Some results on Riemannian $g$-natural metrics generated by classical lifts on the tangent bundle

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 $TM$.