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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
##### Keywords:
natural operation; $$g$$-natural metric; geodesic flow
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