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On natural metrics on tangent bundles of Riemannian manifolds. (English) Zbl 1114.53015
Let \((M,g)\) be a Riemannian manifold. O. Kowalski and M. Sekizawa [Bull. Tokyo Gakugei Univ., Ser. Math. Nat. Sci. 40, 1–29 (1988; Zbl 0656.53021)] constructed a class of \(g\)-natural metrics on \(TM\).
In the paper under review the authors study the geometrical properties of \(g\)-natural metrics and prove that these metrics can be obtained by a construction of E. Musso and F. Tricceri [Ann. (4), 150, Mat. Pura Appl., 1–19 (1988; Zbl 0658.53045)]. The Levi-Civita connection of Riemannian \(g\)-natural metrics is given and, as application, the authors sort out all Riemannian \(g\)-natural metrics such that the fibres of \(TM\) are geodesic or such that the geodesic flow on \(TM\) is incompressible.

53B20 Local Riemannian geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53D25 Geodesic flows in symplectic geometry and contact geometry
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