From fe4faa19c5d544cf828e69600649c4c360efd389 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jan-Bernhard=20Korda=C3=9F?= Date: Tue, 22 Nov 2016 16:44:21 +0100 Subject: [PATCH] Minor corrections. --- contents.tex | 6 +++--- frontmatter.tex | 10 +++++----- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/contents.tex b/contents.tex index 80bed7c..f41ebbb 100644 --- a/contents.tex +++ b/contents.tex @@ -1,7 +1,7 @@ \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. Often they reveal more subtle geometric information. @@ -865,7 +865,7 @@ where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. \begin{align*} \Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij}) \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 \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, @@ -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 $N$. 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*} \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)) diff --git a/frontmatter.tex b/frontmatter.tex index ae0a809..fe5ed09 100644 --- a/frontmatter.tex +++ b/frontmatter.tex @@ -6,10 +6,10 @@ % \includegraphics[scale=0.15]{mathe-logo.jpg} \vspace*{3cm} - \textls[20]{ - {\Large \textsc{Notes taken at the}}\\[0.8cm] - {\huge \textsc{RTG Lectures of the RTG 2227}}\\[2.2cm] - {\Large \textsc{on the subjects of}}\\[1.8cm] + \textls[10]{ + {\large \textsc{Notes taken at the}}\\[0.8cm] + {\Huge \textsc{RTG Lectures}}\\[2.2cm] + {\large \textsc{on the subjects of}}\\[1.8cm] {\Large \textsc{Torsion Invariants}}\\ \textsc{by Prof. Dr. R. Sauer}\\[1.2cm] {\Large\textsc{Harmonic maps}}\\ @@ -17,7 +17,7 @@ \vfill - \textsc{\Large Karlsruhe and Heidelberg}\\[0.4cm] + \textsc{\Large Karlsruhe and Heidelberg}\\[0.5cm] \textsc{\Large Winter 2016/17} } \vspace{3cm} -- GitLab