  In this paper, we show that every collapsed Gromov--Hausdorff limit of
compact Heisenberg manifolds is isometric to a flat torus. Here we say that a
sequence of sub-Riemannian manifolds collapses if their total measure with
respect to the Popp's volume or the minimal Popp's volume converges to zero. In
the appendix, we give the systolic inequality on sub-Riemannian Heisenberg
manifolds, and observe that the exponent of the total measure is equal to the
inverse of the Hausdorff dimension.
