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Killing vector fields on tangent bundles with Cheeger-Gromoll metric. (English) Zbl 1060.53019
Let $$(TM,\overline g)$$ be the tangent bundle of a Riemannian manifold $$(M,g)$$ where $$\overline g$$ is the Cheeger-Gromoll metric. In this paper, the authors focus on the study of Killing vector fields on $$(TM, \overline g)$$ and some of their properties which they compare with the case where instead of $$\overline g$$ the Sasaki metric is used on $$TM$$. First, they determine a general expression for a Killing vector field and then use it to derive some properties in some particular cases, as for example when $$(M,g)$$ is compact, orientable and has vanishing first Betti number or strictly positive Ricci curvature respectively. Finally, they use the explicit expression to prove that the sectional curvature of $$(TM, \overline g)$$ is never constant, a result which has already been proven by Sekizawa for constant curvature spaces $$(M,g)$$. This work is part of a broader study by the authors of the geometry of $$(TM,\overline g)$$ and the unit sphere bundle $$(T_1M,\overline g)$$, also when $$\overline g$$ is replaced by other natural metrics.

##### MSC:
 53B20 Local Riemannian geometry 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 53C20 Global Riemannian geometry, including pinching 53C24 Rigidity results
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