  In this note I study the Sasakian geometry associated to the standard CR
structure on the Heisenberg group, and prove that the Sasaki cone coincides
with the set of extremal Sasakian structures. Moreover, the scalar curvature of
these extremal metrics is constant if and only if the metric has
$\Phi$-sectional curvature $-3.$ I also briefly discuss some relations with the
well-know sub-Riemannian geometry of the Heisenberg group as well as the
standard Sasakian structure induced on compact quotients.
