\documentclass{article}


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

\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 3}
\centerline{\bf Professor M.Shubin}
\centerline{\bf Winter 1997}
%\medskip\noindent
%{\bf Textbooks:}
%\smallskip\noindent
%1) {\it Riemannian Geometry}, by Manfredo Perdig\~ao do Carmo.
%Birkh\"auser, Boston, 1993.
%\smallskip\noindent
%2) {\it General Theory of Relativity}, by P.A.M.Dirac. Princeton
%University Press, 1996.

\medskip\noindent
{\bf 1.} Derive a geometric formula for covariant derivative of a
contravariant vector
via parallel transport:
$$A^\mu(x+dx)-P_{x\mapsto x+dx}A^\mu(x)=
(A^\mu_{\ ,\nu}(x)+\Ga^\mu_{\nu\al}A^\al)dx^\nu=A^\mu_{\ :\nu}dx^\nu\,.$$

\medskip\noindent
{\bf 2.} Show that $A^\mu_{\ :\nu}=\bigl(\nabla_{{\pa\over\pa
x^\nu}}A\bigr)^\mu$.

\medskip\noindent
{\bf 3.} Derive a geometric formula for the covariant derivative of an
arbitrary tensor
via parallel transport.

\medskip\noindent
{\bf 4.} Find all geodesics on the Lobachevsky plane.

\medskip\noindent
{\bf 5.} Let $G$ be a Lie group with biinvariant Riemannian metric. Prove that
geodesics passing through $e\in G$ are exactly one-parametric subgroups, i.e.
smooth maps $g:\RR\to G$ such that $g(0)=e$ and $g(t+s)=g(t)g(s)$ for all
$t,s\in\RR$.

\medskip\noindent
{\bf 6.} Describe all isometric transformations of the Lobachevsky plane
$\HH^2=\{z|\;\Im z>0\}$ (with the metric $ds^2=y^{-2}(dx^2+dy^2)$, $z=x+iy$)
which leave
a given point $p$ fixed. They should include ``rotations" (which actually
induce rotations
in the tangent space $T_p\HH^2$) and reflections with respect to a geodesic
passing
through $p$.

\medskip\noindent
{\bf 7.} Prove that the group of all isometries of $\HH^2$ is a
3-dimensional Lie group
which is generated by the following transformations:

1) ``Translations": $z\mapsto z+a,\ a\in\RR$;

2)``Homotheties" or ``Scalings": $z\mapsto bz,\ b>0$;

3) ``Rotations around the point $i$":
$$z\mapsto {\cos\theta\cdot z +\sin\theta\over -\sin\theta\cdot
z+\cos\theta}\;;$$

4) Reflection $z\mapsto -\bz$.

\medskip\noindent
{\bf 8.} Use the product rule for the covariant derivative with respect to
the tensor product
to derive the formula for $B^\nu_{\ :\rho:\sigma}-B^\nu_{\ :\sigma:\rho}$
in terms of
the curvature tensor.

\medskip\noindent
{\bf 9.} Calculate all the components of the curvature tensor for the
paraboloid
$$\{(x,y,z)|\;z=a(x^2+y^2)\}\subset\RR^3$$
(here $a=\hbox{const}>0$)
with the metric which is induced by the standard Euclidean metric in $\RR^3$.


\medskip\noindent
{\bf 10.} Calculate the curvature tensor on the standard sphere
$$S^2=\{(x,y,z)|\;x^2+y^2+z^2=R^2\}\subset\RR^3$$
where $R>0$.

\noindent
({\it Hint}: Use exercise 9 and symmetries.)
\end{document}