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Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds

Year 2021, Volume: 14 Issue: 1, 46 - 58, 15.04.2021
https://doi.org/10.36890/iejg.776559

Abstract

In this paper, we study the statistical immersion of codimension one from a Sasakian statistical manifold of constant φ− curvature to a holomorphic statistical manifold of constant holomorphic curvature and its converse. We prove that in both cases the constant φ− curvature equals to one and the constant holomorphic curvature must be zero. Moreover, we construct several examples of statistical manifolds, Sasakian statistical manifolds and holomorphic statistical manifolds of constant holomorphic curvature zero.

Supporting Institution

Anhui Normal University

Project Number

Research Found for the Doctoral Program of Anhui Normal University(751841)

References

  • [1] Amari, S.: Differential-geometrical methods in statistics. Springer. Berlin (1985).
  • [2] Amari, S.: Information geometry on hierarchy of probability distributions. IEEE Transactions on Information Theory. 45 (5), 1701-1711 (2001).
  • [3] Aydin, M. E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29 (3), 465-477 (2015).
  • [4] Aydin, M. E., Mihai, A., Mihai, I.: GeneralizedWintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bulletin of Mathematical Sciences. 7 (1), 155–166 (2017).
  • [5] Blair, D. E.: Riemannian geometry of contact and symplectic manifolds, second Edition. Birkhäuser. New York (2010).
  • [6] Chen, B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasgow Mathematical Journal. 41(1), 33-41 (1999).
  • [7] Chen, B. Y., Mihai, A., Mihai, I.: A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Results in Mathematics. 74(4), 165 (2019).
  • [8] Furuhata, H.: Hypersurfaces in statistical manifolds. Differential Geometry and its Applications. 27(3), 420-429 (2009).
  • [9] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, K.: Kenmotsu statistical manifolds and warped product. Journal of Geometry. 108 (3), 1175–1191 (2017).
  • [10] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, S., Shahid, M. H.: Sasakian statistical manifold. Journal of Geometry and Physics. 117, 179-186 (2017).
  • [11] Kassabov, O.: On totally real submanifolds. Bulletin of the Belgian Mathematical Society-Simon Stevin. 38 (2), 136-143 (1986).
  • [12] Kurose, T.: Geometry of statistical manifolds. In: Mathematics in the 21st Century. Nihon-hyouron-sha, Japan (2004).
  • [13] Matsuzoe, H.: Statistical manifolds and affine differential geometry. Advanced Studies in Pure Mathematics. 57, 303-321 (2010).
  • [14] Milijević, M.: Totally real statistical submanifolds. Interdisciplinary Information Sciences. 21 (2), 87-96 (2015).
  • [15] Milijević, M.: CR statistical submanifolds. Kyushu Journal of Mathematics. 73(1), 89-101 (2019).
  • [16] Shima, H., Yagi, K.: Geometry of Hessian manifolds. Differential Geometry and its Applications. 7 (3), 277-290 (1997).
  • [17] Siddiqui, A. N., Shahid, M. H.: On totally real statistical submanifolds. Filomat. 32 (13), 4473-4483 (2018).
  • [18] Tashiro, Y., Tachibana, S.: On Fubinian and C-Fubinian manifolds. Kodai Mathematical Seminar Reports. 15(3), 176–183 (1963).
  • [19] Carmo, M. P. D.: Riemannian geometry. Birkhäuser. Boston (1992).
  • [20] Vos, P. W.: Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Annals of the Institute of Statistical Mathematics. 41 (3), 429-450 (1989).
  • [21] Yano, K., Kon, M.: CR submanifolds of Kaehlerian and Sasakian manifolds. Birkhäuser. Boston (2012).
Year 2021, Volume: 14 Issue: 1, 46 - 58, 15.04.2021
https://doi.org/10.36890/iejg.776559

Abstract

Project Number

Research Found for the Doctoral Program of Anhui Normal University(751841)

References

  • [1] Amari, S.: Differential-geometrical methods in statistics. Springer. Berlin (1985).
  • [2] Amari, S.: Information geometry on hierarchy of probability distributions. IEEE Transactions on Information Theory. 45 (5), 1701-1711 (2001).
  • [3] Aydin, M. E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29 (3), 465-477 (2015).
  • [4] Aydin, M. E., Mihai, A., Mihai, I.: GeneralizedWintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bulletin of Mathematical Sciences. 7 (1), 155–166 (2017).
  • [5] Blair, D. E.: Riemannian geometry of contact and symplectic manifolds, second Edition. Birkhäuser. New York (2010).
  • [6] Chen, B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasgow Mathematical Journal. 41(1), 33-41 (1999).
  • [7] Chen, B. Y., Mihai, A., Mihai, I.: A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Results in Mathematics. 74(4), 165 (2019).
  • [8] Furuhata, H.: Hypersurfaces in statistical manifolds. Differential Geometry and its Applications. 27(3), 420-429 (2009).
  • [9] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, K.: Kenmotsu statistical manifolds and warped product. Journal of Geometry. 108 (3), 1175–1191 (2017).
  • [10] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, S., Shahid, M. H.: Sasakian statistical manifold. Journal of Geometry and Physics. 117, 179-186 (2017).
  • [11] Kassabov, O.: On totally real submanifolds. Bulletin of the Belgian Mathematical Society-Simon Stevin. 38 (2), 136-143 (1986).
  • [12] Kurose, T.: Geometry of statistical manifolds. In: Mathematics in the 21st Century. Nihon-hyouron-sha, Japan (2004).
  • [13] Matsuzoe, H.: Statistical manifolds and affine differential geometry. Advanced Studies in Pure Mathematics. 57, 303-321 (2010).
  • [14] Milijević, M.: Totally real statistical submanifolds. Interdisciplinary Information Sciences. 21 (2), 87-96 (2015).
  • [15] Milijević, M.: CR statistical submanifolds. Kyushu Journal of Mathematics. 73(1), 89-101 (2019).
  • [16] Shima, H., Yagi, K.: Geometry of Hessian manifolds. Differential Geometry and its Applications. 7 (3), 277-290 (1997).
  • [17] Siddiqui, A. N., Shahid, M. H.: On totally real statistical submanifolds. Filomat. 32 (13), 4473-4483 (2018).
  • [18] Tashiro, Y., Tachibana, S.: On Fubinian and C-Fubinian manifolds. Kodai Mathematical Seminar Reports. 15(3), 176–183 (1963).
  • [19] Carmo, M. P. D.: Riemannian geometry. Birkhäuser. Boston (1992).
  • [20] Vos, P. W.: Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Annals of the Institute of Statistical Mathematics. 41 (3), 429-450 (1989).
  • [21] Yano, K., Kon, M.: CR submanifolds of Kaehlerian and Sasakian manifolds. Birkhäuser. Boston (2012).
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Feng Wu 0000-0003-0409-9960

Yan Jıang 0000-0001-8004-983X

Liang Zhang 0000-0002-6091-9313

Project Number Research Found for the Doctoral Program of Anhui Normal University(751841)
Publication Date April 15, 2021
Acceptance Date November 24, 2020
Published in Issue Year 2021 Volume: 14 Issue: 1

Cite

APA Wu, F., Jıang, Y., & Zhang, L. (2021). Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds. International Electronic Journal of Geometry, 14(1), 46-58. https://doi.org/10.36890/iejg.776559
AMA Wu F, Jıang Y, Zhang L. Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds. Int. Electron. J. Geom. April 2021;14(1):46-58. doi:10.36890/iejg.776559
Chicago Wu, Feng, Yan Jıang, and Liang Zhang. “Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 46-58. https://doi.org/10.36890/iejg.776559.
EndNote Wu F, Jıang Y, Zhang L (April 1, 2021) Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds. International Electronic Journal of Geometry 14 1 46–58.
IEEE F. Wu, Y. Jıang, and L. Zhang, “Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 46–58, 2021, doi: 10.36890/iejg.776559.
ISNAD Wu, Feng et al. “Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds”. International Electronic Journal of Geometry 14/1 (April 2021), 46-58. https://doi.org/10.36890/iejg.776559.
JAMA Wu F, Jıang Y, Zhang L. Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds. Int. Electron. J. Geom. 2021;14:46–58.
MLA Wu, Feng et al. “Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 46-58, doi:10.36890/iejg.776559.
Vancouver Wu F, Jıang Y, Zhang L. Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds. Int. Electron. J. Geom. 2021;14(1):46-58.