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\begin{document}
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Universit\"at Regensburg, Mathematik\hfill WS 2012/13\\
Prof. Dr. Bernd Ammann\hfill 7.01.2013\\
Dr. Nicolas Ginoux\\

\begin{center}
{\bf Differential Geometry I \\
  Exercise Sheet no. 11} $ $\\
\end{center}

{\bf Exercise 1}
\ben\item Let $V\to M$ be a complex vector bundle endowed with a Hermitian metric $\langle\cdot\,,\cdot\rangle$ over a smooth manifold.
Show that the dual bundle $V^*\to M$ is isomorphic to the so-called \emph{conjugate} vector bundle $\overline{V}\to M$, where $\overline{V}_x:=V_x$ but where $\lambda\cdot v:=\overline{\lambda}v$ for all $v\in V_x$, $x\in M$ and $\lambda\in\mathbb{C}$.
\item Let $\tau\to\mCP^n$ be the tautological bundle as defined in Exercise 2 of Sheet no. 10.
Using the canonical Hermitian inner product on $\mC^{n+1}$, construct a Hermitian metric on $\tau$.
%Is $\tau^*\to\mCP^n$ isomorphic to $\tau\to\mCP^n$?
\een

{\bf Exercise 2}\\
Let $\nabla$ be any connection on the tangent bundle $TM$ of a smooth manifold $M$ and $T$ be its torsion, that is, 
$T(X,Y):=\nabla_XY-\nabla_YX-[X,Y]$ for all $X,Y\in\mathfrak{X}(M)$.
Show that $T$ is a tensor on $M$, more precisely show that $T$ defines a section of the vector bundle $T^*M\otimes T^*M\otimes TM\to M$.\\

{\bf Exercise 3}\\
Let $M$ be any smooth manifold.
\ben\item Given any vector bundles $E\to M$ and $F\to M$ with connections $\nabla^E$ and $\nabla^F$ respectively, prove that there exists a unique connection $\nabla$ on the tensor product bundle $E\otimes F\to M$ such that $\nabla(s\otimes s')=(\nabla^E s)\otimes s'+s\otimes(\nabla^Fs')$ for all sections $s$ of $E$ and $s'$ of $F$.
\item Let $E\to M$ be a vector bundle with connection $\nabla^E$. 
In each fiber $E_p$ the trace ${\rm tr}_p$ is an element of 
${\rm Hom}_{\mathbb{K}}(E_p^*\otimes E_p,\mathbb{K})\cong E_p\otimes E_p^*$.
Show that $p\mapsto {\rm tr}_p$ is a smooth map from $M$ to  $E\otimes E^*$.
Show that it is a parallel section of $E\otimes E^*\to M$.
 \item Given any real vector bundle $E\to M$ with Riemannian metric $\langle\cdot\,,\cdot\rangle$ and connection $\nabla^E$, show that the Riemannian metric -- as a section of the vector bundle $E^*\otimes E^*\to M$ -- is parallel iff the connection $\nabla^E$ is metric.
\een
%\newpage
{\bf Exercise 4}\\
Let $V\to M$ be a real or complex line bundle over a smooth manifold.
Show that the tensor vector bundle $V^*\otimes V\to M$ is trivial.

\vskip1cm
{\sl Abgabe der L\"osungen:
\textbf{Montag, den 14.1.2013} vor der Vorlesung}.\\

{\sl Wir w\"unschen allen Teilnehmerinnen und Teilnehmern ein erfolgreiches neues Jahr!}
\end{document}



