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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 $C^\infty$--differentiablefunction, having $0<k_0<k(x)\leq K_0$, for any $x\in M_1$ and $g:=g_1-kg_2$ thesemi Riemannian metric on the product manifold $M:=M_1\times M_2$. We associateto $g$ 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 aregeodesics 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 thegeodesics of $g$ joining $z_0$ with a point $(x_1,y_1)$, with $x_1\in A$, areRiemannian. The Riemannian geodesics of $g$ determine a "partial" property ofgeodesic connection on $M$. Finally, we determine two new classes of semiRiemannian metrics (one of which includes some FLRM-metrics), geodesicallyconnected by Riemannian geodesics of $g$.

Riemannian geodesics of semi Riemannian warped metrics