On Riemannian g-natural metrics of the form a.gs+b.gh+c.gv on the tangent bundle of a Riemannian manifold (M,g)

Let (M, g) be a Riemannian manifold and TM its tangent bundle. In [5] we have investigated the family of all Riemannian g-natural metrics G on TM (which depends on 6 arbitrary functions of the norm of a vector u is an element of TM). In this paper, we continue this study under some additional geometric properties, and then we restrict ourselves to the subfamily {G = a.g(s) + b.g(h) + c.g(v), a, b and c are constants satisfying a > 0 and a(a + c) - b(2) > 0}. It is known that the Sasaki metric g(s) is extremely rigid in the following sense: if (TM, g(s)) is a space of constant scalar curvature, then (M, g) is flat. Here we prove, among others, that every Riemannian g-natural metric from the subfamily above is as rigid as the Sasaki metric.