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We obtain a Kähler Einstein structure on the tangent bundle ofa Riemannian manifold of constant negative curvature. Moreover, theholomorphic sectional curvature of this Kähler Einsteinstructure is constant. Similar results are obtained for a tubearound zero section in the tangent bundle, in the case of theRiemannian manifolds of constant positive curvature

A Kähler Einstein structure on the tangent bundle of a space form