Relative radial mass and rigidity of some warped product manifolds

We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a fixed point or orthogonal to a special fixed hypersurface. We assume that one of these metrics is 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.