Riemannian geodesics of semi Riemannian warped metrics

Oriella M. Amici; Biagio C. Casciaro

paper

Riemannian geodesics of semi Riemannian warped metrics

arXiv:1301.5140

Abstract

Let $(M_{1}, g_{1})$ and $(M_{2}, g_{2})$ be two $C^{\infty}$ --differentiable connected, complete Riemannian manifolds, $k : M_{1} \to R$ a $C^{\infty}$ --differentiable function, having $0 < k_{0} < k (x) \leq K_{0}$ , for any $x \in M_{1}$ and $g := g_{1} - k g_{2}$ 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 $g$ .