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Open Alexandrov spaces of nonnegative curvature

Xueping Li, Xiaochun Rong

TL;DR

This work analyzes open $n$-dimensional Alexandrov spaces $X$ with curvature bounded below by $0$ and a soul $S$, focusing on structural parallels with open nonnegatively curved Riemannian manifolds. Central methods include the Sharafutdinov projection $\phi:X\to S$, the theory of (weakly) integrable submetries, and the study of extremal subsets, canonical neighborhoods, and flat-strip phenomena. The authors establish partial verifications and rigidity results: (i) weakly integrable submetries exhibit canonical local trivializations and fiber-bundle structures under integrability; (ii) in codimension two, they prove a canonical bundle or splitting dichotomy for $X$, with a bundle isomorphism $g\exp: C(\Sigma^{\perp}S)\to X$ or a metric-splitting of the universal cover; and (iii) they develop a framework of strictly non-critical maps to obtain local bundle structures. Together, these results advance the Alexandrov-geometry analogues of soul theory, submetry rigidity, and open-manifold decompositions, with implications for generalized Cheeger-Gromoll-type decompositions in singular spaces.

Abstract

Let $X$ be an open (i.e. complete, non-compact and without boundary) Alexandrov $n$-space of nonnegative curvature with a soul $S$. In this paper, we will establish several structural results on $X$ that can be viewed as counterparts of structural results on an open Riemannian manifold with nonnegative sectional curvature.

Open Alexandrov spaces of nonnegative curvature

TL;DR

This work analyzes open -dimensional Alexandrov spaces with curvature bounded below by and a soul , focusing on structural parallels with open nonnegatively curved Riemannian manifolds. Central methods include the Sharafutdinov projection , the theory of (weakly) integrable submetries, and the study of extremal subsets, canonical neighborhoods, and flat-strip phenomena. The authors establish partial verifications and rigidity results: (i) weakly integrable submetries exhibit canonical local trivializations and fiber-bundle structures under integrability; (ii) in codimension two, they prove a canonical bundle or splitting dichotomy for , with a bundle isomorphism or a metric-splitting of the universal cover; and (iii) they develop a framework of strictly non-critical maps to obtain local bundle structures. Together, these results advance the Alexandrov-geometry analogues of soul theory, submetry rigidity, and open-manifold decompositions, with implications for generalized Cheeger-Gromoll-type decompositions in singular spaces.

Abstract

Let be an open (i.e. complete, non-compact and without boundary) Alexandrov -space of nonnegative curvature with a soul . In this paper, we will establish several structural results on that can be viewed as counterparts of structural results on an open Riemannian manifold with nonnegative sectional curvature.
Paper Structure (27 sections, 54 theorems, 22 equations)

This paper contains 27 sections, 54 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stratifications, submetries, of Alexandrov spaces, and lifting gradient flows
  4. Stabilities, and local structures
  5. Rigid structures on $\Sigma_{\bar{x}}X$
  6. Partial flat strips
  7. Partial progress in Conjectures \ref{['0e']}-\ref{['0g']}, and examples
  8. Proof of Theorem A
  9. Proof of (A1)
  10. Proof of (A2)
  11. Proof of (A3)
  12. Proof of Proposition \ref{['0j']}
  13. Proof of Theorem B
  14. Proof of (B2)
  15. Proof of Lemma \ref{['3a']}
  16. ...and 12 more sections

Key Result

Theorem 1

(Soul, Cheeger-Gromoll CG) Let $M$ be an open Riemannian manifold of $\text{sec}\ge 0$. Then $M$ contains a compact totally convex submanifold $S$ (called a soul of $M$), and there is a diffeomorphism, $f: T^\perp S\to M$, where $T^\perp S$ denotes the normal bundle of $S$. Moreover, if $\text{sec}_

Theorems & Definitions (115)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 8: Integrability of submetries
  • Conjecture 9
  • Proposition 10
  • Conjecture 11
  • Lemma 1.1
  • proof : Proof of Lemma \ref{['1a']}
  • ...and 105 more