Diagonal lift in the tensor bundle and its applications. (Q1399741)

Let \((M,g)\) be a Riemannian manifold and \(T^1_q(M)\) the bundle of tensors of type \((1,q)\) on~\(M\). The authors introduce a Riemannian metric \({}^Dg\) on~\(T^1_q(M)\), completely determined by the metric~\(g\) on~\(M\), which they call the \textit{diagonal lift of~\(g\).} Its definition is similar to that of the well-known Sasaki metric on~\(TM\). They compute the Levi-Civita connection of~\({}^Dg\) and they investigate under which conditions the complete or horizontal lift to~\(T^1_q(M)\) of a Killing vector field on~\(M\) is Killing for the metric~\({}^Dg\).