Geodesic
Some results on Riemannian $g$-natural metrics generated by classical lifts on the tangent bundle
Eurasian mathematical journal, Tome 8 (2017) no. 4, pp. 18-34

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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 $TM$. 
@article{EMJ_2017_8_4_a3,
     author = {L. Bilen and A. Gezer},
     title = {Some results on {Riemannian} $g$-natural metrics generated by classical lifts on the tangent bundle},
     journal = {Eurasian mathematical journal},
     pages = {18--34},
     publisher = {mathdoc},
     volume = {8},
     number = {4},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2017_8_4_a3/}
}
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L. Bilen; A. Gezer. Some results on Riemannian $g$-natural metrics generated by classical lifts on the tangent bundle. Eurasian mathematical journal, Tome 8 (2017) no. 4, pp. 18-34. http://geodesic.mathdoc.fr/item/EMJ_2017_8_4_a3/