Diameter and focal radius of submanifolds

IF 0.7 3区数学 Q3 MATHEMATICS

Annals of Global Analysis and Geometry Pub Date : 2025-07-02 DOI:10.1007/s10455-025-10010-7

Ricardo A. E. Mendes

引用次数: 0

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.

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子流形的直径和焦半径

在本文中，我们给出了单连通空间形式的浸没子流形的一个表征，其外在直径与焦点半径之商达到最小可能值2。它们本质上是圆形的球体，或者是投影空间的“维罗纳式”嵌入。该证明结合了K. Sakamoto的平面测地线子流形的分类和a . Schur的空间曲线Bow引理的一个版本。讨论了开放问题及其与M. Gromov和A. Petrunin最近工作的关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Global Analysis and Geometry 数学-数学

CiteScore

1.20

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： This journal examines global problems of geometry and analysis as well as the interactions between these fields and their application to problems of theoretical physics. It contributes to an enlargement of the international exchange of research results in the field. The areas covered in Annals of Global Analysis and Geometry include: global analysis, differential geometry, complex manifolds and related results from complex analysis and algebraic geometry, Lie groups, Lie transformation groups and harmonic analysis, variational calculus, applications of differential geometry and global analysis to problems of theoretical physics.