If (M,∇) is a manifold with a symmetric linear connection, then T*M can be endowed with the natural Riemann extension g¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g¯g¯ initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure PP on (T*M, g¯) and prove that P is harmonic (in the sense of E.Garciá-Río, L.Vanhecke and M. E.Vázquez-Abal (1997)) if and only if g¯ reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952)., Cornelia-Livia Bejan, Şemsi Eken., and Obsahuje bibliografii
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. Studying the compatibility and the anti-compatibility relations between the determined structures and a natural diagonal metric, we find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. Finally, we prove the characterization theorem for the natural diagonal (almost) para-Kählerian structures on the total space of the cotangent bundle.