$g$-natural metrics of constant curvature on unit tangent sphere bundles

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We completely classify Riemannian $g$-natural metrics of constant sectional curvature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$. Since the base manifold $M$ turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian $g$-natural metric on the unit tangent sphere bundle of a Riemannian surface.