2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62309We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.27 pages, Latex2e with several style files; various misprints corrected, positivity theorem for black holes considerably strengthened, to appear in Pacific Jour. of MathematicsDifferential GeometryMathematical Physics53c20; 83c40text