Riemannian geodesics of semi Riemannian warped product metrics

Let (M₁,g₁) and (M₂,g₂) be two differentiable, connected, complete Riemannian manifolds, k a limitated differentiable function among M₁ and R and g:= g₁- kg₂ the semi Riemannian metric on the product manifold M:= M₁ х M₂ . We associate to g a suitable family of Riemannian metrics Gr+ g₂, with an appropriated r, on M and we call Riemannian geodesics of g the geodesics of g which are geodesics of a metric of the previous family, via a suitable reparametrization. Among the properties of these geodesics, we quote: For any point z of M and for any point y of M₂, there exists a not empty subset A of M₁, such that all the geodesics of g joining z with a point (x,y), with x Є A, are Riemannian. The Riemannian geodesics of g determine a "partial" property of geodesic connection on M. Finally, we determine two new classes of semi Riemannian metrics (one of which includes some FLRM-metrics), geodesically connected by Riemannian geodesics of g.