  Let $(M_1,g_1)$ and $(M_2,g_2)$ be two $C^\infty$--differentiable connected,
complete Riemannian manifolds, $k:M_1\to\mathbb R$ a $C^\infty$--differentiable
function, having $0<k_0<k(x)\leq K_0$, for any $x\in M_1$ and $g:=g_1-kg_2$ the
semi Riemannian metric on the product manifold $M:=M_1\times M_2$. We associate
to $g$ a suitable family of Riemannian metrics $G_r+g_2$, with $r>-K_0^{-1}$,
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 $z_0=(x_0,y_0)\in M$
and for any $y_1\in M_2$ there exists a subset $A$ of $M_1$, such that all the
geodesics of $g$ joining $z_0$ with a point $(x_1,y_1)$, with $x_1\in A$, 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 $g$.
