Math 6510 • Differentiable Manifolds

A symmetric quadratic form on \(\mathbb{R}^2\) is given by
\[
  ds^2=\frac{dx^2+dy^2}{(1+x^2+y^2)^2}.
\]
\begin{enumerate}
\item Show that \(ds^2\) is a Riemannian metric for \(\mathbb{R}^2\).
\item Compute the length of the curve \(t\mapsto(0,t)\) for \(a\le t\le b\) in this
metric.
\item Compute the area of \(\mathbb{R}^2\) in this metric.
\item Show that \((x,y)\mapsto\frac{(x,-y)}{x^2+y^2}\) is an isometry on
\((\mathbb{R}^2-\{(0,0)\},ds^2)\).
\end{enumerate}