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On the Riemannian Curvature Invariants of Totally eta-Umbilical Real Hypersurfaces of a Complex Space Form

Year 2021, Volume: 2 Issue: 1, 30 - 41, 29.01.2021

Özlem Deniz Mehmet Gülbahar

Abstract

Some relations involving the Ricci and scalar curvatures of totally $\eta$-umbilical real hypersurfaces of a complex space form are examined. With the help of these relations, some results on totally $\eta$-umbilical real hypersurfaces of a complex space form are given. Furthermore, these results are discussed on totally $\eta$-umbilical real hypersurfaces of the $6$-dimensional complex space form. Some characterizations dealing totally $\eta$-umbilical real hypersurfaces of the $6$-dimensional complex space form are obtained.

Keywords

Curvature Hypersurface Complex space form

References

  • [1] Al-Solamy F.R., Uddin S., An inequality for warped product submanifolds of a locally product Riemannian manifold, Hacettepe Journal of Mathematics and Statistics, 48(2), 351-358, 2019.
  • [2] Arslan K., Ezentas R., Mihai I., Murathan C., Özgur C., B.Y. Chen inequalities for submanifolds in locally conformal almost cosymplectic manifolds, Bulletin Institute of Mathematics, Academia Sinica, 29(3), 231-242, 2001.
  • [3] Cecil T.E., Ryan P.J., Focal sets and real hypersurfaces in complex projective space, Transactions of the American Mathematical Society, 269, 481–499, 1982.
  • [4] Chen B.-Y., Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik, 60(6), 568-578, 1993.
  • [5] Chen B.-Y., Mean curvature and shape operator of isometric immersions in real space forms, Glasgow Mathematical Journal, 38(1), 87-97, 1996.
  • [6] Chen B.-Y., Relations between Ricci curvatureand shape operator for submanifolds with arbitrary codimensions, Glasgow Mathematical Journal, 41(1), 33-41, 1999.
  • [7] Chen B.-Y., Riemannian DNA, inequalities and their applications, Tamkang Journal of Science and Engineering, 3(3), 123-130, 2000.
  • [8] Chen B.-Y., Pseudo-Riemannian Geometry, δ -Invariants and Applications, World Scientific Publishing, 2011.
  • [9] Chen B.-Y., A tour through δ -invariants: From Nash’s embedding theorem to ideal immersions, best way of living and beyond, Publications De L’institut Mathématique, 94(108), 67-80, 2013.
  • [10] Hamada T., Real hypersurfaces of complex space forms in terms of Ricci ‡-tensor, Tokyo Journal of Mathematics, 25(2), 473-483, 2002.
  • [11] Hong S., Matsumoto K., Tripathi M.M., Certain basic inequalities for submanifolds of locally conformal Kaehler space forms, SUT Journal of Mathematics, 41(1), 75-94, 2005.
  • [12] Kılıç E., Tripathi M.M., Gülbahar M., Chen–Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Annales Polonici Mathematici, 116, 37-56, 2016.
  • [13] Kim J.S., Song Y.M., Tripathi M.M., B.-Y. Chen inequalities for submanifolds in generalized complex space forms, Bulletin of the Korean Mathematical Society, 40(3), 411-424, 2003.
  • [14] Kimura M., Maeda S., On real hypersurfaces of a complex projective space, Mathematische Zeitschrift, 202, 299-311, 1989.
  • [15] Kon M., Pseudo-Einstein real hyprersurfaces in complex space forms, Journal of Differential Geometry, 14, 339-354, 1979.
  • [16] Kon M., A characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form, Czechoslovak Mathematical Journal, 58(4), 1279-1287, 2008.
  • [17] Lee J.M., Riemannian Manifolds: An Introduction to Curvature, Springer Science and Business Media, 2006.
  • [18] Lee J., Vîlcu G.E., Inequalities for generalized normalized δ -Casorati curvatures of slant submanifolds in quaternionic space forms, Taiwanese Journal of Mathematics, 19(3), 691-702, 2015.
  • [19] Montiel S., Real hypersurfaces of a complex hyperbolic space, Journal of the Mathematical Society of Japan, 37(3), 515-535, 1985.
  • [20] Şahin B., Chen’s first inequality for Riemannian maps, Annales Polonici Mathematici, 117(3), 249-258, 2016.
  • [21] Tashiro Y., Tachibana S., On Fubinian and C-Fubinian manifolds, Kodai Mathematical Seminar Reports, 15(3), 176-183, 1963.
  • [22] Takagi R., Real hypersurfaces in a complex projective space with constant principal curvatures, Journal of the Mathematical Society of Japan, 27(1), 43-53, 1975.
  • [23] Yano K., Kon M., Structures on Manifolds, World Scientific Publishing, 1984.
  • [24] Zhang P., Zhang L., Inequalities for Casorati curvatures of submanifolds in real space forms, Advances in Geometry, 16(3), 329-335, 2016.

Year 2021, Volume: 2 Issue: 1, 30 - 41, 29.01.2021

Özlem Deniz Mehmet Gülbahar

Abstract

References

  • [1] Al-Solamy F.R., Uddin S., An inequality for warped product submanifolds of a locally product Riemannian manifold, Hacettepe Journal of Mathematics and Statistics, 48(2), 351-358, 2019.
  • [2] Arslan K., Ezentas R., Mihai I., Murathan C., Özgur C., B.Y. Chen inequalities for submanifolds in locally conformal almost cosymplectic manifolds, Bulletin Institute of Mathematics, Academia Sinica, 29(3), 231-242, 2001.
  • [3] Cecil T.E., Ryan P.J., Focal sets and real hypersurfaces in complex projective space, Transactions of the American Mathematical Society, 269, 481–499, 1982.
  • [4] Chen B.-Y., Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik, 60(6), 568-578, 1993.
  • [5] Chen B.-Y., Mean curvature and shape operator of isometric immersions in real space forms, Glasgow Mathematical Journal, 38(1), 87-97, 1996.
  • [6] Chen B.-Y., Relations between Ricci curvatureand shape operator for submanifolds with arbitrary codimensions, Glasgow Mathematical Journal, 41(1), 33-41, 1999.
  • [7] Chen B.-Y., Riemannian DNA, inequalities and their applications, Tamkang Journal of Science and Engineering, 3(3), 123-130, 2000.
  • [8] Chen B.-Y., Pseudo-Riemannian Geometry, δ -Invariants and Applications, World Scientific Publishing, 2011.
  • [9] Chen B.-Y., A tour through δ -invariants: From Nash’s embedding theorem to ideal immersions, best way of living and beyond, Publications De L’institut Mathématique, 94(108), 67-80, 2013.
  • [10] Hamada T., Real hypersurfaces of complex space forms in terms of Ricci ‡-tensor, Tokyo Journal of Mathematics, 25(2), 473-483, 2002.
  • [11] Hong S., Matsumoto K., Tripathi M.M., Certain basic inequalities for submanifolds of locally conformal Kaehler space forms, SUT Journal of Mathematics, 41(1), 75-94, 2005.
  • [12] Kılıç E., Tripathi M.M., Gülbahar M., Chen–Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Annales Polonici Mathematici, 116, 37-56, 2016.
  • [13] Kim J.S., Song Y.M., Tripathi M.M., B.-Y. Chen inequalities for submanifolds in generalized complex space forms, Bulletin of the Korean Mathematical Society, 40(3), 411-424, 2003.
  • [14] Kimura M., Maeda S., On real hypersurfaces of a complex projective space, Mathematische Zeitschrift, 202, 299-311, 1989.
  • [15] Kon M., Pseudo-Einstein real hyprersurfaces in complex space forms, Journal of Differential Geometry, 14, 339-354, 1979.
  • [16] Kon M., A characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form, Czechoslovak Mathematical Journal, 58(4), 1279-1287, 2008.
  • [17] Lee J.M., Riemannian Manifolds: An Introduction to Curvature, Springer Science and Business Media, 2006.
  • [18] Lee J., Vîlcu G.E., Inequalities for generalized normalized δ -Casorati curvatures of slant submanifolds in quaternionic space forms, Taiwanese Journal of Mathematics, 19(3), 691-702, 2015.
  • [19] Montiel S., Real hypersurfaces of a complex hyperbolic space, Journal of the Mathematical Society of Japan, 37(3), 515-535, 1985.
  • [20] Şahin B., Chen’s first inequality for Riemannian maps, Annales Polonici Mathematici, 117(3), 249-258, 2016.
  • [21] Tashiro Y., Tachibana S., On Fubinian and C-Fubinian manifolds, Kodai Mathematical Seminar Reports, 15(3), 176-183, 1963.
  • [22] Takagi R., Real hypersurfaces in a complex projective space with constant principal curvatures, Journal of the Mathematical Society of Japan, 27(1), 43-53, 1975.
  • [23] Yano K., Kon M., Structures on Manifolds, World Scientific Publishing, 1984.
  • [24] Zhang P., Zhang L., Inequalities for Casorati curvatures of submanifolds in real space forms, Advances in Geometry, 16(3), 329-335, 2016.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Özlem Deniz 0000-0001-7627-5981

Mehmet Gülbahar 0000-0001-6950-7633

Publication Date January 29, 2021
Published in Issue Year 2021 Volume: 2 Issue: 1

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19113 FCMS is licensed under the Creative Commons Attribution 4.0 International Public License.