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On some hereditary properties of Riemannian \(g\)-natural metrics on tangent bundles of Riemannian manifolds. (English) Zbl 1068.53016
Summary: It is well known that if the tangent bundle \(TM\) of a Riemannian manifold \((M,g)\) is endowed with the Sasaki metric \(g^s\), then the flatness property on \(TM\) is inherited from the base manifold [O. Kowalski, J. Reine Angew. Math. 250, 124–129 (1971; Zbl 0222.53044)]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace \(g^s\) by the most general Riemannian “\(g\)-natural metric” on \(TM\) [see O. Kowalski and M. Sekizawa, Bull. Tokyo Gakugei Univ., Sect. IV, Ser. Math. Nat. Sci. 40, 1–29 (1988; Zbl 0656.53021), M. T. K. Abbassi and M. Sarih, Arch. Math. (Brno), submitted for publication]. In this direction, we prove that if \((TM,G)\) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then \((M,g)\) possesses the same property, respectively. We also give explicit examples of \(g\)-natural metrics of arbitrary constant scalar curvature on \(TM\).

MSC:
53B20 Local Riemannian geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53A55 Differential invariants (local theory), geometric objects
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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