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Diameter and focal radius of submanifolds

Ricardo A. E. Mendes

TL;DR

The paper proves that for any closed immersed submanifold M in a simply-connected space form, the extrinsic diameter satisfies $\mathrm{diam}_{\mathrm{ext}}(M) \ge 2\, r_{\mathrm{foc}}(M)$, with equality characterizing highly symmetric embeddings. Equality occurs exactly for round spheres with totally umbilical embeddings, Veronese embeddings of real/complex/quaternionic/octonionic projective spaces, or compositions of sphere coverings with Veronese embeddings; the proof combines Bow Lemma-based geodesic comparison with Sakamoto's planar-geodesic classification. The results connect classical differential geometry with symmetric-space embeddings and have ties to recent work of Gromov and Petrunin. The work also outlines open problems on extremal immersions and variant notions such as circumradius, offering a clear roadmap for identifying extremal submanifolds in Euclidean and non-Euclidean settings.

Abstract

In this note, we give a characterization of immersed submanifolds of simply-connected space forms for which the quotient of the extrinsic diameter by the focal radius achieves the minimum possible value of $2$. They are essentially round spheres, or the ``Veronese'' embeddings of projective spaces. The proof combines the classification of submanifolds with planar geodesics due to K. Sakamoto with a version of A. Schur's Bow Lemma for space curves. Open problems and the relation to recent work by M. Gromov and A. Petrunin are discussed.

Diameter and focal radius of submanifolds

TL;DR

The paper proves that for any closed immersed submanifold M in a simply-connected space form, the extrinsic diameter satisfies , with equality characterizing highly symmetric embeddings. Equality occurs exactly for round spheres with totally umbilical embeddings, Veronese embeddings of real/complex/quaternionic/octonionic projective spaces, or compositions of sphere coverings with Veronese embeddings; the proof combines Bow Lemma-based geodesic comparison with Sakamoto's planar-geodesic classification. The results connect classical differential geometry with symmetric-space embeddings and have ties to recent work of Gromov and Petrunin. The work also outlines open problems on extremal immersions and variant notions such as circumradius, offering a clear roadmap for identifying extremal submanifolds in Euclidean and non-Euclidean settings.

Abstract

In this note, we give a characterization of immersed submanifolds of simply-connected space forms for which the quotient of the extrinsic diameter by the focal radius achieves the minimum possible value of . They are essentially round spheres, or the ``Veronese'' embeddings of projective spaces. The proof combines the classification of submanifolds with planar geodesics due to K. Sakamoto with a version of A. Schur's Bow Lemma for space curves. Open problems and the relation to recent work by M. Gromov and A. Petrunin are discussed.

Paper Structure

This paper contains 7 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bow Lemma
  4. Focal radius and normal curvatures
  5. Veronese embeddings of projective spaces
  6. Proof of Theorem \ref{['MT']}
  7. Remarks and open questions

Key Result

Theorem 1

Let $M$ be a closed, smooth, connected, immersed submanifold of $M^n(\kappa)$. Then the extrinsic diameter of $M$ is at least $2$ times its focal radius. Moreover, equality holds if and only if: $M$ is a round sphere, and the immersion is a totally umbilical embedding; or $M$ is a (real, complex, qu

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 3
  • Example 6