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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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