Relative radial mass and rigidity of some warped product manifolds

TL;DR

This paper develops a Riccati-type formula for warped product manifolds with shared geodesic rays, leading to new geometric results like positive mass theorems and rigidity, with applications to standard models.

Contribution

It introduces a novel Riccati formula for metrics with common geodesic rays, extending geometric analysis in warped product manifolds.

Findings

Derived a Riccati type formula for specific metrics

Established positive mass type theorems

Proved rigidity results for warped product manifolds

Abstract

We 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.