Some Results on Riemannian \( g \)-Natural Metrics Generated by Classical Lifts on the Tangent Bundle

Abstract

Let \( (M, g) \) be an \( n \)-dimensional Riemannian manifold and \( TM \) its tangent bundle equipped with Riemannian \( g \)-natural metrics which are linear combinations of the three classical lifts of the base metric with constant coefficients. The purpose of the present paper is three-fold. Firstly, to study conditions for the tangent bundle \( TM \) to be locally conformally flat. Secondly, to define a metric connection on the tangent bundle \( TM \) with respect to the Riemannian \( g \)-natural metric and study some its properties. Finally, to classify affine Killing and Killing vector fields on the tangent bundle \( TM \).