  We derive geometric formulas of the mass for asymptotically hyperbolic
manifolds using large coordinate horospheres, and state relevant rigidity
results. The formulas use the difference between the mean curvatures with
respect to a given metric and the hyperbolic metric, so some rigidity results
are stated in terms of the mean curvature inequality on a certain portion of
horospheres. We also present a way to characterize the region near infinity
that does not contribute to the mass as an application.
