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B.-Y. Chen's Inequality for Kähler-like Statistical Submersions

Year 2022, , 277 - 286, 31.10.2022
https://doi.org/10.36890/iejg.1006287

Abstract

In this paper, we first define the notion of Lagrangian statistical submersion from a K\"ahler-like statistical manifold onto a statistical manifold. Then we prove that the horizontal distribution of a Lagrangian statistical submersion is integrable. Next, we establish Chen-Ricci inequality for Lagrangian statistical submersions from K\"ahler-like statistical manifolds onto statistical manifolds and discuss the equality case of the obtained inequality through a basic tensor introduced by O'Neill that plays the role of the second fundamental form of an isometric immersion. At the end, we give a nontrivial example of a K\"ahler-like statistical submersion.

References

  • [1] Abe, N., and Hasegawa, K.: An affine submerion with horizontal distribution and its application. Diff. Geom. Appl. 14, 235-250 (2001).
  • [2] Amari, S.: Differential Geometric Methods in Statistics. Lecture Notes in Statistics. Springer. New York. 28, (1985).
  • [3] Aytimur, H., Ozgur, C.: On Cosymplectic-Like Statistical Submersions. Mediterr. J. Math. 16 70, (2019).
  • [4] Aytimur, H., Ozgur, C.: Sharp Inequalities For Anti-Invariant Riemannian Submersions From Sasakian Space Forms. J. Geom. Phy. 166 104251, (2021).
  • [5] Aytimur, H., Kon, M. Mihai, A., Ozgur, C., Takano, K.: Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds. Mathematics. 7, 1202 (2019).
  • [6] Aydin, M.E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29(3), 465-477 (2015).
  • [7] Gray, A.: Pseudo-Riemannian almost product manifolds and submersion. J. Math. Mech. 16, 715-737 (1967) .
  • [8] Gulbahar, M., Meri ç, S.E., Kılıç, E.: Sharp inequalities involving the Ricci curvature for Riemannian submersions. Kragujevac J. Math. 41 (2), 279-293 (2017).
  • [9] Meriç, S.. E., Gulbahar, M., Kılıç, E.: Some Inequalities for Riemannian Submersions. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 63, 1-12 (2017).
  • [10] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13, 459-469 (1966).
  • [11] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press. New York-London (1983).
  • [12] Şahin, B.: Anti-invariant Riemannian submersions from almost Hermitian manifolds. Cent. Eur. J. Math. 8 (3), 437-447 (2010).
  • [13] Siddiqui, A.N., Shahid, M.H.: A lower bound of normalized scalar curvature for bi-slant submanifolds in generalized Sasakian space forms using Casorati curvatures. Acta Math. Univ. Comenianae 87 (1), 127-140 (2018).
  • [14] Siddiqui, A.N., Shahid, M.H.: On totally real statistical submanifolds. Filomat. 32 (13), pp. 11 (2018).
  • [15] Siddiqui, A.N., Shahid, M.H., Lee, J.W.: On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature. AIMS Mathematics. 5 (4), 3495-3509 (2020).
  • [16] Siddiqui, A.N., Chen, B.-Y., Bahadir, O.: Statistical solitons and inequalities for statistical warped product submanifolds. Mathematics. 7 (9), 797 (2019).
  • [17] Siddiqui, A.N., Chen, B.-Y., Siddiqi, M.D.: Chen inequalities for statistical submersions between statistical manifolds. Inter. J. Geom. Methods in Modern Phy. 18 (04), 2150049 (2021).
  • [18] Takano, K.: Statistical manifolds with almost complex structures and its statistical submerions. Tensor (N.S.) 65, 123-137 (2004).
  • [19] Takano, K.: Examples of the statistical submerions on the statistical model. Tensor (N.S.) 65, 170-178 (2004).
  • [20] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85 (1-2), 171-187 (2006).
  • [21] Tastan, H.M.: On Lagrangian submersions. Hacettepe J. Math. and Stat. 43 (6), (2014).
  • [22] Vilcu, G.E.: Almost product structures on statistical manifolds and para-Khler-like statistical submersions. Bulletin des Sciences Mathematiques. 171, 103018 (2021).
  • [23] Vilcu, A.D., Vilcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kahler-like statistical submersions. Entropy.17, 6213-6228 (2015).
Year 2022, , 277 - 286, 31.10.2022
https://doi.org/10.36890/iejg.1006287

Abstract

References

  • [1] Abe, N., and Hasegawa, K.: An affine submerion with horizontal distribution and its application. Diff. Geom. Appl. 14, 235-250 (2001).
  • [2] Amari, S.: Differential Geometric Methods in Statistics. Lecture Notes in Statistics. Springer. New York. 28, (1985).
  • [3] Aytimur, H., Ozgur, C.: On Cosymplectic-Like Statistical Submersions. Mediterr. J. Math. 16 70, (2019).
  • [4] Aytimur, H., Ozgur, C.: Sharp Inequalities For Anti-Invariant Riemannian Submersions From Sasakian Space Forms. J. Geom. Phy. 166 104251, (2021).
  • [5] Aytimur, H., Kon, M. Mihai, A., Ozgur, C., Takano, K.: Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds. Mathematics. 7, 1202 (2019).
  • [6] Aydin, M.E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29(3), 465-477 (2015).
  • [7] Gray, A.: Pseudo-Riemannian almost product manifolds and submersion. J. Math. Mech. 16, 715-737 (1967) .
  • [8] Gulbahar, M., Meri ç, S.E., Kılıç, E.: Sharp inequalities involving the Ricci curvature for Riemannian submersions. Kragujevac J. Math. 41 (2), 279-293 (2017).
  • [9] Meriç, S.. E., Gulbahar, M., Kılıç, E.: Some Inequalities for Riemannian Submersions. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 63, 1-12 (2017).
  • [10] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13, 459-469 (1966).
  • [11] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press. New York-London (1983).
  • [12] Şahin, B.: Anti-invariant Riemannian submersions from almost Hermitian manifolds. Cent. Eur. J. Math. 8 (3), 437-447 (2010).
  • [13] Siddiqui, A.N., Shahid, M.H.: A lower bound of normalized scalar curvature for bi-slant submanifolds in generalized Sasakian space forms using Casorati curvatures. Acta Math. Univ. Comenianae 87 (1), 127-140 (2018).
  • [14] Siddiqui, A.N., Shahid, M.H.: On totally real statistical submanifolds. Filomat. 32 (13), pp. 11 (2018).
  • [15] Siddiqui, A.N., Shahid, M.H., Lee, J.W.: On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature. AIMS Mathematics. 5 (4), 3495-3509 (2020).
  • [16] Siddiqui, A.N., Chen, B.-Y., Bahadir, O.: Statistical solitons and inequalities for statistical warped product submanifolds. Mathematics. 7 (9), 797 (2019).
  • [17] Siddiqui, A.N., Chen, B.-Y., Siddiqi, M.D.: Chen inequalities for statistical submersions between statistical manifolds. Inter. J. Geom. Methods in Modern Phy. 18 (04), 2150049 (2021).
  • [18] Takano, K.: Statistical manifolds with almost complex structures and its statistical submerions. Tensor (N.S.) 65, 123-137 (2004).
  • [19] Takano, K.: Examples of the statistical submerions on the statistical model. Tensor (N.S.) 65, 170-178 (2004).
  • [20] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85 (1-2), 171-187 (2006).
  • [21] Tastan, H.M.: On Lagrangian submersions. Hacettepe J. Math. and Stat. 43 (6), (2014).
  • [22] Vilcu, G.E.: Almost product structures on statistical manifolds and para-Khler-like statistical submersions. Bulletin des Sciences Mathematiques. 171, 103018 (2021).
  • [23] Vilcu, A.D., Vilcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kahler-like statistical submersions. Entropy.17, 6213-6228 (2015).
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Aliya Naaz Sıddıquı 0000-0003-3895-7548

Siraj Uddin

Mohammad Hasan Shahid This is me

Publication Date October 31, 2022
Acceptance Date May 21, 2022
Published in Issue Year 2022

Cite

APA Sıddıquı, A. N., Uddin, S., & Shahid, M. H. (2022). B.-Y. Chen’s Inequality for Kähler-like Statistical Submersions. International Electronic Journal of Geometry, 15(2), 277-286. https://doi.org/10.36890/iejg.1006287
AMA Sıddıquı AN, Uddin S, Shahid MH. B.-Y. Chen’s Inequality for Kähler-like Statistical Submersions. Int. Electron. J. Geom. October 2022;15(2):277-286. doi:10.36890/iejg.1006287
Chicago Sıddıquı, Aliya Naaz, Siraj Uddin, and Mohammad Hasan Shahid. “B.-Y. Chen’s Inequality for Kähler-Like Statistical Submersions”. International Electronic Journal of Geometry 15, no. 2 (October 2022): 277-86. https://doi.org/10.36890/iejg.1006287.
EndNote Sıddıquı AN, Uddin S, Shahid MH (October 1, 2022) B.-Y. Chen’s Inequality for Kähler-like Statistical Submersions. International Electronic Journal of Geometry 15 2 277–286.
IEEE A. N. Sıddıquı, S. Uddin, and M. H. Shahid, “B.-Y. Chen’s Inequality for Kähler-like Statistical Submersions”, Int. Electron. J. Geom., vol. 15, no. 2, pp. 277–286, 2022, doi: 10.36890/iejg.1006287.
ISNAD Sıddıquı, Aliya Naaz et al. “B.-Y. Chen’s Inequality for Kähler-Like Statistical Submersions”. International Electronic Journal of Geometry 15/2 (October 2022), 277-286. https://doi.org/10.36890/iejg.1006287.
JAMA Sıddıquı AN, Uddin S, Shahid MH. B.-Y. Chen’s Inequality for Kähler-like Statistical Submersions. Int. Electron. J. Geom. 2022;15:277–286.
MLA Sıddıquı, Aliya Naaz et al. “B.-Y. Chen’s Inequality for Kähler-Like Statistical Submersions”. International Electronic Journal of Geometry, vol. 15, no. 2, 2022, pp. 277-86, doi:10.36890/iejg.1006287.
Vancouver Sıddıquı AN, Uddin S, Shahid MH. B.-Y. Chen’s Inequality for Kähler-like Statistical Submersions. Int. Electron. J. Geom. 2022;15(2):277-86.