Harmonic vector fields on pseudo-Riemannian manifolds

TL;DR

This paper extends the theory of harmonic vector fields from Riemannian to pseudo-Riemannian manifolds, classifying specific harmonic fields on pseudo-Euclidean hyperquadrics and quadrics, and exploring their relationships via para-Kaehler structures.

Contribution

It generalizes harmonic vector field theory to pseudo-Riemannian manifolds and classifies harmonic conformal gradient and Killing fields on pseudo-Euclidean quadrics.

Findings

Classification of harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics

Classification of harmonic Killing fields on pseudo-Riemannian quadrics

Use of para-Kaehler twisted anti-isometry to relate harmonic vector fields

Abstract

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature.