Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CHARACTERIZATIONS OF SOME ISOMETRIC IMMERSIONS IN TERMS OF CERTAIN FRENET CURVES

  • Choi, Jin-Ho (DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCES KYUNGPOOK NATIONAL UNIVERSITY) ;
  • Kim, Young-Ho (DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCES KYUNGPOOK NATIONAL UNIVERSITY) ;
  • Tanabe, Hiromasa (YONAGO HIGASHI HIGHSCHOOL)
  • Received : 2009.05.07
  • Published : 2010.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

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

Acknowledgement

Supported by : KOSEF

References

  1. E. Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1-23. https://doi.org/10.2307/1969817
  2. U.-H. Kim and S. Maeda, Notes on extrinsic spheres, Bull. Korean Math. Soc. 35 (1998), no. 3, 433-439.
  3. S. Maeda, A characterization of constant isotropic immersions by circles, Arch. Math. (Basel) 81 (2003), no. 1, 90-95. https://doi.org/10.1007/s00013-003-4677-1
  4. H. Nakagawa and K. Ogiue, Complex space forms immersed in complex space forms, Trans. Amer. Math. Soc. 219 (1976), 289-297. https://doi.org/10.1090/S0002-9947-1976-0407756-3
  5. K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann. 210 (1974), 163-170. https://doi.org/10.1007/BF01360038
  6. K. Ogiue, Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974), 73-114. https://doi.org/10.1016/0001-8708(74)90066-8
  7. B. O'Neill, Isotropic and Kahler immersions, Canad. J. Math. 17 (1965), 907-915. https://doi.org/10.4153/CJM-1965-086-7
  8. H. Tanabe, Characterization of totally geodesic submanifolds in terms of Frenet curves, Sci. Math. Jpn. 63 (2006), no. 1, 83-88.