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

Yıl 2022, , 277 - 286, 31.10.2022

Aliya Naaz Sıddıquı , Siraj Uddin , Mohammad Hasan Shahid

https://doi.org/10.36890/iejg.1006287

Cited By: 3

Öz

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.

Anahtar Kelimeler

Statistical submersions, Lagrangian statistical submersions, Kähler-like statistical submersions, statistical manifolds, Chen-Ricci Inequality

Kaynakça

• [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).