Levi harmonic maps of contact Riemannian manifolds

We study Levi harmonic maps i.e. $C^\infty$ solutions $f : M \to M^\prime$ to $\tau_\mathcal{H} (f) \equiv \mathr{trace}_{g} ( \Pi_\mathcal{H}\beta_f ) = 0$, where $(M , \eta, g)$ is an (almost) contact (semi) Riemannian manifold, $M^\prime$ is a (semi) Riemannian manifold, $\beta_f$ is the second fundamental form of $f$, and $\Pi_\mathcal{H} \beta_f$ is the restriction of $\beta_f$ to the Levi distribution $\mathcal{H} = \mathrm{Ker}(\eta )$. Many examples are exhibited e.g. the Hopf vector field on the unit sphere $S^{2n+1}$, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannnian manifold of constant curvature $1$ are Levi harmonic maps. A CR map $f$ of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if $f$ is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.