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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 14.01.2013\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 12} $ $\\
\end{center}

{\bf Exercise 1}
Let $(M^n,g)$ be a Riemannian manifold and $(U_\varphi,\varphi)$ be a chart on $M$.
\ben\item Show that the components $R_{ijk}^l:=d\varphi^l\left(R(\frac{\partial}{\partial \varphi^i},\frac{\partial}{\partial \varphi^j})\frac{\partial}{\partial \varphi^k}\right)$ of the curvature tensor $R$ of $g$ on $U_\varphi$ are given in terms of the Christoffel symbols associated to $(U_\varphi,\varphi)$ by
\[ R_{ijk}^l=\frac{\partial \Gamma_{kj}^l}{\partial \varphi^i}-\frac{\partial \Gamma_{ki}^l}{\partial \varphi^j}+\sum_{m=1}^n \left(\Gamma_{mi}^l\Gamma_{kj}^m-\Gamma_{mj}^l\Gamma_{ki}^m\right).
\]
\item Deduce that, if ${\rm ric}=\sum_{i,j=1}^n{\rm ric}_{ij}d\varphi^i\otimes d\varphi^j$ denotes the decomposition of the Ricci-tensor of $g$, then
\[{\rm ric}_{ij}=\sum_{k=1}^nR_{kij}^k=\sum_{k=1}^n\left(\frac{\partial \Gamma_{ji}^k}{\partial \varphi^k}-\frac{\partial \Gamma_{jk}^k}{\partial \varphi^i}+\sum_{m=1}^n \left(\Gamma_{mk}^k\Gamma_{ji}^m-\Gamma_{mi}^k\Gamma_{jk}^m\right)\right).\]
\een



{\bf Exercise 2}\\
Let $(M:=M_1\times M_2,g:=g_1\oplus g_2)$ be the product of two Riemannian manifolds as in Exercise 3.(b) of Sheet no. 10.
\ben\item Show that the Levi-Civita connection $\nabla$ of $(M,g)$ is given by
\[\nabla_{(X_1,X_2)}(Y_1,Y_2)=\nabla_{X_1}^{M_1}Y_1+\nabla_{X_2}^{M_2}Y_2+\partial_{X_1}Y_2+\partial_{X_2}Y_1,\]
for all $X_1,Y_1\in\Gamma(\pi_1^*TM_1)$, $X_2,Y_2\in\Gamma(\pi_2^*TM_2)$ and where $\partial_{X_1}Y_2$ (resp. $\partial_{X_2}Y_1$) denotes the usual derivative (make sense of this).
\item Deduce that the curvature tensor $R$ of $(M,g)$ is given by
\[R_{X,Y}^MZ=R_{X_1,Y_1}^{M_1}Z_1+R_{X_2,Y_2}^{M_2}Z_2,\]
for all $X_i,Y_i,Z_i\in T_{x_i}M_i$, $i=1,2$, where $X:=X_1+X_2$, $Y:=Y_1+Y_2$ and $Z:=Z_1+Z_2$. (Here we write $R_{X,Y}$ instead of $R(X,Y)$ as this is better
for typesetting in this context.)
\item Calculate the Ricci tensor and the scalar curvature of $M$ in terms 
of the Ricci tensor and the scalar curvature of $M_1$ and $M_2$.
\een

{\bf Exercise 3}\\
Let $E,F\to M$ be (real or complex) vector bundles with connections $\nabla^E,\nabla^F$ over a given manifold $M$ and $x\in M$ be a point.
Prove the following identities:
\ben\item The curvature tensor of the connection $\nabla^E\oplus\nabla^F$ on $E\oplus F\to M$ is given by 
\[R_{X,Y}^{E\oplus F}=R_{X,Y}^E\oplus R_{X,Y}^F,\]
for all $X,Y\in T_xM$.
\item The curvature tensor of the tensor connection on $E\otimes F\to M$ as defined in Exercise 3 of Sheet no. 11 is given by
\[R_{X,Y}^{E\otimes F}=R_{X,Y}^{E}\otimes{\rm Id}_{F}+{\rm Id}_{E}\otimes R_{X,Y}^{F},\]
for all $X,Y\in T_xM$.
\item The curvature tensor of the dual bundle $E^*\to M$ endowed with the induced connection is given by
\[(R_{X,Y}^{E^*}\alpha)(V)=-\alpha(R_{X,Y}^EV),\]
for all $X,Y\in T_xM$, for all $V\in E_x$ and $\alpha\in E_x^*$.
\een

%\newpage
{\bf Exercise 4}\\
Let $(M^n,g)$ be a Riemannian manifold and denote by $\nabla$ resp. $R$ the Levi-Civita connection resp. the Riemannian curvature tensor of $(M^n,g)$. 
Let the $(0,4)$-tensor $\tilde R$ be defined by $\tilde R (X,Y,Z,W):=g(R(X,Y)Z,W)$, for all $X,Y,Z,W\in T_xM$ and $x\in M$.
\ben\item Let $x\in M$ be a point.
Prove that the following identities are satisfied: for all $X,Y,Z,T,U\in T_pM$,
\[\hspace{-1.5cm}(\nabla_X\tilde R)(Y,Z,T,U)=-(\nabla_X\tilde R)(Z,Y,T,U)=-(\nabla_X\tilde R)(Y,Z,U,T)=(\nabla_X\tilde R)(T,U,Y,Z).\]
\item For a given tensor field $A\in \Gamma(T^*M\otimes T^*M)$ let the \emph{divergence} of $A$ be defined by
  $$\mathrm{div}(A)(X):=\sum_{j=1}^n(\nabla_{E_j}A)(E_j,X)
\qquad \forall X\in TM,$$ 
where $\{E_j\}_{1\leq j\leq n}$ is a local orthonormal basis of $TM$, that is, $g(E_i,E_j)=\delta_{i}^j$.
Prove using the second Bianchi identity:
\[\mathrm{div}(\mathrm{ric})=\frac{1}{2}d\,\mathrm{scal}.\]
{\it (Hint: for a given point $x\in M$, the basis $\{E_j\}_{1\leq j\leq n}$ can be chosen such that $(\nabla E_i)_{|_x}=0$ holds; how can this be done?)}
\item \emph{Application}: Assuming $n\geq3$, the manifold $M$ connected and the existence of a smooth function $f:M\to \mR$ with ${\rm ric}=f\cdot g$ on $M$, prove that $f$ is constant on $M$.
Such a Riemannian manifold is then called \emph{Einstein}.
%$$\mathrm{div}(\mathrm{ric}-\frac12 \mathrm{scal}\;g).$$
\een
\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 21.1.2013} vor der Vorlesung}.
\end{document}



