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\title{Homework 2 --- Euclidean, hyperbolic and conformal geometry}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

\begin{prob}
A subset $C \subset \E^3$ is {\em convex} iff for each pair of distinct points
$x,y \in C$, the line segment joining $x$ to $y$ is contained in $C$.
A {\em convex combination} of points $v_1 \dots v_m$ of $\E^n$ is a linear
combination of the form $t_1v_1 + \dots + t_mv_m$ such that $t_1 + \dots + t_m = 1$
and $t_i \ge 0$ for all $i$.
\begin{enumerate}
\item{Show that a subset $C$ of $\E^n$ is convex iff $C$ contains every convex
combination of points in $C$.}
\item{The {\em convex hull} of a subset $S \subset \E^n$ is the intersection $C(S)$
of all the convex subsets of $\E^n$ containing $S$. Prove that $C(S)$ is
the set of all convex combinations of points of $S$.}
\item{Let $G$ be the group of homeomorphisms (i.e. $1$--$1$, continuous with
a continuous inverse) of $\E^n$ which take convex sets to convex sets. Show that
$G$ is exactly the group of {\em affine transformations}: that is, the
group of transformations of the form
$$x \to Ax + t$$
where $A \in GL(3,\R)$ and $t$ is a translation.}
\end{enumerate}
\end{prob}

\begin{prob}
Let $S \subset \E^3$ be a sphere, and $\pi$ a plane tangent to $S$ at $x$.
{\em Stereographic projection} from $S$ to $\pi$ is the map taking
each point $p \in S$ to the intersection $q$ of the line $x'p$ with $\pi$,
where $x'$ is the point on $S$ opposite $x$. Show that stereographic
projection can be extended to an inversion.
\end{prob}

\begin{prob}
Let $A,B,C$ be mutually tangent spheres. Let $X_0$ be a fourth sphere
tangent to $A,B$ and $C$. Construct a chain of spheres, beginning with
$X_0$, where $X_i$ is tangent to $A,B,C$ and to $X_{i-1}$. Show that
the chain closes up on the sixth sphere. Note: assume that the spheres are
chosen so that their interiors are all disjoint.
\end{prob}

\begin{prob}
Show that for $x,y \in \D3$, the identity
$$\cosh(d_{\text{hyp}}(x,y)) = 1 + \frac {2|x-y|^2} {(1-|x|^2)(1-|y|^2)}$$
holds, where $d_{\text{hyp}}$ denotes the distance between points
with the conformally invariant hyperbolic metric on $\D3$,
and $|x-y|$ denotes the Euclidean distance between $x$ and $y$, and
$|x|$ denotes the Euclidean distance from $0$ to $x$.

Hint: show that the expression on the right hand side is invariant
under a conformal transformation of $\D3$.
\end{prob}

\begin{prob}
Suppose $S,T$ are two distinct spheres perpendicular to the unit sphere. Let
$i_S$ and $i_T$ be the isometries of $\D3$ in the hyperbolic metric which
are defined by inversion in $S$ and $T$ respectively. Show that the fixed
point set in $\D3$ of the composition $i_S i_T$ is
exactly the intersection $S \cap T$.
\end{prob}

\begin{prob}
A {\em hyperbolic sphere} is the set of points in hyperbolic space equidistant
from some point, the center. Show that the hyperbolic spheres in the
conformal model $\D3$ are exactly the Euclidean spheres completely contained
in $\D3$.
\end{prob}

\begin{prob}
Recall that a {\em vector field} $w$ on $U \subset \R^n$
is a smoothly varying choice of tangent vector $w(p) \in T_p\R^n$
for each $p \in \R^n$. A vector field is
{\em conformal} if it is tangent to a $1$--parameter family of conformal
transformations. That is, if there is a family of conformal maps
$\phi_t:U \to \R^n$ such that each $\phi_t$ is conformal, and if for each
$p$, the tangent vector
$$\frac {d\phi_t(p)} {dt}\Bigg|_{t=0} = w(p)$$
Show that a vector field $w$ on $\R^n$ is conformal iff
$$\frac {\partial w_i} {\partial x_j} = - \frac {\partial w_j} {\partial x_i}$$
for $i \ne j$, and
$$\frac {\partial w_i} {\partial x_i} = \frac 1 n \sum_k \frac {\partial w_k}
{\partial x_k}$$
Here $w_i$ denotes the $i$th component of the vector $w_i$, which can be
thought of as a function on $U$. Deduce that for a conformal vector field $w$,
$$(n-1) \frac {\partial^2 w_i} {\partial x_i^2} + \sum_{i\ne j} \frac
{\partial ^2 w_i} {\partial x_j^2} = 0$$
\end{prob}

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