  In [7] Klainerman introduced the hyperboloidal method to prove the global
existence results for nonlinear Klein-Gordon equations by using commuting
vector fields. In this paper, we extend the hyperboloidal method from Minkowski
space to Lorentzian spacetimes. This approach is developed in [14] for proving,
under the maximal foliation gauge, the global nonlinear stability of Minkowski
space for Einstein equations with massive scalar fields, which states that, the
sufficiently small data in a compact domain, surrounded by a Schwarzschild
metric, leads to a unique, globally hyperbolic, smooth and geodesically
complete solution to the Einstein Klein-Gordon system.
  In this paper, we set up the geometric framework of the intrinsic hyperboloid
approach in the curved spacetime. By performing a thorough geometric comparison
between the radial normal vector field induced by the intrinsic hyperboloids
and the canonical $\p_r$, we manage to control the hyperboloids when they are
close to their asymptote, which is a light cone in the Schwarzschild zone. By
using such geometric information, we not only obtain the crucial boundary
information for running the energy method in [14], but also prove that the
intrinsic geometric quantities including the Hawking mass all converge to their
Schwarzschild values when approaching the asymptote.
