Abstract:
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.