=0$ in $\{g_{ij}\}$ is constant)\todo{repair} and $\{g_{ij}\} = g_{11} = f(\partial_t,\partial_t) = \dop t^2(\partial_t,\partial_t) = 1$. ($=0$ if $\{g_{ij}\}$ is constant) and $\{g_{ij}\} = g_{11} = f(\partial_t,\partial_t) = \dop t^2(\partial_t,\partial_t) = 1$.
Hence Hence
\begin{align*} \begin{align*}
\tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0, \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0,
...@@ -1098,7 +1098,7 @@ Then we can calculate ...@@ -1098,7 +1098,7 @@ Then we can calculate
where $\Ric^g \colon \T M \otimes \T M \to \R$ is the Ricci tensor, $R^h$ is the full curvature tensor of $(N,h)$ and $\{e_1, \ldots, e_m\}$ is an orthonormal frame of $N$. where $\Ric^g \colon \T M \otimes \T M \to \R$ is the Ricci tensor, $R^h$ is the full curvature tensor of $(N,h)$ and $\{e_1, \ldots, e_m\}$ is an orthonormal frame of $N$.
The latter summand is $\operatorname{const} \sec^h(\operatorname{span}(\dop f_t(e_i),\dop f_t(e_j)))$. The latter summand is $\operatorname{const} \sec^h(\operatorname{span}(\dop f_t(e_i),\dop f_t(e_j)))$.
We continue We continue\footnote{TODO: According to Andy, who erased this part very quickly, there should be some mistake somewhere here...}
\begin{align*} \begin{align*}
\leq -\sum_{i = 1}^m h(\sum_{j=1}^m \dop f_t(\Ric^g(e_i,e_j)e_j),\dop f_t(e_i)) \leq -\sum_{i = 1}^m h(\sum_{j=1}^m \dop f_t(\Ric^g(e_i,e_j)e_j),\dop f_t(e_i))
\leq C \sum_{i,j = 1}^m h(\dop f_t(e_i, \dop f_t(e_j)) \leq C \sum_{i,j = 1}^m h(\dop f_t(e_i, \dop f_t(e_j))
......
...@@ -6,10 +6,10 @@ ...@@ -6,10 +6,10 @@
% \includegraphics[scale=0.15]{mathe-logo.jpg} % \includegraphics[scale=0.15]{mathe-logo.jpg}
\vspace*{3cm} \vspace*{3cm}
\textls[20]{ \textls[10]{
{\Large \textsc{Notes taken at the}}\\[0.8cm] {\large \textsc{Notes taken at the}}\\[0.8cm]
{\huge \textsc{RTG Lectures of the RTG 2227}}\\[2.2cm] {\Huge \textsc{RTG Lectures}}\\[2.2cm]
{\Large \textsc{on the subjects of}}\\[1.8cm] {\large \textsc{on the subjects of}}\\[1.8cm]
{\Large \textsc{Torsion Invariants}}\\ {\Large \textsc{Torsion Invariants}}\\
\textsc{by Prof. Dr. R. Sauer}\\[1.2cm] \textsc{by Prof. Dr. R. Sauer}\\[1.2cm]
{\Large\textsc{Harmonic maps}}\\ {\Large\textsc{Harmonic maps}}\\
...@@ -17,7 +17,7 @@ ...@@ -17,7 +17,7 @@
\vfill \vfill
\textsc{\Large Karlsruhe and Heidelberg}\\[0.4cm] \textsc{\Large Karlsruhe and Heidelberg}\\[0.5cm]
\textsc{\Large Winter 2016/17} \textsc{\Large Winter 2016/17}
} }
\vspace{3cm} \vspace{3cm}
......
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