Para-Kahler tangent bundles of constant para-holomorphic sectional curvature

We characterize the natural diagonal almost product (locally‎ ‎product) structures on the tangent bundle of a Riemannian‎ ‎manifold‎. ‎We obtain the conditions under which the tangent bundle‎ ‎endowed with the determined structure and with a metric of natural‎ ‎diagonal lift type is a Riemannian almost product (locally‎ ‎product) manifold‎, ‎or an (almost) para-Hermitian manifold‎. ‎We find‎ ‎the natural diagonal (almost) para-K\"ahlerian structures on the‎ ‎tangent bundle‎, ‎and we study the conditions under which they have‎ ‎constant para-holomorphic sectional curvature‎.