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Semisymmetric submanifolds

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This article surveys new work on semisymmetric and k-parallel submanifolds Mm in En and Mn(c).

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References

  1. V. R. Kaigorodov, “Semisymmetric Lorentz spaces with complete holonomy groups,” Gravit. Teor. Otnosit.,14–15, 113–120 (1978).

    Google Scholar 

  2. V. R. Kaigorodov, “The curvature of space-time,” Itogi Nauki Tekh., Probl. Geom.,14, 177–204 (1983).

    Google Scholar 

  3. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience, New York (1963).

    Google Scholar 

  4. P. I. Kovalev, “Lie triple systems and spaces of affine connection,” Mat. Zametki,14, No. 1, 107–112 (1973).

    Google Scholar 

  5. Yu. G. Lumiste, “Irreducible submanifolds of small dimension with parallel third fundamental form,” Uch. Zap. Tartus. Univ., No. 734, 50–62 (1986).

    Google Scholar 

  6. Yu. G. Lumiste, “Submanifolds with flat van der Waerden-Bortolotti connection and parallemess of third fundamental forms,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 1, 18–27 (1987).

    Google Scholar 

  7. Yu. G. Lumiste, “Reducibillity of submanifolds with parallel third fundamental form,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 32–41 (1987).

    Google Scholar 

  8. Yu. G. Lumiste, “Irreducible normally flat semisymmetric submanifolds. I,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 45–53 (1990).

    Google Scholar 

  9. Yu. G. Lumiste, “Irreducible normaly flat semisymmetric submanifolds. II,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 31–40 (1990).

    Google Scholar 

  10. Yu. G. Lumiste, “Classification of submanifolds with small dimensionality and parallel higher-order fundamental forms,” in: Abstracts of the All-Union Congress on Differential Geometry in Honor of the 80-th Birthday of N. V. Efimov, Sept. 29–Oct. 5, 1990, Rostov-on-Don (1990), p. 58.

  11. Yu. G. Lumiste and V. A. Mirzoyan, “Submanifolds with parallel third fundamental form,” Uch. Zap. Tartus. Univ., No. 665, 42–54 (1984).

    Google Scholar 

  12. I. Maazikas, “Congruences of 2-planes with totally geodesic Grassman images,” Uch. Zap. Tartus. Univ., No. 355, 76–85 (1975).

    Google Scholar 

  13. V. A. Mirzoyan, “Submanifolds with parallel second fundamental form in spaces of constant curvature,” Uch. Zap. Tartus. Univ., No. 464, 59–74 (1978).

    Google Scholar 

  14. V. A. Mirzoyan, “Submanifolds with higher-order parallel fundamental forms,” Dokl. Akad. Nauk ArmSSR,66, No. 2, 71–75 (1978).

    Google Scholar 

  15. V. A. Mirzoyan, “Submanifolds with higher-order parallel fundamental forms,” VINITI Report No. 2074-78, Dep. in VINITI June 20, 1978.

  16. V. A. Mirzoyan, “Submanifolds with parallel third fundamental form,” in: Abstracts of the Seventh All-Union Conference on Contemporary Problems in Geometry, Minsk, October 1979 [in Russian], Minsk (1979).

  17. V. A. Mirzoyan, “Submanifolds with commuting normal vector fields,” Dokl. Akad. Nauk ArmSSR,72, No. 1, 14–17 (1981).

    Google Scholar 

  18. V. A. Mirzoyan, “Submanifolds with commuting normal vector fields,” Uch. Zap. Erevan. Univ.,148, No. 3, 9–16 (1982).

    Google Scholar 

  19. V. A. Mirzoyan, “Canonical imbeddings of R-spaces,” Mat. Zametki,33, No. 2, 255–260 (1983).

    Google Scholar 

  20. V. A. Mirzoyan, “Submanifolds with commuting normal vector fields,” Itogi Nauki Tekh., Probl. Geom.,14, 73–100 (1983).

    Google Scholar 

  21. V. A. Mirzoyan, “Intrinsically symmetric submanifolds,” in: Abstracts, Conference on Problems of Theoretical and Applied Mathematics, Tartu, Sept. 21–22, 1990 [in Russian], Tartu (1990).

  22. V. A. Mirzoyan, “Ric-semisymmetric submanifolds,” Itogi Nauki Tekh., Probl. Geom.,23, 29–66 (1991).

    Google Scholar 

  23. K. Riives, “The submanifold V3 with parallel third fundamental form in Euclidean space E5,” Uch. Zap. Tartus. Univ., No. 734, 102–110 (1986).

    Google Scholar 

  24. K. Riives, “Two classes of semisymmetric submanifolds,” Uch. Zap. Tartus. Univ., No. 803, 95–102 (1988).

    Google Scholar 

  25. N. S. Sinyukov, “Geodesic mapping of Riemann spaces,” in: Proc. Third All-Union Mathematical Congress [in Russian], Vol. 1, Moscow (1956), pp. 167–168.

    Google Scholar 

  26. N. S. Sinyukov, Geodesic Mapping of Riemann Spaces [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  27. S. Akiba, “Submanifolds with flat normal connection and parallel second fundamental tensor,” Sci. Repts. Yokohama Nat. Univ., Sect. 1, No. 23, 7–14 (1976).

    Google Scholar 

  28. E. Backes, “Geometric applications of Euclidean Jordan triple systems,” Manuscr. Math.,42, Nos. 2–3, 265–272 (1983).

    Google Scholar 

  29. E. Backes and H. Reckziegel, “On symmetric submanifolds of spaces of constant curvature,” Math. Ann.,263, No. 4, 419–433 (1983).

    Google Scholar 

  30. B.-Y. Chen and S. Yamaguchi, “Submanifolds with totally geodesic Gauss image,” Geom. Dedic,15, No. 3, 313–322 (1984).

    Google Scholar 

  31. J. Deprez, “Semiparallel surfaces in Euclidean space,” J. Geom.,25, No. 2, 192–200 (1985).

    Google Scholar 

  32. J. Deprez, “Semiparallel hypersurfaces,” Rend. Semin. Mat. Univ. Politecn. Torino,44, No. 2, 303–316 (1987).

    Google Scholar 

  33. J. Deprez, “Semiparallel immersions,” in: Geom. and Topol. of Submanifolds: Proc. Meeting at Lurniny, Marseille, May 18–23, 1987, Singapore (1989), pp. 73–88.

  34. F. Dillen, “The classification of hypersurfaces of a Euclidean space with parallel higher order fundamental form,” Math. Z.,203, 635–643 (1990).

    Google Scholar 

  35. F. Dillen, “Sur les hypersurfaces parallèles d'ordre supérieur,” Compt. Rend. Acad. Sci., Ser. 1,311, 185–187 (1990).

    Google Scholar 

  36. D. Ferus, “Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele,” Manuscr. Math.,12, No. 2, 153–162 (1974).

    Google Scholar 

  37. D. Ferus, “Produkt-Zerlegung von Immersionen mit paralleler zweiter Fundamentalform,” Math. Ann.,211, No. 1, 1–5 (1974).

    Google Scholar 

  38. D. Ferus, “Immersions with parallel second fundamental form,” Math. Z.,140, No. 1, 87–92 (1974).

    Google Scholar 

  39. D. Ferus, “Symmetric submanifolds of Euclidean space,” Math. Ann.,247, No. 1, 81–93 (1980).

    Google Scholar 

  40. S. Fujimura, “On Riemannian manifolds satisfying the condition R(X,Y)·R = 0,” J. Fac. Sci., Hokkaido Univ., Ser. 1,22, Nos. 1–2, 1–8 (1972).

    Google Scholar 

  41. C.-S. Houh, “Pseudo-umbilical surfaces with parallel second fundamental form,” Tensor,26, 262–266 (1972).

    Google Scholar 

  42. T. Itoh, “On Veronese manifolds,” J. Math. Soc. Jap,27, No. 3, 497–506 (1975).

    Google Scholar 

  43. E. Kelly, “Tight equivariant imbeddings of symmetric spaces,” J. Differ. Geom.,7, Nos. 3–4, 535–548 (1972).

    Google Scholar 

  44. U.-H. Ki and J. S. Pak, “Submanifolds of a Euclidean m-space with totally umbilical Gauss image,” Tensor,44, No. 3, 233–239 (1987).

    Google Scholar 

  45. Y. Kitagawa and Y. Ohnita, “On the mean curvature of R-spaces,” Math. Ann.,262, No. 2, 239–243 (1983).

    Google Scholar 

  46. S. Kobayashi, “Isometric imbeddings of compact symmetric spaces,” Tohoku Math. J.,20, 21–25 (1968).

    Google Scholar 

  47. M. Kon, “Totally real minimal submanifolds with parallel second fundamental form,” Atti Accad. Naz. Linzei Rend., Sci., Fis., Mat., Natur.,57, Nos. 3–4, 187–189 (1974).

    Google Scholar 

  48. U. Lumiste, “Decomposition and classification theorems for semsymmetric immersions,” Izv. Akad. Nauk. Est.SSR, Fiz., Mat.,36, No. 4, 414–417 (1987).

    Google Scholar 

  49. U. Lumiste, “Decomposition of semisymmetric submanifolds,” Uch. Zap. Tart. Univ., No. 803, 69–78 (1988).

    Google Scholar 

  50. U. Lumiste, “Classification of two-codimensional semisymmetric submanifolds,” Uch. Zap. Tart. Univ., No. 803, 79–94 (1988).

    Google Scholar 

  51. U. Lumiste, “Normally flat semisymmetric submanifolds,” in: Differ. Geom. and Its Appl.: Proc. Conf. Dubrovnik June 26–July 3, 1988, Novi Sad (1989), pp. 159–171.

  52. U. Lumiste, “Normally flat submanifolds with parallel third fundamental form,” Izv. Aka. Nauk Est.SSR, Fiz., Mat.,38, No. 2, 129–138 (1989).

    Google Scholar 

  53. U. Lumiste, “Semisymmetric submanifolds with maximal first normal space,” Izv. Akad. Nauk. Est.SSR, Fiz., Mat.,38, No. 4, 453–457 (1989).

    Google Scholar 

  54. U. Lumiste, “Semisymmetric submanifold as the second-order envelope of symmetric submanifolds,” Izv. Akad. Nauk. Est.SSR, Fiz. Mat.,39, No. 1, 1–8 (1990).

    Google Scholar 

  55. U. Lumiste, “Classification of three-dimensional semisymmetric submanifolds in Euclidean spaces,” Uch. Zap. Tart. Univ., No. 899, 29–44 (1990).

    Google Scholar 

  56. U. Lumiste, “Three-dimensional submanifolds with parallel third fundamental form in Euclidean spaces,” Uch. Zap. Tart. Univ., No. 899, 45–56 (1990).

    Google Scholar 

  57. U. Lumiste, “Submanifolds with parallel higher-order fundamental form,” in: Probl. of Pure and Appl. Math.: Abstr. of Conf. Tartu Sept. 21–22, 1990, Tartu (1990), pp. 24–26.

  58. U. Lumiste and K. Riives, “Three-dimensional semisymmetric submanifolds with axial, planar, or spatial points in Euclidean spaces,” Uch. Zap. Tart. Univ., No. 899, 13–28 (1990).

    Google Scholar 

  59. S. Maeda, “Isotropic immersions with parallel second fundamental form,” Can. Math. Bull.,26, No. 3, 291–296 (1983).

    Google Scholar 

  60. S. Maeda, “Isotropic immersions with parallel second fundamental form. II,” Yokahama Math. J.,3, Nos. 1–2, 131–138 (1983).

    Google Scholar 

  61. M. A. Magid, “Isometric immersions of Lorentz space with parallel second fundamental form,” Tsukuba J. Math.,8, No. 1, 31–54 (1984).

    Google Scholar 

  62. J. Mikesh, “Geodesic mappings of special Riemannian spaces,” in: Top. Differ. Geom.: Colloq. Debrecn. Aug. 26–Sept. 1, 1984, Vol. 2, Amsterdam (1988), pp. 793–813.

  63. H. Naitoh, “Isotropic submanifolds with parallel second fundamental forms in symmetric spaces,” Osaka J. Math.,17, 95–100 (1980).

    Google Scholar 

  64. H. Nakagawa and R. Takagi, “On locally symmetric Kaehler submanifolds in a complex projective space,” J. Math. Soc. Jap.,28, No. 4, 638–667 (1976).

    Google Scholar 

  65. K. Nomizu, “On hypersurfaces satisfying a certain condition on the curvature tensor,” Tohoku Math. J.,20, No. 1, 46–59 (1968).

    Google Scholar 

  66. K. Nomizu and M. A. Magid, “On affine surfaces whose cubic forms are parallel relative to the affine metric,” Proc. Jap. Acad. A.,65, No. 7, 215–222 (1989).

    Google Scholar 

  67. K. Nomizu and U. Pinkall, “Cayley surfaces in affine differential geometry,” Tohoku Math. J.,41, No. 4, 589–596 (1989).

    Google Scholar 

  68. Y. Ohnita, “The degrees of the standard imbeddings of R-spaces,” Tohoku Math. J.,35, No. 4, 499–502 (1983).

    Google Scholar 

  69. J. S. Pak and J. J. Kim, “Isotropic immersions with totally geodesic Gauss image,” Tensor,43, No. 2, 167–174 (1986).

    Google Scholar 

  70. J. S. Pak and K. Sakamoto, “Constant isotropic submanifolds with 4-planar geodesies,” Trans. Am. Math. Soc.,307, No. 1, 317–333 (1988).

    Google Scholar 

  71. H. Reckziegel, “A class of distinguished isometric immersions with parallel second fundamental form,” Result. Math.,6, No. 1, 56–63 (1983).

    Google Scholar 

  72. K. Riives, “On geometry of the second order envelope of a family of Veronese surfaces,” in: Probl. of Pure and Appl. Math.: Abstr. of Conf. Tartu Sept. 21–22, Tartu (1990), pp. 27–28.

  73. K. Sakamoto, “Submanifolds satisfying the condition K(X,Y)·K=0,” Kodai Math. Semin. Rept.,25, No. 2, 143–152 (1973).

    Google Scholar 

  74. K. Sakamoto, “Constant isotropic surfaces in 5-dimensional space forms,” Geom. Dedic.,29, No. 2, 293–306 (1989).

    Google Scholar 

  75. K. Sekigawa, “On some hypersurfaces satsifying R(X,Y)·R = 0,” Tensor,25, 133–136 (1972).

    Google Scholar 

  76. U. Simon and A. Weinstein, “Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie,” Manuscr. Math.,1, No. 2, 139–146 (1969).

    Google Scholar 

  77. W. Strübing, “Symmetric submanifolds of Riemannian manifolds,” Math. Ann.245, No. 1, 37–44 (1979).

    Google Scholar 

  78. Z. I. Szabo, “Structure theorems on Riemannian spaces satisfying R(X,Y)·R=0. I. The local version,” J. Differ. Geom.,17, 531–532 (1982).

    Google Scholar 

  79. Z. I. Szabo, “Classification and construction of complete hypersurfaces satisfying R(X,Y)·R=0,” Acta Sci. Math.,47, Nos. 3–4, 321–348 (1984).

    Google Scholar 

  80. Z. I. Szabo, “Structure Theorems on Riemannian spaces satisfying R(X,Y)·R=0. II. Global version,” Geom. Dedic,19, 65–108 (1985).

    Google Scholar 

  81. H. Takagi, “An example of Riemannian manifolds satisfying R(X,Y)·R=0 but not ∇R=0,” Tohoku Math. J.,24, No. 1, 105–108 (1972).

    Google Scholar 

  82. M. Takeuchi, “Parallel submanifolds of space forms,” in: Manifolds and Lie Groups: Papers in Honour of Y. Matsushima, Basel (1981), pp. 429–447.

  83. M. Takeuchi and S. Kobayashi, “Minimal imbeddings of R-spaces,” J. Differ. Geom.,2, No. 2, 203–215 (1968).

    Google Scholar 

  84. K. Tsukada, “Parallel Kaehler submanifolds of Hermitian symmetric spaces,” Math. Z.,190, No. 1, 129–150 (1985).

    Google Scholar 

  85. K. Tsukada, “Parallel submanifolds in a quaternion projective space,” Osaka J. Math.,22, 187–241 (1985).

    Google Scholar 

  86. K. Tsukada, “Parallel submanifolds of Cayley plane,” Sci. Repts. Niigata Univ., A., No. 21, 19–32 (1985).

    Google Scholar 

  87. J. Vilms, “Submanifolds of Euclidean space with parallel second fundamental form,” Proc. Am. Math. Soc.,32, No. 1, 263–267 (1972).

    Google Scholar 

  88. R. Walden, “Untermannigfaltigkeiten mit paralleler zweiter Fundamentalform in euklidischen Räumen und Sphären,” Manuscr. Math.,10, No. 1, 91–102 (1973).

    Google Scholar 

  89. K. Yano and S. Ishihara, “Submanifolds with parallel mean curvature vector,” J. Differ. Geom.,6, No. 1, 95–118 (1971).

    Google Scholar 

  90. K. Yano and S. Ishihara, “Submanifolds of codimension 2 or 3 with parallel second fundamental tensor,” J. Korean Math. Soc.,9, 1–11 (1972).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki Seriya Problemy Geometrii, Vol. 23, pp. 3–28, 1991.

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Lumiste, Y.G. Semisymmetric submanifolds. J Math Sci 70, 1609–1623 (1994). https://doi.org/10.1007/BF02110592

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