  We derive geometric formulas for the mass of asymptotically hyperbolic
manifolds using coordinate horospheres. As an application, we obtain a new
rigidity result of hyperbolic space: if a complete asymptotically hyperbolic
manifold has scalar curvature lower bound -n(n-1) and is isometric to
hyperbolic space outside a coordinate horosphere, then the manifold is
isometric to hyperbolic space. In addition, we apply our formula to investigate
regions near infinity that do not contribute to the mass quantity, which leads
to improved rigidity results of hyperbolic space.
