2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/155484We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a point or orthogonal to an hypersurface, one of these metrics being a warped product if the dimension $n$ is greater than or equal to 3. This formula has non-trivial geometric consequences such as a positive mass type theorem and other rigidity results. We also apply our result to some standard models.14 pagesDifferential Geometry53C24; 53C21text