2025/04/17 by Mendes, Ricardo A. E.
#53A07 (Primary) 53C35 #53C40 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.13270
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.