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

Yıl 2021, , 46 - 58, 15.04.2021

Feng Wu Yan Jıang Liang Zhang

https://doi.org/10.36890/iejg.776559

Öz

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.

Anahtar Kelimeler

Sasakian statistical manifolds, holomorphic statistical manifolds, constant φ−curvature, constant holomorphic curvature, statistical hypersurface

Destekleyen Kurum

Anhui Normal University

Proje Numarası

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

Kaynakça

• [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).