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

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\text{mb}{S}^{\text{tn}=\text{tn}(n)},\mathrm{tg})$, we consider the total exterior scalar curvature ${\mathrm{\Theta }}_{f_g}(M)$, and squared $L^2$ norm of the mean curvature vector ${\mathrm{\Phi }}_{f_g}(M)$ and second fundamental form ${\mathrm{\Pi }}_{f_g}(M)$ functionals of $f_g$, respectively. Then $\text{mc}{W}_{f_g}(M)=(1-{\delta }_{n,1})(n\mathrm{/}(n-1)){\mathrm{\Theta }}_{f_g}(M)+{\mathrm{\Phi }}_{f_{g)}}(M)$ and $\text{mc}{D}_{f_g}(M)=(1-{\delta }_{n,1})(1\mathrm{/}(n-1)){\mathrm{\Theta }}_{f_g}(M)+{\mathrm{\Pi }}_{f_{g)}}(M)$ are functionals intrinsically defined in the space of metrics in the conformal class of $g$, and $\text{mc}{S}_g(M):=\int s_{g}d{\mu }_g=\text{mc}{W}_{f_g}(M)-\text{mc}{D}_{f_g}(M)$. We extend the notions of $\sigma$ invariant and Kazdan-Warner type to manifolds of dimension $n\ge 1$. $M$ is a manifold of type II if, and only if, it admits a Ricci flat metric $g$ with minimal isometric embedding $f_g$ that minimizes $\text{mc}{W}_{f_{g^{\mathrm{\prime }}}}(M)$ and $\text{mc}{D}_{f_{g^{\mathrm{\prime }}}}(M)$ among metrics $g^{\mathrm{\prime }}$ in conformal classes $[g^{\mathrm{\prime }}]$ with scalar flat representatives. We show that the torus $T^n$, 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 $\sigma$ invariant and said minimal value $\text{mc}{W}_{f_g}(M)=\text{mc}{D}_{f_g}(M)$, with Euclidean 3d manifolds of isomorphic ${\pi }_1$ being diffeomorphic iff the values of $\text{mc}{W}_{f_g}(M)$ for their canonical $g$s are the same. An elliptic 3d manifold $(M,{\mathrm{\Gamma }}_M)$ of underlying group ${\pi }_1(M)\cong {\mathrm{\Gamma }}_M\subset \text{mb}S\text{mb}O(4)$ has $\sigma (M)=6(2{\pi }^2{)}^{\frac{2}{3}}\mathrm{/}\mathrm{\mid }{\pi }_1(M){\mathrm{\mid }}^{\frac{2}{3}}$, and if $(M,{\mathrm{\Gamma }}_M)$ and $(M^{\mathrm{\prime }},{\mathrm{\Gamma }}_{M^{\mathrm{\prime }}})$ are two of them of isomorphic ${\pi }_1$, $M$ is diffeomorphic to $M^{\mathrm{\prime }}$ iff the spaces of ${\mathrm{\Gamma }}_M$ and ${\mathrm{\Gamma }}_{M^{\mathrm{\prime }}}$ invariant homogeneous spherical harmonics of degree $\mathrm{\mid }{\pi }_1\mathrm{\mid }$ are the same.