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<p>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.</p>

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Diameter and focal radius of submanifolds

  • 期刊论文

  • 发表日期：

    2025年7月2日

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  • Ricardo A. E. Mendes

摘要

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.