Geometric relativity is mostly referred to the study of spatial manifolds in space-time manifolds as main objects in general relativity. Our goal in this paper, is to study spatial models of the universe that are themselves tanegnt bundles. In Arnowitt, Deser and Misner's formulation of general relativity as a Hamiltonian system, the main quantitties that are studied are total energy and mass of a system and in this paper, we will look at the ADM mass for some asymptotically Euclidean tangent bundles. First, we will completely characterize asymptotically Euclidean interpolation metrics on tangent bundles of simple manifolds. We will then consider a family of interpolation metrics that we will call admissible metrics. We will define the notion of lower ADM mass (that is weaker than ADM mass) and estimate the lower ADM mass of admissible metrics. From the said estimate, we show that a stronger version of Schoen and Yau's positive mass and rigidity of positive mass holds for admissible metrics.
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