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

TL;DR

This paper introduces a new generalized mass for asymptotically flat manifolds based on higher order symmetric functions of the curvature tensor, establishing conditions for non-negativity and rigidity.

Contribution

It defines a novel mass involving higher order symmetric functions of the curvature tensor and proves its non-negativity and rigidity under specific geometric conditions.

Findings

Mass is non-negative when the manifold is locally conformally flat with vanishing $\sigma_k$ curvature at infinity.

Zero mass implies the manifold is isometric to a Euclidean end near infinity.

The generalized mass extends classical concepts to higher order curvature invariants.

Abstract

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