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We define a generalized mass for asymptotically flat manifolds using somehigher order symmetric function of the curvature tensor. This mass isnon-negative when the manifold is locally conformally flat and the $\sigma_k$curvature vanishes at infinity. In addition, with the above assumptions, if themass is zero, then, near infinity, the manifold is isometric to a Euclideanend.

A generalized mass involving higher order symmetric function of the curvature tensor