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Bull. Korean Math. Soc. 2010; 47(6): 1285-1296

Printed November 1, 2010

https://doi.org/10.4134/BKMS.2010.47.6.1285

Copyright © The Korean Mathematical Society.

Characterizations of some isometric immersions in terms of certain Frenet curves

Jin Ho Choi, Young Ho Kim, and Hiromasa Tanabe

Kyungpook National University, Kyungpook National University, Yonago Higashi Highschool

Abstract

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We give criterions for a submanifold to be an extrinsic sphere and to be a totally geodesic submanifold by observing some Frenet curves of order 2 on the submanifold. We also characterize constant isotropic immersions into arbitrary Riemannian manifolds in terms of Frenet curves of proper order 2 on submanifolds. As an application we obtain a characterization of Veronese embeddings of complex projective spaces into complex projective spaces.

Keywords: Frenet curves of proper order 2, extrinsic spheres, totally geodesic submanifolds, constant isotropic immersions, Veronese embeddings

MSC numbers: Primary 53B25; Secondary 53C40

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