A characterization of the Riemann extension in terms of harmonicity

If (M,ac) is a manifold with a symmetric linear connection, then T*M can be endowed with the natural Riemann extension (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure on (T*M, ) and prove that is harmonic (in the sense of E.Garcia-Rio, L.Vanhecke and M. E.Vazquez-Abal (1997)) if and only if reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952).