论文标题
与局部RICCI有界覆盖几何形状的折叠歧管
Collapsed manifolds with local Ricci bounded covering geometry
论文作者
论文摘要
对于$ρ，v> 0 $，我们说一个$ n $ -manifold $ m $满足本地$（ρ，v）$ - 绑定的几何形状，如果ricci曲率$ \ text {ric} _m \ ge ge-（n-1）$，以及m $ x \ in m $ x \ ge ge e ge e ge-ge（n-1）$ v> 0 $，其中$ \ tilde x $是$ x $ $ x $的（本地）Riemannian通用封面上的$ x $的倒数图像。 在本文中，我们将Cheeger-fukaya-Gromov的nilpotent纤维捆绑捆绑在倒塌的$ n $ n $ -manifold $ m $限制的截面曲率上到本地$ $ m $（ρ，v）$ - 绑定的ricci ricci覆盖几何形状，涵盖了$ m $，$ m $与非collapsed riemannynian porliemannian sorementian porliementian sove of polliemannian dimentian。 Nilpotent纤维束定理显着改善了[HU]中的纤维束定理，并加强了从[NZ]中看到的nilpotent纤维束，并意味着[HW]中的圆环束，这些圆环束在其他局部或全球拓扑条件下获得。 Our construction of a nilpotent fibration requires a new proof for a result in [HKRX]: if an $n$-manifold $M$ with local $(1,v)$-bound Ricci covering geometry has diameter $<ε(n,v)$, a constant depends on $n$ and $v$, then $M$ is diffeomorphic to an infra-nilmanifold. [HKRX]中的证明是表明RICCI流量产生了几乎平坦的度量，因此结果来自Gromov的定理几乎平坦的歧管。新的证明独立于格罗莫夫定理，因此具有推论。如果第一个betti number $ b_1（m）= n $，则$ m $满足$（1，v）$ - 限制的ricci覆盖几何形状，因此$ m $对标准圆环（[CO2]）是不同的。
For $ρ, v>0$, we say that an $n$-manifold $M$ satisfies local $(ρ,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_ρ(\tilde x))\ge v>0$, where $\tilde x$ is an inverse image of $x$ on the (local) Riemannian universal cover of the $ρ$-ball at $x$. In this paper, we extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed $n$-manifold $M$ of bounded sectional curvature to $M$ of a local $(ρ,v)$-bound Ricci covering geometry, and $M$ is close to a non-collapsed Riemannian manifold of lower dimension. The nilpotent fiber bundle theorem significantly improves fiber bundle theorem in [Hu], and it strengthens a nilpotent fiber bundle seen from [NZ] and implies the torus bundle in [HW], which are obtained under additional local or global topological conditions, respectively. Our construction of a nilpotent fibration requires a new proof for a result in [HKRX]: if an $n$-manifold $M$ with local $(1,v)$-bound Ricci covering geometry has diameter $<ε(n,v)$, a constant depends on $n$ and $v$, then $M$ is diffeomorphic to an infra-nilmanifold. The proof in [HKRX] is to show that the Ricci flows produces an almost flat metric, thus the result follows from the Gromov's theorem on almost flat manifolds. The new proof is independent of the Gromov's theorem, thus has which as a corollary. If the first Betti number $b_1(M)=n$, then $M$ satisfies a $(1,v)$-bound Ricci covering geometry, thus $M$ is diffeomorphic to a standard torus ([Co2]).
