AI 中文总结
研究通过等距嵌入观察度量，探讨不同条件下闭流形的Kazdan - Warner类型及sigma不变量，利用相关结果确定了具有特定模型几何的流形的KW类型和sigma不变量，如双曲模型流形及其他几种流形的相关性质。
AI 中文摘要
我们通过等距嵌入$f_g:(M^n,g)\rightarrow (\mathbb{S}^n,\tau_g)$及其变形来观察度量。若$M$具有常数量曲率$s_g$且Ricci张量$r_g \leq 0$的度量$g$，且不存在除Ricci平坦度量外的标量平坦度量，则$M$不是Kazdan - Warner（KW）I型流形；若Ricci平坦度量空间非空，则$M$是KW II型流形，否则是KW III型流形。若$M$具有非平凡数量曲率$s_{g'}\geq 0$的度量$g'$和满足$r_{g_{-}}<0$的Einstein度量$g_{-}$，则$M$必有标量平坦非Ricci平坦和Ricci平坦度量，若可定向则可自旋。在特定维度条件下不存在这样特有的流形。基于这些结果，我们找到了几个具有Thurston模型几何$({\rm Isom}(X,g),X)$的流形$M=X/\Gamma_M$的KW类型和sigma不变量。特别地，证明了双曲模型$(\mathbb{H}^n,g_{\mathbb{H}^n})$的$M^n=\mathbb{H}^n/\Gamma_M$是KW III型流形，在一定条件下其$\Gamma_M$不变双曲度量$g_M$和类实现其sigma不变量，双曲度量空间是路径连通的，由等体积度量$g_t$的等距变形$f_{g_t}$组成，且$(M,g_t)$与$(M,g_M)$等距。同时，3维幂零、可解、$\widetilde{\mathbb{PSL}}(2,\mathbb{R})$和$\mathbb{R}\times \mathbb{H}^2$流形也都是KW III型流形，但不可实现的sigma不变量为零。
英文摘要
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 $M=X/Γ_M$ with model geometry $({\rm Isom}(X,g),X)$ of Thurston. Notably, we show that an $M^n=\mb{H}^n/Γ_M$ of hyperbolic model $(\mb{H}^n,g_{\mb{H}^n})$ is of KW type III, that if $n\geq 3$ its $Γ_M$ 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.
