Heisenberg metric Kaluza-Klein reduction ($V=\alpha +\chi y^2$)

For $h=\alpha d{x}^2+\beta d{y}^2+\chi (dz-ydx{)}^2$, the Kaluza-Klein decomposition along a Killing field ${\partial }_x$ has $V=\alpha +\chi y^2$, $A=-\chi ydz/V$ and $\gamma =\beta d{y}^2+\alpha \chi d{z}^2/V$. For a Riemannian metric, positive constants make the fibre norm and quotient metric positive everywhere.