\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 1}
\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.} Let $\La:\RR^4\to\RR^4$ be a Lorentz transformation i.e. a linear
transformation
which preserves  the Lorentz metric $(dx^0)^2-(dx^1)^2-(dx^2)^2-(dx^3)^2$.
Prove that
$\det \La =\pm 1$.

\medskip\noindent
{\bf 2.} Let $x^{\prime\mu}=\La^\mu_\nu x^\nu$ be a Lorentz transformation, so
$A^{\prime\mu}=\La^\mu_\nu A^\nu$ for any contravariant vector $A^\mu$. Write
the transformation law for any covariant vector.

\medskip\noindent
{\bf 3.} Let $\langle,\rangle$ be a symmetric Lorentzian inner product in
$\RR^4$
(i.e. it is non-degenerate and the corresponding quadratic form is of the type
$+---$, that is has exactly one positive
and 3 negative squares). A vector $x$ is called a {\it time-like} vector if
$\langle x,x\rangle>0$.
Prove that for any two time-like vectors $x,y$  we have the inverse
Cauchy-Schwarz inequality
$\langle x,y\rangle^2\ge \langle x,x\rangle\langle y,y\rangle\;.$

\medskip\noindent
{\bf 4.} Check that the elements of $V\otimes \ldots\otimes V\otimes
V'\otimes\ldots\otimes V'$
($r$ times $V$ and $s$ times $V'$, where $V'$ is  the dual space to $V$)
are tensors of
type $r,s$ in the sense that they can be presented as
sets of numbers associated with any choice of a basis in $V$ with
appropriate transormation law
when the basis is replaced by another basis.

\medskip\noindent
{\bf 5.} Check that $\de^\rho_\mu$ is a tensor.

\medskip\noindent
{\bf 6.} Check that $g_{\mu\nu}$ and $g^{\mu\nu}$ are tensors.

\medskip\noindent
{\bf 7.} Calculate the Planck units with 10\% precision.

\medskip\noindent
{\bf 8.} Find a manifold $M$, $\dim_{\RR} M=4$, such that there exists no
Lorentzian metric on $M$.

\medskip\noindent
{\bf 9.} ({\it A construction of  the Lobachevsky plane}). Consider the
group $G$ of all
affine orientation preserving transformations of $\RR$ i.e. transformations
$g_{x,y}:t\mapsto yt+x$, $x\in\RR$, $y>0$. Write a left-invariant
Riemannian metric
on $G$, such that it coincides with the Euclidean metric $dx^2+dy^2$ at the
unit element
$e\in G$. (Here $e$ corresponds to $x=0, y=0$).

\noindent
({\it Answer:} $ds^2={dx^2+dy^2\over y^2}$ up to a positive constant
factor, or in other words
$g_{12}=0$, $g_{11}=g_{22}=1/y^2$.)

\medskip\noindent
{\bf 10.} In the previous exercise denote $z=x+iy$, $i=\sqrt{-1}$, and show
that
any transformation $z\mapsto {az+b\over cz+d}$ with $ad-bc=1$ is an
isometry of $G$ with the
metric $ds^2$.

\noindent
({\it Hint}: Observe that $ds^2=-{4dzd\bar z\over (z-\bar z)^2}$.)
\end{document}