On the geometry of the Heisenberg group with a balanced metric (Q6100865)

The authors study the geometry of the 3-dimension Heisenberg group \(\mathrm{Nil}_3\), equipped with a balanced metric, i.e., the sum of a left and a right invariant metric. By solving an asymptotic Dirichlet problem they determine a totally geodesic surface \(\mathbb{T}\) and prove that \(\mathrm{Nil}_3\) is isomorphic to the product \(\mathbb{T}\times \mathbb{Z}\), where \(\mathbb{Z}\) is the center of \(\mathrm{Nil}_3\). The Heisenberg group \(\mathrm{Nil}_3\) is the space of the \(3\times 3\) real matrices \[ \begin{pmatrix} 1 & x & z \cr 0 & 1 & y \cr 0 & 0 & 1 \\ \end{pmatrix} \quad (x,y,z)\in \mathbb{R}. \] The surface is given by the equation \(z=\frac{1}{2}xy\), i.e., \(\mathbb{T}\) is a hyperbolic paraboloid. The map \[ \bigl((x,y),t\bigr)\mapsto \begin{pmatrix} 1 & x & \frac{1}{2}xy+t \\ 0 & 1 & y \\ 0 & 0 & 1 \\ \end{pmatrix}\] defines the isomorphism of \(\mathbb{T}\times \mathbb{Z}\) onto \(\mathrm{Nil}_3\).