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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.

MSC:
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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