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Extrinsic Bonnet-Myers Theorem and almost rigidity

Weiying Li, Guoyi Xu

TL;DR

The paper addresses an extrinsic version of the Bonnet-Myers theorem and the almost rigidity of extrinsic diameter for compact manifolds embedded in Euclidean spaces. It develops an extrinsic diameter bound: for complete \( (M^n,g) \) with Rc \ge (n-1), the extrinsic diameter \( \mathrm{Diam}_{\mathbb{R}^m}(M^n,g) \) is strictly less than \( \pi \) for every isometric embedding, and demonstrates sharpness via sphere-like examples with \( K(g) \ge 1 \) approaching \( \pi \). The work also yields a partial rigidity result when \( K(g) \ge 1 \) and the ambient codimension is \( 1 \), and establishes a quantitative almost rigidity bound in the Gromov-Hausdorff sense: \( \frac{d_{GH}((M^n,g), [0,\pi])}{\sqrt{\pi-\mathrm{Diam}_{\mathbb{R}^{n+1}}(M^n,g)}} \le 4\pi^{3/2} \). Through height/width estimates of Euclidean triangles arising from the embedding and Toponogov-type comparisons, the paper links intrinsic curvature to extrinsic width and shows that near-maximal extrinsic diameter forces the embedded manifold to lie near a line segment, enabling a precise GH-approximation to the interval.

Abstract

We establish the extrinsic Bonnet-Myers Theorem for compact Riemannian manifolds with positive Ricci curvature. And we show the almost rigidity for compact hypersurfaces, which have positive sectional curvature and almost maximal extrinsic diameter in Euclidean space.

Extrinsic Bonnet-Myers Theorem and almost rigidity

TL;DR

The paper addresses an extrinsic version of the Bonnet-Myers theorem and the almost rigidity of extrinsic diameter for compact manifolds embedded in Euclidean spaces. It develops an extrinsic diameter bound: for complete \( (M^n,g) \) with Rc \ge (n-1), the extrinsic diameter \( \mathrm{Diam}_{\mathbb{R}^m}(M^n,g) \) is strictly less than for every isometric embedding, and demonstrates sharpness via sphere-like examples with \( K(g) \ge 1 \) approaching . The work also yields a partial rigidity result when \( K(g) \ge 1 \) and the ambient codimension is , and establishes a quantitative almost rigidity bound in the Gromov-Hausdorff sense: \( \frac{d_{GH}((M^n,g), [0,\pi])}{\sqrt{\pi-\mathrm{Diam}_{\mathbb{R}^{n+1}}(M^n,g)}} \le 4\pi^{3/2} \). Through height/width estimates of Euclidean triangles arising from the embedding and Toponogov-type comparisons, the paper links intrinsic curvature to extrinsic width and shows that near-maximal extrinsic diameter forces the embedded manifold to lie near a line segment, enabling a precise GH-approximation to the interval.

Abstract

We establish the extrinsic Bonnet-Myers Theorem for compact Riemannian manifolds with positive Ricci curvature. And we show the almost rigidity for compact hypersurfaces, which have positive sectional curvature and almost maximal extrinsic diameter in Euclidean space.
Paper Structure (4 sections, 15 theorems, 59 equations, 7 figures)

This paper contains 4 sections, 15 theorems, 59 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The extrinsic Bonnet-Myers Theorem
  3. The height of triangles
  4. The almost rigidity for extrinsic diameter

Key Result

Theorem 1.1

For complete Riemannian manifold $(M^n, g)$ with $Rc\geq (n- 1)$, we have Furthermore (diam wrt f strict) is sharp in the following sense: there exists a sequence of $(S^n, g_k)$ with $K(g_k)\geq 1$ and $\mathscr{I}_k\in\mathcal{IE}((S^n,g_k)$, $\mathbb{R}^{n+1})$, such that $\lim_{k\rightarrow\infty}\mathrm{Diam}_{\mathscr{I}_k}(S^n, g_k)= \pi$.

Figures (7)

  • Figure 1: The figure of h(t)
  • Figure 2: The figure of f(t)
  • Figure 3: Scaling the slice spheres to smooth the two ends singularities
  • Figure 4: The Euclidean Triangle
  • Figure 5: Cut $\mathscr{I}(M^n)$ by $P^{-1}(t)$
  • ...and 2 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 15 more