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g-Natural metrics on tangent bundles and Jacobi operators
S. Degla and L. Todjihounde
Received: December 10, 2010; Revised: June 3, 2011; Accepted: June 16, 2011
Abstract. Let (M, g) be a Riemannian manifold and G a nondegenerate g-natural metric on its tangent bundle TM. In this paper we establish a relation between the Jacobi operators of (M, g) and that of (TM, G).
In the case of a Riemannian surface (M, g), we compute explicitly the spectrum of some Jacobi operators of (TM, G) and give necessary and sufficient conditions for (TM, G) to be an Osserman manifold.
Keywords: F-tensor fields; g-natural metrics; Jacobi operators; Osserman manifolds.
AMS Subject classification: Primary: 53B20,53C07 Secondary: 53A55, 53C25.
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