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In this paper we achieve a first concrete step towards a better understandingof 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 symmetricsecond fundamental form are hyperplanes. This result relies on a sub-Riemanniancharacterization of a higher dimensional ruling property, as well as on thestudy of sub-Riemannian geodesics on Heisenberg hypersurfaces.

A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups