Abstract
This paper consists of two parts. One is to construct a class of helical geodesic equivariant immersions of orderd(⩾3), which are neither Kaehler nor totally real immersions, into complex projective spaces. The other is to show the basic results about a helix in complex space forms.
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References
Bando, S. and Ohnita, Y., ‘Minimal 2-spheres with Constant Curvature inP n (ℂ)’,J. Math. Soc. Japan 39 (1987), 477–488.
Besse, A., ‘Manifolds all of whose Geodesics are Closed’,Ergebnisse der Mathematik, Bd. 93, Springer Berlin, Heiderberg, New York, 1978.
Calabi, E., ‘Isometric Imbedding of Complex Manifolds’,Ann. Math. 58 (1953), 1–23.
Ferus, D. and Schirrmacher, S., ‘Submanifolds in Euclidean Space with Simple Geodesics’,Math. Ann. 260 (1982), 57–62.
Hong, Y., ‘Helical Submanifolds in Euclidean Spaces’,Indiana Univ. Math. J. 35 (1986), 29–43.
Maebashi, T. and Maeda, S., ‘Constant Mean Curvature Submanifolds of Higher Codimensions’,Kodai Math. J. 9 (1986), 175–178.
Maeda, S., ‘Imbedding of a Complex Projective Space Similar to Segre Imbedding’,Archiv. Math. 37 (1981), 556–560.
Maeda, S. and Sato, N., ‘On Submanifolds all of whose Geodesics are Circles in a Complex Space Form’,Kodai Math. J. 6 (1983), 157–166.
Naitoh, H., ‘Isotropic Submanifolds with Parallel Second Fundamental Form inP m(c)’,Osaka J. Math. 18 (1981), 427–464.
Nakagawa, H., and Ogiue, K., ‘Complex Space Forms Immersed in Complex Space Forms’,Trans. Amer. Math. Soc. 219 (1976), 289–297.
Ogiue, K., ‘Differential Geometry of Kaehler Submanifolds’,Advances Math. 13 (1974), 73–114.
Ohnita, Y., ‘Homogeneous Harmonic Maps into Complex Projective Spaces’, (Preprint), Max-Planck-Institut für Math., Bonn, 1988.
Pak, J. S. and Sakamoto, K., ‘Submanifolds with Properd-Planar Geodesics Immersed in Complex Projective Spaces’,Tohoku Math. J. 38 (1986), 297–311.
Sakamoto, K., ‘Helical Immersions into a Unit Sphere’,Math. Ann. 261 (1982), 63–80.
Sakamoto, K., ‘Helical Minimal Immersions of Compact Riemannian Manifolds into a Unit Sphere’,Trans. Amer. Math. Soc. 288 (1985), 765–790.
Tsukada, K., ‘Helical Geodesic Immersions of Compact Rank One Symmetric Spaces into Spheres’,Tokyo J. Math. 6 (1983), 267–285.
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This research was partially supported by Grants-in-Aid for Scientific Research (Nos 62740070 and 62740054), Ministry of Education, Science and Culture and by the Max-Planck-Institut für Mathematik in Bonn.
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Maeda, S., Ohnita, Y. Helical geodesic immersions into complex space forms. Geom Dedicata 30, 93–114 (1989). https://doi.org/10.1007/BF02424315
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DOI: https://doi.org/10.1007/BF02424315