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Diagonal lift in the tensor bundle and its applications. (English) Zbl 1034.53016

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 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 \({}^Dg\).

MSC:

53B20 Local Riemannian geometry
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