\documentclass{article}

\def\ZZ{\hbox{\bf Z}}
\def\RR{\hbox{\bf R}}
\def\CC{\hbox{\bf C}}
\def\QQ{\hbox{\bf Q}}

\def\Ga{\Gamma}
\def\ga{\gamma}
\def\al{\alpha}
\def\De{\Delta}
\def\de{\delta}
\def\La{\Lambda}
\def\la{\lambda}
\def\eps{\varepsilon}
\def\ka{\kappa}
\def\oM{\overline{M}}
\def\oX{\overline{X}}
\def\onabla{\overline{\nabla}}
\def\om{\omega}
\def\Om{\Omega}
\def\sup{\rm sup}
\def\grad{\rm grad}
\def\pa{\partial}
\def\bpa{\bar{\partial}}
\def\spec{\hbox{spec}}
\def\Dom{\hbox{Dom}}
\def\bb{\bar{b}}
\def\bB{\bar{B}}
\def\bM{\overline{M}}
\def\bx{\bar{x}}
\def\bw{\bar{w}}
\def\bxi{\bar{\xi}}
\def\bz{\bar{z}}
\def\dist{{\rm dist}}
\def\bpar{\bar{\partial}}
\def\dimg{\dim_{\Gamma}}
\def\codimg{\hbox{codim}_{\Gamma}}
\def\cO{{\cal O}}
\def\Re{\hbox{Re}\,}
\def\Im{\hbox{Im}\,}
\def\Ker{\hbox{Ker}\,}

\begin{document}

\centerline {\bf Riemannian Geometry and General Relativity}
\centerline{\bf (Differential Geometry-2, MTH 3412)}
\centerline{\bf Home Assignement 2}
\centerline{\bf Professor M.Shubin}
\centerline{\bf Winter 1997}
\medskip\noindent
{\bf Textbooks:}

1) {\it Riemannian Geometry}, by Manfredo Perdig\~ao do Carmo.
Birkh\"auser, Boston, 1993.

2) {\it General Theory of Relativity}, by P.A.M.Dirac. Princeton University
Press, 1996.

\bigskip\noindent
{\bf 1.} Derive the transformation rule for the Christoffel symbols
$\Ga^\nu_{\mu\alpha}$
under a change of curvilinear coordinates.

\medskip\noindent
{\bf 2.} Show that $\Ga^\nu_{\mu\alpha}$ is not a tensor.

\medskip\noindent
{\bf 3.} On $\RR^n$ show that $\Ga^\nu_{\mu\alpha}$ can be arbitrary
$C^\infty$ functions
i.e. for any choice of such $n^3$ functions  there exists  a unique
affine connection with the Christoffel symbols $\Ga^\nu_{\mu\alpha}$
in the canonical coordinates on $\RR^n$.

\medskip\noindent
{\bf 4.} Prove that for any connection with the Christoffel symbols
$\Ga^\nu_{\mu\alpha}$
the quantities
$T^\nu_{\mu\alpha}=\Ga^\nu_{\mu\alpha}-\Ga^\nu_{\alpha\mu}$ form a tensor.
(It is  called the {\it torsion tensor} of the given connection.)

\medskip\noindent
{\bf 5.} Prove that for any connection with the Christoffel symbols
$\Ga^\nu_{\mu\alpha}$
the quantities
$\tilde\Ga^\nu_{\mu\alpha}={1\over 2}(\Ga^\nu_{\mu\alpha}+\Ga^\nu_{\alpha\mu})$
are the Christoffel symbols of a symmetric connection.

\medskip\noindent
{\bf 6.} On a Riemannian manifold $M$ with the metric $g_{\mu\nu}$ prove
that for any
contravariant vector $A^\mu$ and the corresponding covariant vector $A_\mu$
we have
$g^{\mu\nu}A_\mu A_\nu=g_{\mu\nu}A^\mu A^\nu$ i.e. the map of $T_xM$ to
$T^*_xM$ defined by the
metric is isometry on every tangent space provided the metric on $T^*_xM$
is given by
$g^{\mu\nu}$.

\medskip\noindent
{\bf 7.} Describe geometrically the parallel transport along a parallel
$c$ of latitude $\theta$ on the standard unit sphere $S^2\subset \RR^3$.

\noindent
{\it Hint:} Consider the cone $C$ tangent to $S^2$ along $c$ and show that the
parallel transport of any tangent vector along $c$ is the same whether taken
relative to $S^2$ or to $C$.

\medskip\noindent
{\bf 8.} Calculate the Christoffel symbols on the Lobachevsky plane
$$\RR^2_+=\{(x,y)\in\RR^2|\,y>0\}$$
(with the metric $ds^2=y^{-2}(dx^2+dy^2)$).

\medskip\noindent
{\bf 9.} On the Lobachevsky plane (see Problem 8 above) describe the
parallel transport
along the ``curve" $x=t, y=1$.

\noindent
{\it Hint:} The transported vector rotates with constant angular velocity.

\medskip\noindent
{\bf 10.} Show that for any Riemannian or pseudo-Riemannian metric
$g=(g_{\mu\nu})$
and arbitrary tensor $T=(T^\alpha_{\mu\nu})$ with
$T^\alpha_{\mu\nu}=-T^\alpha_{\nu\mu}$
there exists unique affine connection which is compatible with $g$ and has the given
torsion tensor $T$ (see Problem 4).


\end{document}