Twelve characterisations of Killing vector fields in Finsler geometry

Abstract

It is well known that infinitesimal isometries preserve many of the canonical geometric structures of a Riemannian or Finslerian manifold. In this paper, we identify those geometric structures of a Finslerian manifold that are preserved by an infinitesimal isometry (a Killing vector field) and whose preservation characterises such fields. Using the Frölicher-Nijenhuis formalism as a unified geometric framework, we establish twelve equivalent characterisations of the Killing property in the Finslerian setting, utilizing the Finsler norm, the sphere bundle, contact and symplectic structures, Reeb and Hamiltonian vector fields, first integrals, the homogenized almost complex structure, and the Sasaki metrics. Furthermore, we derive the local expressions of these characterisations, yielding several equivalent forms of the Killing equations, including scalar, one-form and tensorial formulations. In our approach, none of these formulations is regarded as primary; each reflects a different geometric aspect of the Killing condition.