We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.
"Some Relationships between the Geometry of the Tangent Bundle and the Geometry of the Riemannian Base Manifold." Tokyo J. Math. 35 (1) 1 - 15, June 2012. https://doi.org/10.3836/tjm/1342701340