Motion on tangent bundles which preserves both the orthogonal and tangent structures (Q705695)

Let \(M\) be a differentiable manifold and \(G\) a pseudo-Riemannian structure on the tangent manifold \(TM\) which is nondegenerate on the fibres of \(TM\). Let \(V\) be the vertical distribution and \(H\) the orthogonal distribution to \(V\) with respect to \(G\). Denote by \(F\) the almost product structure with the eigendistributions \(V\) and \(H\), and by \(J\) the canonical almost tangent structure on \(TM\). The author determines the maximal order of the transformation group on \(TM\) which preserves the structures \(G\), \(F\) and \(J\).