On Riemannian g-natural Metrics of the Form a.gs + b.gh + c.gv on the Tangent Bundle of a Riemannian Manifold (M, g)

Abstract

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 ∈ 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) − b2 > 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.