Properties of Closed Manifolds, Model Geometries, and Differentiable Invariants

Overview

This research investigates closed manifolds, focusing on their classification according to Kazdan-Warner (KW) types and their sigma invariants. The classification hinges on specific properties of metrics, particularly their scalar curvature ($s_g$) and Ricci tensor ($r_g$), and the existence of scalar flat or Ricci flat metrics. The study also examines manifolds with established model geometries.

Research Context

The study views metrics through their isometric embeddings $f_g:(M^n,g)\rightarrow (\mb{S}^{\tn},\tg)$ and subsequent deformations. A central concept is the classification of manifolds based on the Kazdan-Warner (KW) types, which are defined by conditions related to scalar flat and Ricci flat metrics. The research differentiates three KW types: Type I, Type II, and Type III, each corresponding to specific metric characteristics.

Approach

The research defines conditions for classifying manifolds into Kazdan-Warner (KW) types based on their metric properties. These conditions include the presence of a metric $g$ with constant scalar curvature $s_g$ and Ricci tensor $r_g \leq 0$. Further distinctions are made based on whether the manifold carries scalar flat metrics other than Ricci flat ones, or if the space of Ricci flat metrics is non-empty or empty.

For a manifold $M$ carrying a metric $g$ with constant scalar curvature $s_g$ and Ricci tensor $r_g \leq 0$, the following Kw-type definitions are applied:

• If $M$ does not carry scalar flat metrics other than Ricci flat ones, then $M$ is not a manifold of KW type I.
• If, in the above case, the space of Ricci flat metrics is not empty, then $M$ is a manifold of KW type II.
• If, in the above case, the space of Ricci flat metrics is empty, then $M$ is of KW type III.

The study also considers scenarios where $M$ carries a metric $g'$ of nontrivial scalar curvature $s_{g'}\geq 0$ and an Einstein metric $g_{-}$ such that $r_{g_{-}} \leq 0$. In such cases, if $M$ is orientable, it is spinnable.

Subsequently, the research examines specific manifolds $M=X/\Gamma_M$ with model geometry $({\rm Isom}(X,g),X)$ of Thurston, determining their KW type and sigma invariant based on the established criteria.

Findings

• If a manifold $M$ carries a metric $g$ of constant scalar curvature $s_g$ and Ricci tensor $r_g \leq 0$, and if $M$ does not carry scalar flat metrics other than Ricci flat ones, then $M$ is not a manifold of KW type I. If, additionally, the space of Ricci flat metrics is not empty, $M$ is of KW type II. Otherwise, if the space of Ricci flat metrics is empty, $M$ is of KW type III.
• If a manifold $M$ carries a metric $g'$ of nontrivial scalar curvature $s_{g'}\geq 0$ and an Einstein metric $g_{-}$ such that $r_{g_{-}} \leq 0$, then $M$ must carry both scalar flat non-Ricci flat metrics and Ricci flat metrics. If such $M$ is orientable, it is spinnable.
• No such manifold (as described immediately above) exists if $n\leq 3$.
• If the Ricci tensor $r_{g'}$ is further assumed to be positive, no such manifold exists if $n\leq 4$. In these dimensions ($n\leq 4$), $M^n$ 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.
• An $M^n=\mb{H}^n/\Gamma_M$ of hyperbolic model $(\mb{H}^n,g_{\mb{H}^n})$ is of KW type III.
• For hyperbolic manifolds $M^n=\mb{H}^n/\Gamma_M$, if $n\geq 3$, its $\Gamma_M$ invariant hyperbolic metric $g_M$ and class realize its sigma invariant.
• For hyperbolic manifolds $M^n=\mb{H}^n/\Gamma_M$, 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 $t$.
• Three-dimensional nil manifolds, solv manifolds, $\widetilde{\mb{P}\mb{S}\mb{L}}(2,\mb{R})$ manifolds, and $\mb{R}\times \mb{H}^2$ manifolds are all of KW type III, but have a vanishing nonachievable sigma invariant.

Why This Matters

The classification of closed manifolds into Kazdan-Warner types provides a framework for understanding their fundamental geometric properties, particularly concerning the existence and characteristics of specific metrics like scalar flat and Ricci flat metrics. This categorization helps to delineate the landscape of possible manifold geometries under various curvature conditions. The findings concerning specific model geometries, such as hyperbolic manifolds and certain 3D homogeneous manifolds, contribute to a deeper characterization of these well-studied mathematical objects.