Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on isometric immersions of the Cayley projective plane and Frenet curves

Hiromasa Tanabe

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We give a characterization of the first standard imbedding of the Cayley projective plane into a real space form in terms of a particular class of Frenet curves of order 2.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 1 (2005), 12-16.

First available in Project Euclid: 18 May 2005

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Zentralblatt MATH identifier

Primary: 53B25: Local submanifolds [See also 53C40] 53C40: Global submanifolds [See also 53B25]

Cayley projective plane parallel isometric immersions Frenet curves


Tanabe, Hiromasa. A note on isometric immersions of the Cayley projective plane and Frenet curves. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 1, 12--16. doi:10.3792/pjaa.81.12.

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  • T. Adachi, S. Maeda and K. Ogiue, Extrinsic shape of circles and the standard imbedding of a Cayley projective plane, Hokkaido Math. J. 26 (1997), no. 2, 411–419.
  • D. Ferus, Immersions with parallel second fundamental form, Math. Z. 140 (1974), 87–93.
  • M. Kôzaki and S. Maeda, A characterization of extrinsic spheres in a Riemannian manifold, Tsukuba J. Math. 26 (2002), no. 2, 291–297.
  • S. Maeda and H. Tanabe, Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves. (Preprint).
  • K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann. 210 (1974), 163–170.
  • B. O'Neill, Isotropic and Kähler immersions, Canad. J. Math. 17 (1965), 907–915.
  • K. Sakamoto, Planar geodesic immersions, Tôhoku Math. J. (2) 29 (1977), no. 1, 25–56.
  • M. Takeuchi, Parallel submanifolds of space forms, in Manifolds and Lie groups (Notre Dame, Ind., 1980), 429–447, Progr. Math., 14, Birkhäuser, Boston, Mass, 1981.
  • H. Tanabe, Characterization of totally geodesic submanifolds in terms of Frenet curves. (Preprint).
  • H. Tanabe, Quaternionic Frenet curves and totally geodesic immersions. (Preprint).