The sigma invariant of the n torus, the K3 surface, and Euclidean and elliptic 3d manifolds

On the space of isometric embeddings fg of metrics g on a manifold Mⁿ into the standard (S^= (n), ), we consider the total exterior scalar curvature Θ₅₆ (M), and squared L² norm of the mean curvature vector Φ₅₆ (M) and second fundamental form Π₅₆ (M) functionals of fg, respectively. Then W₅₆ (M) = (1-δ₍, ₁) (n/ (n-1) ) Θ₅_₆ (M) + Φ₅_₆) (M) and D₅₆ (M) = (1-δ₍, ₁) (1/ (n-1) ) Θ₅₆ (M) +Π₅_₆) (M) are functionals intrinsically defined in the space of metrics in the conformal class of g, and Sg (M): = sg dμg=W₅₆ (M) - D₅₆ (M). We extend the notions of σ invariant and Kazdan-Warner type to manifolds of dimension n 1. M is a manifold of type II if, and only if, it admits a Ricci flat metric g with minimal isometric embedding fg that minimizes W₅_₆' (M) and D₅_₆' (M) among metrics g' in conformal classes g' with scalar flat representatives. We show that the torus Tⁿ, the K3 surface, and any Euclidean 3d manifold are manifolds of Kazdan-Warner type II, exhibiting in each case the canonical Ricci flat g that realizes the vanishing σ invariant and said minimal value W₅₆ (M) = D₅₆ (M), with Euclidean 3d manifolds of isomorphic π₁ being diffeomorphic iff the values of W₅₆ (M) for their canonical gs are the same. An elliptic 3d manifold (M, ΓM) of underlying group π₁ (M) ΓM SO (4) has σ (M) =6 (2π²) ^2{3}/|π₁ (M) |^2{3}, and if (M, ΓM) and (M', Γ₌') are two of them of isomorphic π₁, M is diffeomorphic to M' iff the spaces of ΓM and Γ₌' invariant homogeneous spherical harmonics of degree |π₁| are the same.