Harmonic vector fields on pseudo-Riemannian manifolds

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. The congruence structure of conformal gradient fields on pseudo-Riemannian hyperquadrics and Killing fields on pseudo-Riemannian quadrics is elucidated, and harmonic vector fields of these two types are classified up to congruence. A para-Kähler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature.