Some results on Kenmotsu statistical manifolds

Yıl 2022, Cilt: 51 Sayı: 3, 800 - 816, 01.06.2022

Yan Jıang Feng Wu Liang Zhang

https://doi.org/10.15672/hujms.954555

Cited By: 2

Öz

In this paper, we first investigate the Kenmotsu statistical structures built on a Kenmotsu space form and determine some special Kenmotsu statistical structures under two curvature conditions. Secondly, we show that if the holomorphic sectional curvature of the hypersurface orthogonal to the structure vector in a Kenmotsu statistical manifold is constant, then the $\phi-$sectional curvature of the ambient Kenmotsu statistical manifold must be constant $-1$, and the constant holomorphic sectional curvature of the hypersurface is $0$. In addition, some non-trivial examples are given to illustrate the results of this paper.

Anahtar Kelimeler

Kenmotsu statistical manifold, constant ϕ-sectional curvature, constant ϕ-curvature, hypersurface

Kaynakça

• [1] S. Amari, Differential-geometrical methods in statistics. Lecture Notes in Statistics, Springer-Verlag, New York, 1985.
• [2] S. Decu, S. Haesen, L. Verstraelen and G. E. Vîlcu, Curvature invariants of statistical submanifolds in Kenmotsu statistical manifolds of constant $\phi$-sectional curvature, Entropy 20 (7), 529, 2018.
• [3] I.K. Erken, C. Murathan and A. Yazla, Almost cosympletic statistical manifolds, Quaest. Math. 43 (2), 265–282, 2020.
• [4] H. Furuhata, Hypersurfaces in statistical manifolds, Differential Geom. Appl. 27 (3), 420–429, 2009.
• [5] H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical mani- folds, in: Geometry of Cauchy-Riemann submanifolds, 179–215, Springer, Singapore, 2016.
• [6] H. Furuhata, I. Hasegawa, Y. Okuyama and K. Sato, Kenmotsu statistical manifolds and warped product, J. Geom. 108 (3), 1175–1191, 2017.
• [7] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato and M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys. 117, 179–186, 2017.
• [8] J. B. Jun, U. C. De and G. Pathak, On Kenmotsu manifolds, J. Korean Math. Soc. 42 (3), 435–445, 2005.
• [9] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24, 93–103, 1972.
• [10] T. Kurose, Dual connections and affine geometry, Math. Z. 203 (1), 115–121, 1990.
• [11] H. Matsuzoe, Statistical manifolds and affine differential geometry, Adv. Stud. Pure Math. 57, 303–321, 2010.
• [12] C.R. Min, S.O. Choe and Y.H. An, Statistical immersions between statistical manifolds of constant curvature, Glob. J. Adv. Res. Class. Mod. Geom. 3(2), 66–75, 2014.
• [13] G. Pitis, Geometry of Kenmotsu manifolds, Publishing House of Transilvania University of Brasov, Brasov, 2007.
• [14] G. Pitis, Contact forms in geometry and topology, in: Topics in Modern Differential Geometry, Atlantis Trans. Geom., 2017.
• [15] H. Shima and K. Yagi, Geometry of Hessian manifolds, Differential Geom. Appl. 7 (3), 277–290, 1997.
• [16] A.N. Siddiqui, M.H. Shahid, On totally real statistical submanifolds, Filomat, 32 (13), 4473–4483, 2018.
• [17] A.N. Siddiqui, Y.J. Suh and O. Bahadr Extremities for Statistical Submanifolds in Kenmotsu Statistical Manifolds, Filomat, 35 (2), 591–603, 2021.
• [18] S. Tanno, The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J. 21, 21–38, 1969.
• [19] J.A. Vickers, Distributional geometry in general relativity, J. Geom. Phys. 62 (3), 692–705, 2012.
• [20] G.E. Vîlcu, Almost product structures on statistical manifolds and para-Kähler-like statistical submersions, Bull. Sci. Math. 171, 103018, 2021.
• [21] P.W. Vos, Fundamental equations for statistical submanifolds with applications to the Bartlett correction, Ann. Inst. Statist. Math. 41 (3), 429–450, 1989.
• [22] K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Progress in Mathematics, 30. Birkhäuser, Boston, Mass., 1983.