In this paper we introduce a mass for asymptotically flat manifolds by using
the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is
a geometric invariant, if the Gauss-Bonnet curvature is integrable and the
decay order τ satisfies τ>3n−4​. Then we show a positive
mass theorem for asymptotically flat graphs over Rn. Moreover we
obtain also Penrose type inequalities in this case.Comment: 32 pages. arXiv:1211.7305 was integrated into this new version as an
applicatio