Levi Harmonic Maps of Contact Riemannian Manifolds

Abstract

We study Levi harmonic maps, i.e., C∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\), where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, βf is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of βf to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\). Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S2n+1, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian 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.