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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 19.11.2012\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 6} $ $\\
\end{center}

{\bf Exercise 1}\\
Let $F:M\lra N$ be a smooth map between smooth manifolds $M$ and $N$.
Let $X,Y$ (resp. $\tilde{X},\tilde{Y}$) be (smooth) vector fields on $M$ (resp. $N$).
We say that $X$ is \emph{$F$-related} to $\tilde{X}$ iff $dF\circ X=\tilde{X}\circ F$ holds on $M$. \\
Show that, if $X$ is $F$-related to $\tilde{X}$ and $Y$ is $F$-related to
$\tilde{Y}$, then $[X,Y]$ is $F$-related to $[\tilde{X},\tilde{Y}]$.\\

{\bf Exercise 2}\\
Let $M^n$ be a smooth $n$-dimensional submanifold of $\mR^k$.
Given vector fields $X,Y$ on $M$, we extend them to vector fields $\overline{X},\overline{Y}$ in an open neighbourhood of $M$ in $\mR^k$ and set $(\nabla_XY)_{|_p}:={\rm pr}_p\left((\partial_{\overline{X}}\overline{Y})_{|_p}\right)$ for every $p\in M$, where ${\rm pr}_p:\mR^k\to T_pM$ denotes the orthogonal projection on $T_pM$ (identified to a vector subspace of $\mR^k$).\\
Show that $\nabla$ defines a metric and torsion-free connection on $M$.\\

{\bf Exercise 3}\\
Recall that an \emph{isometry} between two smooth $n$-dimensional Riemannian manifolds $(M,g)$ and $(N,h)$ is a diffeomorphism $\varphi:M\to N$ which preserves the metric, that is, with $\varphi^*h=g$.
\begin{enumerate}
 \item Show that, in the case $(M,g)=(N,h)$, the isometries of $(M,g)$ form a group w.r.t. the composition of maps.
\item Let ${\rm Aff}(M,g)$ be the set of diffeomorphisms of $M$ preserving the Levi-Civita connection $\nabla$ of $(M,g)$, that is, 
\[\hspace{-1.2cm}{\rm Aff}(M,g):=\left\{\varphi:M\to M\textrm{ diffeo. with }\nabla_{\varphi_*X}\varphi_*Y=\varphi_*(\nabla_XY)\;\forall X,Y\in\mathfrak{X}(M)\right\},\]
where, for any $X\in\mathfrak{X}(M)$, the vector field $\varphi_*X\in\mathfrak{X}(M)$ is defined by $(\varphi_*X)(x):=d_{\varphi^{-1}(x)}\varphi(X(\varphi^{-1}(x)))$ for all $x\in M$.
Show that ${\rm Aff}(M,g)$ is a group containing the group of isometries of $(M,g)$.
\item Let $M=\mR^n$ be endowed with the standard Riemannian metric $g$, i.e.\ the Euclidean metric.
Determine all elements $\varphi\in{\rm Aff}(M,g)$ with $\varphi(0)=0$.
\end{enumerate}
 

\newpage
{\bf Exercise 4}\\
Let $M$ be a smooth manifold.
Given a $1$-parameter group of diffeomorphisms $\varphi:M\times\mR\to M$, $(x,t)\mapsto\varphi_t(x)$ on $M$, let $X$ be associated tangent vector field on $M$ as in Exercise no. 3 of Sheet 4.
Show that, for any smooth tangent vector field $Y$ on $M$,
\[\frac{d}{dt}_{|_{t=0}}(\varphi_t)_*Y=-[X,Y],\]
where, for any $t\in\mR$, $(\varphi_t)_*Y$ denotes the push-out tangent vector field of $Y$ defined in the last exercise above.

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 26.11.2012} vor der Vorlesung}.\\
\end{document}



