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Transactions of the American Mathematical Society

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Helical minimal immersions of compact Riemannian manifolds into a unit sphere


Author: Kunio Sakamoto
Journal: Trans. Amer. Math. Soc. 288 (1985), 765-790
MSC: Primary 53C42; Secondary 53C40
DOI: https://doi.org/10.1090/S0002-9947-1985-0776403-9
MathSciNet review: 776403
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Abstract: An isometric immersion of a Riemannian manifold $ M$ into a Riemannian manifold $ \overline M $ is called helical if the image of each geodesic has constant curvatures which are independent of the choice of the particular geodesic. Suppose $ M$ is a compact Riemannian manifold which admits a minimal helical immersion of order $ 4$ into the unit sphere. If the Weinstein integer of $ M$ equals that of one of the projective spaces, then $ M$ is isometric to that projective space with its canonical metric.


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  • [1] A. Allamigeon, Propriétés globales des espaces de Riemann harmoniques, Ann. Inst. Fourier (Grenoble) 15 (1965), 91-132. MR 0198391 (33:6549)
  • [2] A. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math Grenzgeb., Band 93, Springer-Verlag, Berlin and New York, 1978. MR 496885 (80c:53044)
  • [3] R. Escobales, Riemannian submersions with totally geodesic fibers, J. Differential Geom. 10 (1975), 253-276. MR 0370423 (51:6650)
  • [4] W. Klingenberg, Riemannian geometry, Gruyter Stud. in Math., vol. 1; de Gruyter, Berlin, 1982. MR 666697 (84j:53001)
  • [5] J. A. Little, Manifolds with planar geodesics, J. Differential Geom. 11 (1976), 265-285. MR 0417992 (54:6037)
  • [6] H. Nakagawa, On a certain minimal immersion of a Riemannian manifold into a sphere, Kōdai Math. J. 3 (1980), 321-340. MR 604477 (82k:53077)
  • [7] K. Sakamoto, Planar geodesic immersions, Tôhoku Math. J. 29 (1977), 25-56. MR 0470913 (57:10657)
  • [8] -, Helical immersions into a unit sphere, Math. Ann. 26 (1982), 63-80. MR 675208 (84f:53054)
  • [9] -, On a minimal helical immersions into a unit sphere, Geometry of Geodesics and Related Topics, Advanced Studies in Pure Math. 3 (1984), 193-211. MR 758654 (86a:53064)
  • [10] K. Tsukada, Isotropic minimal immersions of spheres into spheres, J. Math. Soc. Japan 35 (1983), 355-379. MR 692333 (84g:53090)
  • [11] -, Helical geodesic immersions of compact rank one symmetric spaces into spheres, Tokyo J. Math. 6 (1983), 267-285. MR 732082 (85h:53044)
  • [12] N. R. Wallach, Symmetric spaces, (W. M. Boothby and G. L. Weiss, Eds.), Marcel Dekker, New York, 1972. MR 0407774 (53:11545)
  • [13] A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geom. 9 (1974), 513-517. MR 0390968 (52:11791)
  • [14] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, London and New York, 1965. MR 1424469 (97k:01072)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0776403-9
Keywords: Helical immersions, strongly harmonic manifolds, geodesics, Blaschke structure, cut loci, second fundamental forms
Article copyright: © Copyright 1985 American Mathematical Society

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