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\title{Homework 3 --- Spherical geometry}

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

\maketitle

This homework is due October 20th at the start of class. Recall that $O(3,\R)$
denotes the group of $3\times 3$ matrices with real entries satisfying
$A^tA=\text{id}$, and $SO(3,\R)$ denotes the subgroup with determinant $1$.

\begin{prob}
For vectors $x,y,z,w$ in $3$--space, prove the identity
$$(x \times y) \cdot (z \times w) = \left|\begin{matrix}
x \cdot z & x \cdot w \\
y \cdot z & y \cdot w \\ \end{matrix}\right|$$
\end{prob}

\begin{prob}
Prove that two spherical triangles are congruent if and only if they have the
same angles.
\end{prob}

\begin{prob}
Let $D$ be a regular dodecahedron. Calculate the dihedral angles of
$D$ --- that is, the angles between the faces.

Hint: think about the projection of $D$ from its circumcenter to its
circumscribing sphere.
\end{prob}

\begin{prob}
Let $A \in SO(3,\R)$. Show that $A$ has a real eigenvalue equal to $1$.
Show $A$ is conjugate to a matrix of the form $\left[\begin{matrix}
1 & 0 & 0 \\
0 & \cos(\alpha) & \sin(\alpha) \\
0 & -\sin(\alpha) & \cos(\alpha) \\
\end{matrix}\right]$.
\end{prob}

\begin{prob}
Let $\S^3$ be the group of length $1$ quaternions; that is, elements of the
form $a + bi + cj + dk$ where multiplication of quaternions is determined by
$$i^2 = j^2 = k^2 = -1, ij = -ji = k, jk = -kj = i, ki = -ik = j$$
and the length of the quaternion $a + bi + cj + dk$ is $\sqrt{a^2 + b^2 + c^2 + d^2}$.

Check first that this is a group, and that the elements of the form
$a + bi$ form a subgroup isomorphic to $S^1$ (the group of length $1$ complex
numbers).

Show that there is an isomorphism $\S^3 \cong SU(2)$ where $SU(2)$ is the
group of $2\times 2$ complex matrices $A$ with determinant $1$ such that
$\overline{A}^t A = \text{id}$. (Here $\overline{A}$ denotes the matrix whose entries are
the complex conjugates of the entries of $A$)

Hint: think of $\R^4$ alternately as the quaternions, or as $\C^2$.
\end{prob}

\begin{prob}
Define an equivalence relation on points $(z,w)$ in $S^3$ by
$(z_1,w_1) \sim (z_2,w_2)$ if and only if there is a nonzero complex number
$\lambda$ with $|\lambda|=1$
such that $(\lambda z_1, \lambda w_1) = (z_2,w_2)$. Show that the
equivalence classes of this equivalence relation are exactly the cosets of
some $S^1$ in $SU(2)$. What is the coset space?
\end{prob}

\begin{prob}
Let $E$ be an invertible $3\times 3$ real matrix, and let $J=E^tE$. Let
$O(J,\R)$ be the group of $3\times 3$ real matrices $A$ such that
$A^t J A = J$. Show that $O(J,\R) \cong O(3,\R)$, and find an explicit
isomorphism.
\end{prob}

\vfill
\pagebreak

\begin{prob}
Using the spherical law of cosines and the area formula, calculate the
area and the shape of the largest spherical quadrilateral with perimeter
equal to some fixed value $t$. Do the same for a spherical pentagon with 
perimeter equal to some fixed $t$. Assume $t$ is reasonably small.

Conversely, determine the shape of a
spherical quadrilateral or pentagon with fixed area of smallest perimeter.
\end{prob}

\begin{prob}[Foucault's pendulum --- hard]
A {\em pendulum} is a weight on the end of a string whose other end is fastened
rigidly. Imagine a person walking around on a planet which is not rotating, 
for example on Earth's moon, carrying a pendulum. If the person stands still,
the pendulum swings back and forth along a fixed axis. If the person walks in a
straight line on the surface of the planet, the pendulum swings back and forth
along the same apparent axis --- that is, the axis makes a constant angle
with the direction the person is facing. Show that if a person walks around a (polygonal)
loop on the surface of the planet enclosing an area $\alpha r^2$ where $r$ is
the radius of the planet, then the axis of the pendulum at the end of the
loop has rotated from the axis of the pendulum at the start of the loop by
an angle $\alpha$.

Now imagine the person on Earth, holding the pendulum. Now the person stands
still and the Earth rotates. Show that the axis of the pendulum appears to
rotate, relative to the stationary observer. In $24$ hours, the pendulum
rotates through an angle $2\pi \sin(\beta)$ where $\beta$ is the latitute.
\end{prob}

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