Math 6510 • Differentiable Manifolds

For \((x,y,z)\in\mathbb{R}^3\) let
\(\eta=dx\otimes dx+dy\otimes dy-dz\otimes dz\) denote a symmetric 2-form on
\(\mathbb{R}^3\). Let
\[
  M=\{(x,y,z)\in\mathbb{R}^3:1+x^2+y^2=z^2,\ z>0\}
\]
be one nappe of the hyperboloid.
\begin{enumerate}
\item Compute
\[
  \int_\Omega dx\wedge dy+(4-2z)dx\wedge dz+(z-2)dy\wedge dz
\]
where \(\Omega\subset M\) is the region inside \(x^2+y^2\le3\).
\item Show that the restriction \(\eta|_M\) is a Riemannian metric for \(M\).
\item What is the length of the curve \(t\mapsto(\sinh t,0,\cosh t)\) for
\(a\le t\le b\) in this metric?
\end{enumerate}