Diameter and focal radius of submanifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909584026763264 |
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| author | Mendes, Ricardo A. E. |
| author_facet | Mendes, Ricardo A. E. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_13270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diameter and focal radius of submanifolds Mendes, Ricardo A. E. Differential Geometry 53A07 (Primary) 53C35, 53C40 (Secondary) 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. |
| title | Diameter and focal radius of submanifolds |
| topic | Differential Geometry 53A07 (Primary) 53C35, 53C40 (Secondary) |
| url | https://arxiv.org/abs/2504.13270 |