We consider the Positive Mass Theorem for Riemannian manifolds $$(M^{n},g)$$ ( M n , g ) with the asymptotic end $$(\mathbb {R}^{k}\times X^{n-k}, g_{\mathbb {R}^{k}}+g_{X})$$ ( R k × X n - k , g R k + g X ) ($$k\ge 3$$ k ≥ 3 ) by studying the corresponding compactification problem. Here $$(X, g_X)$$ ( X , g X ) is a compact scalar flat manifold. We show that the Positive Mass Theorem holds if certain generalized connected sum admits no metric of positive scalar curvature. Moreover we establish the rigidity result, namely, the mass is zero iff M is isometric to $$\mathbb {R}^{k}\times X^{n-k}$$ R k × X n - k .

Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group G are extremal (in fact rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact the same result holds for a compact connected homogeneous manifold G/H with G compact connect and semi-simple.