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Riemannian geometry
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header.tex
\usepackage{amsthm} %\topmargin -1 cm %\oddsidemargin 0.4 cm %\evensidemargin 0.4 cm %\textwidth 18 cm %\textheight 24 cm \newcommand{\IR}{\mathbb{R}} \newcommand{\IC}{\mathbb{C}} \newcommand{\IH}{\mathbb{H}} \newcommand{\IM}{\mathbb{M}} \newcommand{\IS}{\mathbb{S}} \newcommand{\IZ}{\mathbb{Z}} \newcommand{\IO}{\mathbb{O}} \newcommand{\IK}{\mathbb{K}} \newcommand{\eps}{\varepsilon} \newcommand{\ov}[1]{\overline{#1}} \newcommand{\un}[1]{\underline{#1}} \DeclareMathOperator{\Ric}{Ric} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\Isom}{Isom} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\inj}{inj} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\vol}{vol} \DeclareMathOperator{\const}{const} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\im}{Im} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Stab}{Stab} \newcommand{\dd}{\textup{d}} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\spann}{span} \DeclareMathOperator{\ctanh}{ctanh} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\DIV}{div} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\sh}{sh} \DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\newtanh}{th} \DeclareMathOperator{\cth}{cth} \DeclareMathOperator{\Ad}{Ad} \renewcommand{\sinh}{\sh} \renewcommand{\cosh}{\ch} \renewcommand{\tanh}{\newtanh} \renewcommand{\ctanh}{\cth} \newcommand{\dotcup}{\ensuremath{\mathaccent\cdot\cup}} \newcommand{\EMPTY}[1]{} \newcommand{\TT}[3]{\Big( \begin{matrix} #1 \; #2 \\ #3 \end{matrix} \Big)} \newcommand{\tTT}[3]{\big( \begin{smallmatrix} #1 \; #2 \\ #3 \end{smallmatrix} \big)} \newcommand{\REMARK}[1]{\marginpar{\tiny #1}} \usepackage{amsmath, amssymb} \newtheorem{Theorem}{Theorem} \newtheorem{Lemma}{Lemma} \newtheorem{Corollary}{Corollary} \newtheorem{Proposition}{Proposition} \newtheorem{Definition}{Definition} \newcommand{\IR}{\mathbb{R}}