  In this paper we achieve a first concrete step towards a better understanding
of the so-called Bernstein problem in higher dimensional Heisenberg groups.
Indeed, in the sub-Riemannian Heisenberg group $\mathbb{H}^n$, with $n\geq 2$,
we show that the only entire hypersurfaces with vanishing horizontal symmetric
second fundamental form are hyperplanes. This result relies on a sub-Riemannian
characterization of a higher dimensional ruling property, as well as on the
study of sub-Riemannian geodesics on Heisenberg hypersurfaces.
