On the geometry of tangent hyperquadric bundles: CR and pseudo harmonic vector fields.

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T (M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudo-harmonic vector fields on a compact strictly pseudo-convex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ∈ H(M) whose horizontal lift X↑ to the canonical circle bundle S1 →C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X↑ project on a nonlinear system of subelliptic PDEs on M.