\documentclass[11pt]{article}

% Packages

% Packages langues et encodage
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{csquotes}

% Packages maths
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[g]{esvect}
\usepackage[mathscr]{eucal}
\usepackage{stmaryrd}
\usepackage{subfig}
\usepackage[all]{xy}

% Packages mise en page
\usepackage{fancyhdr}
\usepackage[top=3cm, left=3cm, right=3cm, bottom=2cm]{geometry}
\usepackage[pdftex,pdfborder={0 0 0}]{hyperref}

% Packages graphiques
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}

% Packages autres
\usepackage{color}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{pdfpages}
\usepackage{soul}


% Definition des environnements
\theoremstyle{plain}
	\newtheorem*{thm}{Theorem}
	\newtheorem*{cor}{Corollary}
	\newtheorem*{lem}{Lemma}
	\newtheorem*{prop}{Proposition}

\theoremstyle{definition}
	\newtheorem*{dfn}{Definition}
	\newtheorem*{ntn}{Notation}
	\newtheorem*{dfns}{Definitions}
	\newtheorem*{ntns}{Notations}
	\newtheorem*{rem}{Remark}
	\newtheorem*{rems}{Remarks}
	\newtheorem*{ex}{Example}
	\newtheorem*{cex}{Counter example}
	\newtheorem*{exs}{Examples}
	\newtheorem*{cexs}{Counter-examples}
	\newtheorem{exo}{Exercise}


% Definiition des commandes
\newcommand{\B}{\mathbb{B}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\Z}{\mathbb{Z}}

\renewcommand{\H}{\mathbb{H}}
\renewcommand{\P}{\mathbb{P}}
\renewcommand{\S}{\mathbb{S}}

\newcommand{\acts}{\curvearrowright}
\newcommand{\dx}{\dmesure\!}
\newcommand{\deron}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\derond}[2]{\frac{\partial^2 #1}{\partial #2 ^2}}
\newcommand{\deronc}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}
\newcommand{\grad}{\vv{\gradient}}
\newcommand{\lint}{\llbracket}
\newcommand{\rint}{\rrbracket}
\newcommand{\leqclosed}{\trianglelefteqslant}
\newcommand{\mvert}{\mathrel{}\middle|\mathrel{}}
\newcommand{\norm}[1]{\left\lvert #1 \right\rvert}
\newcommand{\Norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\prsc}[2]{\left\langle #1\,, #2 \right\rangle}\newcommand{\rmes}[1]{\norm{\dmesure\! V_{#1}}}

\renewcommand{\bar}{\overline}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\tilde}{\widetilde}
\renewcommand{\triangleleft}{\vartriangleleft}

%\renewcommand{\FrenchLabelItem}{\textbullet}


% Definition des operateurs
\DeclareMathOperator{\aff}{Aff}
\DeclareMathOperator{\aut}{Aut}
\DeclareMathOperator{\bij}{Bij}
\DeclareMathOperator{\card}{Card}
\DeclareMathOperator{\dmesure}{d}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\stab}{Stab}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\tr}{Tr}
\DeclareMathOperator{\vect}{Span}

\setlength{\parindent}{0pt}
\setlength{\parskip}{5pt}
\setlength{\headsep}{10pt}
\setlength{\headheight}{16pt}

\renewcommand{\headrulewidth}{0.5pt}
\renewcommand{\footrulewidth}{0pt}

\begin{document}

\thispagestyle{fancy}
\lhead{Differential Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Curvatures}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}


\begin{exo}
Let $(M,g)$ be a Riemannian manifold of dimension $n$.
\begin{enumerate}
\item If $n=1$, what is its curvature?
\item If $n=2$, how many degrees of freedom are there in the Riemann tensor? Give the expression in local coordinates of the Riemann, Ricci and scalar curvature of $M$.
\item How many degrees of freedom are there in the Riemann tensor for $n=3$ and $n=4$.
\end{enumerate}
\end{exo}

\begin{exo}
Compute the Riemann, Ricci and scalar curvatures of the following Riemannian manifolds with their standard metric.
\begin{multicols}{2}
\begin{enumerate}
\item $\R^n$
\item $\T^n$
\item $\S^n$
\item $\D$ with $g_\D = \frac{4}{(1-\Norm{x}^2)^2}\prsc{\cdot}{\cdot}$.
\end{enumerate}
\end{multicols}
\end{exo}

\begin{exo}
\begin{enumerate}
\item Compute the sectional curvature of $\S^n_\rho$, the Euclidean sphere of radius $\rho$.
\item Compute the sectional curvature of $(\D,g_\D)$.
\end{enumerate}
\end{exo}

\begin{exo}[Positive versus negative curvature]
\begin{enumerate}
\item Let $N$ denote the North pole of $\S^2$.
\begin{enumerate}
\item For $\rho \in (0,\pi)$, compute the volume of the geodesic ball $B(N,\rho)$. How does it compare to the volume of the ball of radius $\rho$ in $\R^2$?
\item Compute the length of the circle $C(N,\rho)$. When $\rho$ is small enough, how does it compare to its Euclidean analogue?
\item Let $\gamma_1$ and $\gamma_2$ be two geodesics on $\S^2$ such that $\gamma_i(0)=N$. We denote $v_i = \gamma_i'(0)$ and assume that $\Norm{v_i}=1$. What is the distance between $\gamma_1(t)$ and $\gamma_2(t)$ for $t\in(-\pi,\pi)$?
\item When $t$ is small, how does this distance compare to its Euclidean counterpart?
\end{enumerate}
\item Same questions around $0$ in the Poincaré disc $\D$, for $\rho$ small.
\end{enumerate}
\end{exo}

\begin{exo}
Let us consider normal coordinates $(x^1,\dots,x^n)$ around some point $p$ in a Riemannian manifold $(M,g)$. Let us denote as usual $(g_{ij})$ the matrix of $g$ in these coordinates, $(\Gamma_{ij}^k)$ the Christoffel symbols of the Levi--Civita connection and $(R_{ijkl})$ the components of the Riemann tensor (as a $\binom{4}{0}$-tensor). We admit that the following holds in these coordinates:
\begin{equation*}
\forall i,j \in \{1,\dots,n\}, \qquad g_{ij}(x) = \delta_{ij} -\frac{1}{3} \sum_{1\leq k,l\leq n}R_{iklj}(0)x^kx^l +O(\Norm{x}^3).
\end{equation*}
\begin{enumerate}
\item Give a two terms expansion of the Riemannian volume $\dx V$ around $0$ in these coordinates.
\item Give a two terms expansion of the volume of the geodesic ball of center $p$ and radius $\rho$ as $\rho \to 0$.
\end{enumerate}
\end{exo}

\end{document}