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\begin{document}

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


Let $(M,g)$ be a Riemannian manifold and $\nabla$ denote its Levi--Civita connection.


\section{Definitions}


\paragraph{Riemann curvature.} The \emph{Riemann curvature} of $(M,g)$ is the $\binom{3}{1}$-tensor $R$ defined by:
\begin{equation}
\forall X,Y,Z \in \Gamma(TM), \qquad R(X,Y)Z = \nabla_X\nabla_YZ- \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z.
\end{equation}

We denote by $\tilde{R}$ the \emph{fully-covariant} version of the Riemann tensor, that is the $\binom{4}{0}$-tensor defined by:
\begin{equation}
\forall X,Y,Z,T \in \Gamma(TM), \qquad \tilde{R}(X,Y,Z,T) = g(R(X,Y)Z,T).
\end{equation}
On other terms, $\tilde{R}(X,Y,Z,\cdot) = (R(X,Y)Z)^\flat$, where $^\flat: T_xM \to T_x^*M$ is the isomorphism induced by the metric.


\paragraph{Ricci curvature.} The \emph{Ricci curvature} of $(M,g)$ is obtained by contracting the first covariant variable in $R$ with the only contravariant variable. That is, $\ric$ is a $\binom{2}{0}$-tensor defined by:
\begin{equation}
\label{def Ric}
\forall Y,Z \in \Gamma(TM), \qquad \ric(Y,Z) = \tr\left(X \mapsto R(X,Y)Z\right) = \contr_1^1(R)(Y,Z).
\end{equation}


\paragraph{Scalar curvature.} The \emph{scalar curvature} $S$ of $(M,g)$ is the trace of $\ric$ seen as a bilinear map on $T_xM$. That is, the trace of its matrix in any orthonormal basis of $T_xM$. In order to define it intrinsincally, we first need to use one of the musical isomorphisms defined by the metric in order to get a $\binom{1}{1}$-tensor (i.e. a section of $\End(TM)$) and then take the only possible contraction. Thus,
\begin{equation}
S = \contr_1^1(\ric^\sharp) = \tr(Y \mapsto \ric(Y,\cdot)^\sharp),
\end{equation}
where $^\sharp : T_x^*M \to T_xM$ is the isomorphism induced by $g$ and we applied it to one of the variables in $\ric$ (which one not important since $\ric$ and $g$ are symmetric).


\paragraph{Sectional curvature.} If two vectors $u,v \in T_xM$ are linearly independent, then the \emph{sectional curvature} of the plane $P$ spanned by $u$ and $v$ is:
\begin{equation}
K(P) = \frac{R(u,v,v,u)}{g(u,u)g(v,v) -g(u,v)^2}.
\end{equation}
It depends only on $P$ and not on a choice of basis.


\section{Symmetries}

The Riemann curvature is skew-symmetric in the first two variables and satisfies the first Bianchi identity. That is, for any vector fields $X,Y$ and $Z$, we have:
\begin{align}
\label{R asym}
&R(X,Y)Z = -R(Y,X)Z,\\
\label{Bianchi}
&R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0.
\end{align}

Its fully-covariant version presents additional symmetries. For any vector fields $X,Y,Z$ and~$T$, we have:
\begin{align}
\label{R tilde asym}
&\tilde{R}(X,Y,Z,T) = -\tilde{R}(Y,X,Z,T)= -\tilde{R}(X,Y,T,Z) = \tilde{R}(Z,T,X,Y),\\
\label{R tilde Bianchi}
&\tilde{R}(X,Y,Z,T) + \tilde{R}(Y,Z,X,T) + \tilde{R}(Z,X,Y,T) = 0.
\end{align}

Finally, the Ricci tensor is symmetric. For any $Y,Z \in \Gamma(TM)$, we have:
\begin{equation}
\ric(Y,Z)=\ric(Z,Y).
\end{equation}


\section{Expression in local coordinates}

Let $(x_1,\dots,x_n)$ denote local coordinates on some open subset of $M$. As usual, we denote by $\left(\deron{}{x_1},\dots,\deron{}{x_n}\right)$ and $(dx^1,\dots,dx^n)$ the associated local frames of $TM$ and $T^*M$ respectively. The matrix of $g$ in these coordinates is $(g_{ij})_{1\leq i,j \leq n}$, where $g_{ij} = g\left(\deron{}{x_i},\deron{}{x_j}\right)$. We denote by $(g^{ij})_{1\leq i,j \leq n}$ the coefficients of the inverse matrix.


\paragraph{Christoffel symbols.} The \emph{Christoffel symbols} $(\Gamma_{ij}^k)_{1\leq i,j,k\leq n}$ of the connection $\nabla$ in these coordinates are defined by the following relations:
\begin{equation}
\forall i,j \in \lint 1,n\rint, \qquad \nabla_{\deron{}{x_i}}\deron{}{x_j} = \sum_{k=1}^n \Gamma_{ij}^k \deron{}{x_k}.
\end{equation}

Note that, since the Levi--Civita connection is torsion-free, we have $\Gamma_{ij}^k = \Gamma_{ji}^k$ for any $i,j,k$. The Christoffel symbols are given in local coordinates by:
\begin{equation}
\label{Chr symb}
\forall i,j,k \in \lint 1,n\rint, \qquad \Gamma_{ij}^k = \frac{1}{2} \sum_{l=1}^n g^{kl} \left(\deron{g_{il}}{x_j} + \deron{g_{jl}}{x_i} - \deron{g_{ij}}{x_l}\right).
\end{equation}


\paragraph{Riemann tensor.} Let us define the coefficients $(R_{ijk}^l)_{1 \leq i,j,k,l\leq n}$ by:
\begin{equation}
\label{def R ijk l}
\forall i,j,k \in \lint 1,n\rint, \qquad R\left(\deron{}{x_i},\deron{}{x_j}\right)\deron{}{x_k} = \sum_{l=1}^n R_{ijk}^l \deron{}{x_l}.
\end{equation}
By Eq.~\eqref{R asym} and~\eqref{Bianchi}, for any $i,j,k$ and $l$, we have:
\begin{align}
&R_{ijk}^l = - R_{jik}^l\\
&R_{ijk}^l + R_{jki}^l + R_{kij}^l = 0.
\end{align}
Then we can write $R$ locally as:
\begin{equation}
R = \sum_{1\leq i,j,k,l \leq n} R_{ijk}^l dx^i \otimes dx^j \otimes dx^k \otimes \deron{}{x_l} = \sum_{\substack{1\leq i<j\leq n\\1\leq k,l \leq n}} R_{ijk}^l (dx^i \wedge dx^j) \otimes dx^k \otimes \deron{}{x_l}.
\end{equation}

We proved that for any $i,j,k$ and $l \in \lint 1,n\rint$, we have:
\begin{equation}
\label{rel R Chr symb}
R_{ijk}^l = \deron{\Gamma_{jk}^l}{x_i} - \deron{\Gamma_{ik}^l}{x_j} + \sum_{m=1}^r \Gamma_{jk}^m \Gamma_{im}^l - \sum_{m=1}^r \Gamma_{ik}^m \Gamma_{jm}^l.
\end{equation}

Similarly, we define $(R_{ijkl})_{1 \leq i,j,k,l \leq n}$ by:
\begin{equation}
\label{def R ijkl}
\forall i,j,k,l \in \lint 1,n\rint, \qquad R_{ijkl} = \tilde{R}\left(\deron{}{x_i},\deron{}{x_j},\deron{}{x_k},\deron{}{x_l}\right).
\end{equation}
By Eq.~\eqref{R tilde asym} and~\eqref{R tilde Bianchi}, for any $i,j,k$ and $k$, we have:
\begin{align}
&R_{ijkl} = -R_{jikl} = -R_{ijlk} = R_{klij}\\
&R_{ijkl} + R_{jkil} + R_{kijl} = 0.
\end{align}
Then $\tilde{R}$ can be written locally as:
\begin{equation}
\label{R tilde}
\tilde{R} = \sum_{1\leq i,j,k,l \leq n} R_{ijkl} dx^i \otimes dx^j \otimes dx^k \otimes dx^l = \sum_{\substack{1\leq i<j\leq n\\1\leq k<l \leq n}} R_{ijkl} (dx^i \wedge dx^j) \otimes (dx^k \wedge dx^l).
\end{equation}

By Eq.~\eqref{def R ijk l} and~\eqref{def R ijkl}, we see that, for any $i,j,k$ and $l$, we have:
\begin{align}
\label{rel R R tilde}
R_{ijkl} &= \sum_{m=1}^n R_{ijk}^mg_{ml} & &\text{and} & R_{ijk}^l = \sum_{m=1}^n R_{ijkm}g^{ml}.
\end{align}


\paragraph{Ricci tensor.} As above, we can define the coefficient of $\ric$ by:
\begin{equation}
\forall j,k \in \lint 1,n\rint, \qquad \ric_{jk} = \ric\left(\deron{}{x_j},\deron{}{x_k}\right),
\end{equation}
so that in local coordinates we have:
\begin{equation}
\ric = \sum_{1\leq j,k \leq n} \ric_{jk} dx^j \otimes dx^k.
\end{equation}

We deduce from the definition of $\ric$ (cf.~Eq.~\eqref{def Ric}) and Eq.~\eqref{rel R R tilde} that, for any $j$ and $k$, we have:
\begin{equation}
\label{rel Ric R}
\ric_{jk} = \sum_{i=1}^n R_{ijk}^i = \sum_{1 \leq i,m \leq n} R_{ijkm}g^{mi}.
\end{equation}


\paragraph{Scalar curvature.} Since $g\left(\deron{}{x_i},\cdot \right) = \sum_{j=1}^n g_{ij} dx^j$ for any $i \in \lint 1,n\rint$, we have:
\begin{align}
\forall i \in \lint 1,n\rint, &\left(\deron{}{x_i}\right)^\flat = \sum_{j=1}^n g_{ij}dx^j & &\text{and} & \forall j \in \lint 1,n\rint, & \left(dx^j\right)^\sharp = \sum_{i=1}^n g^{ji}\deron{}{x_i}.
\end{align}

Then we get:
\begin{equation}
\ric^\sharp = \sum_{1\leq j,k \leq n} \ric_{jk} dx^j \otimes (dx^k)^\sharp = \sum_{1 \leq i,j,k \leq n} \ric_{jk}g^{ki}dx^j \otimes \deron{}{x_i},
\end{equation}
so that
\begin{equation}
\label{rel S ric}
S = \sum_{1 \leq j,k \leq n} \ric_{jk}g^{kj}.
\end{equation}


\paragraph{In nice coordinates.} Let us now assume that our coordinates are such that $\left(\deron{}{x_1},\dots,\deron{}{x_n}\right)$ is orthonormal at $x=0$. Then we have $(g_{ij}(0))_{1\leq i,j \leq n} = I_n = (g^{ij}(0))_{1\leq i,j \leq n}$, where $I_n$ is the identity matrix of size $n$. In these coordinates we can simplify Eq.~\eqref{Chr symb}, \eqref{rel R R tilde}, \eqref{rel Ric R} and~\eqref{rel S ric} in the following way:
\begin{align}
&\forall i,j,k \in \lint 1,n \rint, & \Gamma_{ij}^k(0) &= \frac{1}{2} \left(\deron{g_{ik}}{x_j}(0) + \deron{g_{jk}}{x_i}(0) - \deron{g_{ij}}{x_k}(0)\right),\\
&\forall i,j,k,l \in \lint 1,n \rint, & R_{ijk}^l(0) &= R_{ijkl}(0),\\
&\forall j,k \in \lint 1,n \rint, & \ric_{jk}(0) &= \sum_{i=1}^n R_{ijk}^i(0) = \sum_{i=1}^n R_{ijki}(0),\\
& & S(0) &= \sum_{j=1}^n \ric_{jj}(0).
\end{align}


\paragraph{In normal coordinates.} If we assume that $(x_1,\dots,x_n)$ are normal coordinates around the point of coordinates $x=0$, then we have:
\begin{equation}
(g_{ij}(x))_{1\leq i,j \leq n} = I_n + O\left(\Norm{x}^2\right) = (g^{ij}(x))_{1\leq i,j \leq n}.
\end{equation}
It is then possible to simplify further Eq.~\eqref{Chr symb} and~\eqref{rel R Chr symb} to get:
\begin{align}
&\forall i,j,k \in \lint 1,n \rint, & \Gamma_{ij}^k(0) &= 0,\\
&\forall i,j,k,l \in \lint 1,n \rint, & R_{ijk}^l(0) &= \frac{1}{2}\left(\deronc{g_{jl}}{x_i}{x_k}(0) - \deronc{g_{jk}}{x_i}{x_l}(0) -\deronc{g_{il}}{x_j}{x_k}(0) + \deronc{g_{ik}}{x_j}{x_l}(0)\right).
\end{align}


\section{The case of surfaces}

When $n=2$, we have $\tilde{R} = R_{1212} (dx^1 \wedge dx^2) \otimes (dx^1 \wedge dx^2)$ for any choice of local coordinates (cf.~Eq.~\eqref{R tilde}). For any $p \in M$, let $\kappa(p) = K(T_pM)$ denote the sectional curvature of the tangent plane. In any local coordinates centered at $p$ and such that $\left(\deron{}{x_1}(0),\deron{}{x_2}(0)\right)$ is orthonormal we have $\kappa(p) = \tilde{R}\left(\deron{}{x_1}(0),\deron{}{x_2}(0),\deron{}{x_2}(0),\deron{}{x_1}(0)\right) = -R_{1212}(0)$, so that at the point $p$:
\begin{equation}
\tilde{R}_p = -\kappa(p) (dx^1 \wedge dx^2) \otimes (dx^1 \wedge dx^2).
\end{equation}

A computation gives that, for any vector fields $X,Y,Z,T$ on $M$, we have:
\begin{equation}
\tilde{R}(X,Y,Z,T) = \kappa \left(g(Y,Z)g(X,T) - g(X,Z)g(Y,T)\right).
\end{equation}
By definition of $\tilde{R}$, this means that:
\begin{equation}
R(X,Y)Z = \kappa \left(g(Y,Z)X - g(X,Z)Y\right),
\end{equation}
for any $X,Y$ and $Z \in \Gamma(TM)$. The definition of $\ric$ as a trace yields:
\begin{equation}
\forall Y,Z \in \Gamma(TM), \qquad \ric(Y,Z) = \kappa g(Y,Z).
\end{equation}
Finally, we get:
\begin{equation}
S = 2\kappa.
\end{equation}

\end{document}