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We completely classify Riemannian -natural metrics of constant sectional curvature on the unit tangent sphere bundle of a Riemannian manifold . Since the base manifold turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian -natural metric on the unit tangent sphere bundle of a Riemannian surface.
We consider generalized m-quasi-Einstein metric within the framework of Sasakian and K-contact manifolds. First, we prove that a complete Sasakian manifold M admitting a generalized m-quasi-Einstein metric is compact and isometric to the unit sphere . Next, we generalize this to complete K-contact manifolds with m ≠ 1.
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