Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Semiparallel and Weyl-semiparallel Hypersurfaces of Kaehler Manifolds

  • Received : 2008.02.11
  • Accepted : 2008.04.17
  • Published : 2009.03.31
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Abstract

We study on semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds. We prove that a (2n + 1)-dimensional Sasakian hypersurface M of a (2n+2)-dimensional Kaehler manifold $\widetilde{M}^{2n+2}$ is semiparallel if and only if it is totally umbilical with unit mean curvature, if dimM = 3 and $\widetilde{M}^4$ is a Calabi-Yau manifold, then $\widetilde{M}$ is flat at each point of M. We also prove that such a hypersurface M is Weyl-semiparallel if and only if it is either an ${\eta}$-Einstein manifold or semiparallel. We also investigate the extended classes of semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds.

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References

  1. A. C. Asperti, G. A. Lobos and F. Mercuri, Pseudo-parallel immersions in space forms, Math. Contemp., 17(1999), 59-70.
  2. A. C. Asperti, G. A. Lobos and F. Mercuri, Pseudo-parallel immersions of a space forms, Adv. Geom., 2(2002) 17(1999), 57-71.
  3. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, 2002.
  4. B. Y. Chen, Geometry of submanifolds and its applications, Science University of Tokyo, Tokyo, 1981.
  5. J. Deprez, Semiparallel Surfaces in Euclidean space, J. Geom., 25(1985), 192-200. https://doi.org/10.1007/BF01220480
  6. J. Deprez, Semiparallel Hypersurfaces, Rend. Sem. Mat. Univers. Politecn. Torino, 44(1986), 303-316.
  7. R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44(1)(1992), 1-34.
  8. R. Deszcz, L. Verstraelen and S. Yaprak, Pseudosymmetric hypersurfaces in 4-dimensional space of constant curvature, Bull. Ins. Math. Acad. Sinica, 22(1994), 167-179.
  9. F. Dillen, Semi-parallel hypersurfaces of a real space form, Israel J. Math., 75(1991), 193-202. https://doi.org/10.1007/BF02776024
  10. U. Lumiste, Semi-symmetric submanifolds as the second order envelope of symmetric submanifolds, Proc. Estonian Acad. Sci. Phys. Math., 39(1990), 1-8.
  11. C. Ozgur, Submanifolds satisfying some curvature conditions imposed on the Weyl tensor, Bull. Austral. Math. Soc., 67(1)(2003), 95-101. https://doi.org/10.1017/S0004972700033554
  12. R. Sharma, Contact hypersurfaces of Kaehler manifolds, J. Geom., 78(1-2)(2003), 156-167. https://doi.org/10.1007/s00022-003-1625-y
  13. K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984.