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The Injectivity Radius of Souls of Alexandrov Spaces

Elena Mäder-Baumdicker, Jona Seidel

TL;DR

This paper extends Šarafutdinov’s injectivity-radius bound from smooth Riemannian manifolds to finite-dimensional Alexandrov spaces with nonnegative curvature and an upper curvature bound $K$. It develops two notions of injectivity radius in length spaces, builds a robust theory of conjugate points without smooth structure, and leverages the Cheeger–Gromoll soul along with Šarafutdinov’s retraction to relate the global injectivity radius to that of the soul. The main result states that if $\mathrm{inj}(X)\neq\mathrm{inj}(S)$, then $\mathrm{inj}(X)\geq\frac{\pi}{\sqrt{K}}$, and in the setting considered, $\mathrm{inj}(S)=\mathrm{inj}(X)$, yielding explicit lower bounds and structural insights. These findings have implications for volume growth, non-collapsing, and topological rigidity in Alexandrov spaces, extending classical Riemannian geometry results to a broader metric-measure framework.

Abstract

A sharp lower bound for the injectivity radius in noncompact nonnegatively curved Riemannian manifolds involving their soul goes back to Šarafutdinov. We generalize this bound to the setting of Alexandrov spaces. Our main theorem reads as follows. If the injectivity radius of an Alexandrov space of nonnegative curvature does not coincide with the one of its souls, then it is at least $ πK^{-1/2} $, where $ K $ is an upper curvature bound. We introduce the soul of Alexandrov spaces in some detail and compare two notions of injectivity radii.

The Injectivity Radius of Souls of Alexandrov Spaces

TL;DR

This paper extends Šarafutdinov’s injectivity-radius bound from smooth Riemannian manifolds to finite-dimensional Alexandrov spaces with nonnegative curvature and an upper curvature bound . It develops two notions of injectivity radius in length spaces, builds a robust theory of conjugate points without smooth structure, and leverages the Cheeger–Gromoll soul along with Šarafutdinov’s retraction to relate the global injectivity radius to that of the soul. The main result states that if , then , and in the setting considered, , yielding explicit lower bounds and structural insights. These findings have implications for volume growth, non-collapsing, and topological rigidity in Alexandrov spaces, extending classical Riemannian geometry results to a broader metric-measure framework.

Abstract

A sharp lower bound for the injectivity radius in noncompact nonnegatively curved Riemannian manifolds involving their soul goes back to Šarafutdinov. We generalize this bound to the setting of Alexandrov spaces. Our main theorem reads as follows. If the injectivity radius of an Alexandrov space of nonnegative curvature does not coincide with the one of its souls, then it is at least , where is an upper curvature bound. We introduce the soul of Alexandrov spaces in some detail and compare two notions of injectivity radii.
Paper Structure (15 sections, 39 theorems, 69 equations, 7 figures)

This paper contains 15 sections, 39 theorems, 69 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Cheeger-Gromoll Soul
  4. The Soul Construction
  5. Šarafutdinov's Retraction
  6. The Injectivity Radius in Length Spaces
  7. Two Injectivity Radii
  8. Conjugate Points
  9. The Injectivity Radius Before Conjugate Points
  10. Conjugate Points and an Upper Curvature Bound
  11. The Injectivity Radius of a Soul
  12. Appendix
  13. Hausdorff Distance
  14. Real Functions with One-Sided Derivative
  15. Klingenberg's Injectivity Radius Estimate

Key Result

Theorem 1

Let $K> 0$ and $X$ be a finite-dimensional Alexandrov space of non-negative curvature bounded above by $K$ and $S\subseteq X$ be a Cheeger-Gromoll soul. If $\mathop{\mathrm{inj}}\nolimits(X)<\frac{\pi}{\sqrt{K}}$, then

Figures (7)

  • Figure 1: The soul construction on the cylinder terminating in a soul $S$. $C$ is a superlevel set of the minimum of the Busemann functions corresponding to the two rays issuing in $p_0$
  • Figure 2: The soul $S$ and the sets $C_t$ from Lemma \ref{['lem_soulSuperlevelSets']} for a closed half-plane $X$ when starting in $p_0$. Some directions of rays issuing in $p_0$ are indicated
  • Figure 3: The soul $S$ in Example \ref{['ex_soulTheoremCounterexample']}
  • Figure 4: Two souls on a cylinder with the isometries $f_1$, $f_2$ and the homotopy $(t,x)\mapsto h_t(x)$
  • Figure 5: The setup in the proof of Lemma \ref{['lem_Klingenberg']} leading to a contradiction
  • ...and 2 more figures

Theorems & Definitions (103)

  • Theorem 1
  • Remark 1.1
  • Corollary 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 93 more