Chern’s magic form and the Gauss–Bonnet–Chern mass

Abstract

In this note, we use Chern’s magic form $$\Phi _k$$ in his famous proof of the Gauss–Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss–Bonnet–Chern curvature $$L_k$$ . Moreover, this equivalence implies a simple proof of the equivalence between the ADM mass and the intrinsically defined mass via the Ricci tensor, which was reconsidered by Miao–Tam (Proc Am Math Soc 144:753–761, 2016) and Herzlich (Ann Henri Poincare 17(12):3605–3617, 2016) very recently.