Math 6510 • Differentiable Manifolds

Consider the Lie subgroup \(Z\) of the Heisenberg group \(H\). Find all
connected two dimensional Lie subgroups of \(H\) that contain \(Z\). Describe them
as both submanifolds of \(H\) and as groups.
\[
  H=
  \left\{
    \begin{pmatrix}
      1 & x & z\\
      0 & 1 & y\\
      0 & 0 & 1
    \end{pmatrix}
    : x,y,z\in\R
  \right\},\qquad
  Z=
  \left\{
    \begin{pmatrix}
      1 & 0 & z\\
      0 & 1 & 0\\
      0 & 0 & 1
    \end{pmatrix}
    : z\in\R
  \right\}
\]