# Article

Full entry | PDF   (0.2 MB)
Keywords:
semiparallel submanifolds; flat normal connection; semisymmetric Riemannian manifolds; manifolds of conullity two
Summary:
By means of the bundle of orthonormal frames adapted to the submanifold as in the title an explicit exposition is given for these submanifolds. Two theorems give a full description of the semisymmetric Riemannian manifolds which can be immersed as such submanifolds. A conjecture is verified for this case that among manifolds of conullity two only the planar type (in the sense of Kowalski) is possible.
References:
[B] Boeckx E.: Foliated semi-symmetric spaces. Doctoral Thesis, Katholic Univ. Leuven, 1995. Zbl 0846.53031
[BKV] Boeckx E., Kowalski O., Vanhecke L.: Riemannian Manifolds of Conullity Two. World Scientific, London, 1996. MR 1462887 | Zbl 0904.53006
[BO] Bishop R.L., O'Neill B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 (1969), 1-49. MR 0251664 | Zbl 0191.52002
[D] Deprez J.: Semi-parallel surfaces in Euclidean space. J. Geom. 25 (1985), 192-200. MR 0821680 | Zbl 0582.53042
[DN] Dillen F., Nölker S.: Semi-parallelity, multi-rotation surfaces and the helix-property. J. Reine Angew. Math. 435 (1993), 33-63. MR 1203910
[F1] Ferus D.: Immersionen mit paralleler zweiter Fundamentalform. Manuscripta Math. 12 (1974), 153-162. MR 0339015 | Zbl 0274.53058
[F2] Ferus D.: Immersions with parallel second fundamental form. Math. Z. 140 (1974), 87-93. MR 0370437 | Zbl 0279.53048
[F3] Ferus D.: Symmetric submanifolds of Euclidean space. Math. Ann. 247 (1980), 81-93. MR 0565140 | Zbl 0446.53041
[KN] Kobayashi S., Nomizu K.: Foundations of Differential Geometry. vol. II, Interscience Publ., New York et al., 1969. MR 0238225 | Zbl 0526.53001
[K] Kowalski O.: An explicit classification of $3$-dimensional Riemannian spaces satisfying $R(X,Y)\cdot R=0$. Czech. Math. J. 46 (121) (1996), 427-474; preprint 1991, presented at the Geometry Meeting in Oberwolfach, October 1991. MR 1408298 | Zbl 0879.53014
[L1] Lumiste Ü.: Decomposition and classification theorems for semi-symmetric immersions. Proc. Estonian Acad. Sci. Phys. Math. 36 (1987), 414-417. MR 0925980 | Zbl 0646.53003
[L2] Lumiste Ü.: Semi-symmetric submanifold as the second order envelope of symmetric submanifolds. Proc. Estonian Acad. Sci. Phys. Math. 39 (1990), 1-8. MR 1059755 | Zbl 0704.53017
[L3] Lumiste Ü.: Irreducible normally flat semi-symmetric submanifolds, I and II. Izv. Vyssh. Uchebn. Zaved. Mat., no. 8 (1990), 45-53 and no.9 (1990), 32-40 (in Russian); English transl.: Soviet Math. (Iz. VUZ) 34(8) (1990), 50-59 and 34(9) (1990), 35-47. MR 1087938
[L4] Lumiste Ü.: Semi-symmetric submanifolds. Problems in Geometry, Vol. 23, Acad. Sci. USSR, Allunion Inst. Scient. Techn. Inform., Moscow, 1991, pp.3-28 (in Russian); English transl.: J. Math. Sci. New York 70(2) (1994), 1609-1623. Zbl 0763.53023
[L5] Lumiste Ü.: Semi-symmetric envelopes of some symmetric cylindrical submanifolds. Proc. Estonian Acad. Sci. Phys. Math. 40 (1991), 245-257. MR 1163442 | Zbl 0802.53014
[L6] Lumiste Ü.: Submanifolds with parallel fundamental form. in: F.J.E. Dillen and L.C.A. Verstraelen (Eds.): Handbook of Differential Geometry, vol. I, Elsevier Sc., Amsterdam, 2000, pp.779-864 (Chapter 7). MR 1736858 | Zbl 0964.53002
[L7] Lumiste Ü.: Semiparallel isometric immersions of $3$-dimensional semisymmetric Riemannian manifolds. Czech. Math. J., to appear. MR 2000064 | Zbl 1080.53036
[L8] Lumiste Ü.: Semiparallel submanifolds with plane generators of codimension two in a Euclidean space. Proc. Estonian Acad. Sci. Phys. Math. 50 (2001), 115-123. MR 1864980 | Zbl 1004.53004
[L9] Lumiste Ü.: University of Tartu and geometry of the 19-th century. in: Teaduse Ajaloo Lehekülgi Eestist, Vol. 2, Valgus, Tallinn, 1976, pp.36-68, (in Estonian, Summaries in Russian and German); English transl. in: V. Abramov, M. Rahula, and K. Riives (Eds.): Ülo Lumiste: {Mathematician}, Estonian Math. Soc., Tartu, 1999, pp.68-102. MR 1723392
[MC] McCleary J.: Geometry from a Differentiable Viewpoint. Cambridge Univ. Press, Cambridge, 1994. MR 1314819 | Zbl 0828.53001
[N] Nut J.J. (Nuut, J.): Lobachevskian Geometry in Analytic Treatment. Publ. Acad. Sci. USSR, Moscow, 1961 (in Russian).
[Ph] Phillips E.: Karl M. Peterson: The earliest derivation of the Mainardi-Codazzi equations and the fundamental theorem of surface theory. Historia Math. 6 (1979), 137-163. MR 0530623 | Zbl 0405.01024
[S1] Sinjukov N.S.: Equidistant Riemannian manifolds. Annual Reports of the Odessa University, 1957, pp.133-135 (in Russian).
[S2] Sinjukov N.S.: Geodesic Maps of Riemannian Spaces. Publishing House Nauka", Moscow, 1979 (in Russian). MR 0552022
[St] Sternberg S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs, NJ 1964; 2nd ed. Chelsea Publishing, New York, 1983. MR 0891190 | Zbl 0518.53001
[Str] Strübing W.: Symmetric submanifolds of Riemannian manifolds. Math. Ann. 245 (1979), 37-44. MR 0552577
[Sz] Szabó Z.I.: Structure theorems on Riemannian spaces satisfying $R(X.Y)\cdot R=0$, I. The local version. J. Differential Geom. 17 (1982), 531-582. MR 0683165
[T] Takeuchi M.: Parallel submanifolds of space forms. Manifolds and Lie Groups. Papers in honour of Y. Matsushima, Birkhäuser, Basel, 1981, pp.429-447. MR 0642871 | Zbl 0481.53047
[W] Wolf J.A.: Spaces of Constant Curvature. Univ. of California, Berkeley, 1972; 4th ed. Publish or Perish, Berkeley, 1977. MR 0343213 | Zbl 0556.53033

Partner of