On a classification of Killing vector fields on a tangent bundle with g-natural metric

TL;DR

This paper classifies Killing vector fields on tangent bundles of Riemannian manifolds equipped with non-degenerate g-natural metrics, establishing conditions for their existence and decomposing the tangent bundle into four classes.

Contribution

It provides a new classification of Killing vector fields on tangent bundles with g-natural metrics, linking their existence on the base manifold and the tangent bundle.

Findings

Decomposition of (TM,G) into four classes based on Killing vector fields

Proved the equivalence of Killing vector fields on M and TM

Characterized conditions for the existence of Killing vector fields on tangent bundles

Abstract

The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately. The equivalence of the existence of Killing vector field on M and TM is proved. Key words: Riemannian manifold, tangent bundle, g-natural metric, Killing vector field, non-degenerate metric.