Closed manifolds, model geometries, and volume related differentiable invariants

We view metrics through their isometric embeddigns $f_g:(M^n,g)\rightarrow (\mb{S}^{\tn},\tg)$ and their deformations. If $M$ carries a metric $g$ of constant scalar curvature $s_g$ and Ricci tensor $r_g \leq 0$, and if this $M$ does not carry scalar flat metrics other than Ricci flat ones, then $M$ is not a manifold of Kazdan-Warner (KW) type I, and if the space of Ricci flat metrics is not empty, $M$ is a manifold of KW type II, while otherwise, $M$ is of KW type III. If $M$ carries a metric $g'$ of nontrivial scalar curvature $s_{g'}\geq 0$, and an Einstein metric $g_{-}$ such that $r_{g_{-}}<0$, then $M$ must carry both, scalar flat non Ricci flat and Ricci flat metrics, and if orientable, it is spinnable. No such manifold exists if $n\leq 3$, and if $r_{g'}$ is assumed further to be positive, no such manifold exists if $n\leq 4$, and in these dimensions, $M^{n}$ can admit Einstein metrics of scalar curvature of at most one sign. If $M$ has a contractible universal cover and carries no Ricci flat metrics at all, $M$ is of KW type III. Based on these resulst, we find the KW type and sigma invariant of several manifolds $M=X/\Gamma_M$ with model geometry $({\rm Isom}(X,g),X)$ of Thurston. Notably, we show that an $M^n=\mb{H}^n/\Gamma_M$ of hyperbolic model $(\mb{H}^n,g_{\mb{H}^n})$ is of KW type III, that if $n\geq 3$ its $\Gamma_M$ invariant hyperbolic metric $g_M$ and class realize its sigma invariant, and that the space of hyperbolic metrics on $M$ is path connected and consists of isotopic deformations $f_{g_t}$ of $f_{g_0}:=f_{g_M}$ of equal volume metrics $g_t$ of constant sectional curvature $-1$, with $(M,g_t)$ isometric to $(M,g_M)$ for all $t$, while $3$d nil, solv, $\widetilde{\mb{P}\mb{S}\mb{L}}(2,\mb{R})$ and $\mb{R}\times \mb{H}^2$ manifolds are all of KW type III also, but have vanishing nonachievable sigma invariant.