Geodesic
Left-invariant metrics on a tensor bundle of type $(2,0)$ over a Lie group
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 4, pp. 146-155

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In this paper, we construct vertical and horizontal lifts of left-invariant vector fields. We establish necessary and sufficient conditions for the horizontal lift of a left-invariant vector field to be a left-invariant field. From a left-invariant metric on $G$, we build a left-invariant metric on $T^2_0G$.
Keywords: left-invariant metric, tensor bundle over a Lie group, vertical and horizontal lifts, left-invariant vertical and horizontal distributions. 
@article{UZKU_2012_154_4_a12,
     author = {N. A. Opokina},
     title = {Left-invariant metrics on a~tensor bundle of type $(2,0)$ over {a~Lie} group},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {146--155},
     publisher = {mathdoc},
     volume = {154},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a12/}
}
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N. A. Opokina. Left-invariant metrics on a tensor bundle of type $(2,0)$ over a Lie group. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 4, pp. 146-155. http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a12/