Submetries vs. submersions

Rigidity theorems for submetries in positive curvature

We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq π/2 $. In case of equality, there is a Riemannian submersion $\mathbb{S} \to M$ from a unit sphere, and as a consequence, $f$ is known up to metric congruence. A similar rigidity theorem also holds in the general context of Riemannian foliations.

次度量映射的光滑性

Smoothness of submetries

本文在次度量映射框架下证明非负曲率紧致光滑黎曼流形间的黎曼淹没必为光滑，并证明底空间光滑性可由全空间光滑性推出，解决了 Berestovskii–Guijarro 猜想。

Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below…

关于 Alexandrov 空间坍缩序列的一个纤维化定理

A fibration theorem for collapsing sequences of Alexandrov spaces

针对仅含弱奇点的 Alexandrov 空间坍缩序列，本文证明在方向空间体积满足一致正下界且足够强时，Yamaguchi 的近似 Lipschitz 淹没为局部平凡纤维化，并刻画纤维度量与体积性质。

Homotopy lifting property of an $e^ε$-Lipschitz and co-Lipschitz map

An $e^ε$-Lipschitz and co-Lipschitz map, as a metric analogue of an $ε$-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stability in Gromov-Hausdorff topology. Due to an overlook in the previous version, the part for bounding intrinsic distance of fibers will be talked about in the coming papers.