### Optimizations on Statistical Hypersurfaces with Casorati Curvatures

Download PDF

**Authors:**A. N. SIDDIQUI AND M. H. SHAHID

**DOI:**10.46793/KgJMat2103.449S

**Abstract:**

In the present paper, we study Casorati curvatures for statistical hypersurfaces. We show that the normalized scalar curvature for any real hypersurface (i.e., statistical hypersurface) of a holomorphic statistical manifold of constant holomorphic sectional curvature k is bounded above by the generalized normalized δ−Casorati curvatures and also consider the equality case of the inequality. Some immediate applications are discussed.

**Keywords:**

δ−Casorati curvatures, holomorphic statistical manifold, statistical hypersurfaces, normalized scalar curvature, dual connections.

**References:**

[1] S. Amari, Diﬀerential Geometric Methods in Statistics, Lecture Notes in Statistics 28, Springer, New York, 1985.

[2] M. Aquib and M. H. Shahid, Generalized normalized δ−Casorati curvature for statistical submanifolds in quaternion Kahler-like statistical space forms, J. Geom. 109 (2018), 13 pages.

[3] M. E. Aydin, A. Mihai and I. Mihai, Some Inequalities on submanifolds in statistical manifolds of constant curvature, Filomat 29(3) (2015), 465–477.

[4] M. E. Aydin, A. Mihai and I. Mihai, Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature, Bull. Math. Sci. (2016), DOI 10.1007/s13373-016-0086-1.

[5] M. N. Boyom, A. N. Siddiqui, W. A. Mior Othman and M. H. Shahid, Classiﬁcation of totally umbilical CR-statistical submanifolds in holomorphic statistical manifolds with constant holomorphic curvature, in: F. Nielsen, F. Barbaresco (Eds.), Geometric Science of Information, Lecture Notes in Computer Science 10589, Springer, Cham, 2017.

[6] F. Casorati, Mesure de la courbure des surface suivant 1’idee commune. Ses rapports avec les mesures de coubure gaussienne et moyenne, Acta Math. 14, (1999), 95–110.

[7] B.-Y. Chen, Some pinching and classiﬁcation theorems forminimal submanifolds, Arch. Math. 60 (1993), 569–578.

[8] S. Decu, S. Haesen, L. Verstraelen and G. E. Vilcu, Curvature invariants of statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ−sectional curvature, Entropy 20(7) (2018), 529.

[9] S. Decu, S. Haesen and L. Verstraelen, Optimal inequalities characterising quasi-umbilical submanifolds, Journal of Inequalities in Pure and Applied Mathematics 9(3) (2008), 7 pages.

[10] S. Decu, A. Pantic, M. Petrovic-Torgasev and L. Verstraelen, Ricci and Casorati principal directions of δ(2) Chen ideal submanifolds, Kragujevac J. Math. 37(1) (2013), 25–31.

[11] H. Furuhata, Hypersurfaces in statistical manifolds, Diﬀerential Geom. Appl. 27 (2009), 420–429.

[12] H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical manifolds, In: S. Dragomir, M. H. Shahid and F. R. Al-Solamy (Eds.), Geometry of Cauchy-Riemann Submanifolds, Springer, Singapore, 2016, 179–215.

[13] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato and M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys. 117 (2017), 179–186.

[14] H. Furuhata, I. Hasegawa, Y. Okuyama and K. Sato, Kenmotsu statistical manifolds and warped product, J. Geom. (2017), DOI 10.1007/s00022-017-0403-1.

[15] C. W. Lee, J. W. Lee, G. E. Vilcu and D. W. Yoon, Optimal inequalities for the Casorati curvatures of the submanifolds of generalized space form endowed with semi-symmetric metric connections, Bull. Korean Math. Soc. 52 (2015), 1631–1647.

[16] C. W. Lee and G. E. Vilcu, Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in quaternion space forms, Taiwanese J. Math. 19 (2015), 691–702.

[17] C. W. Lee, D. W. Yoon and J. W. Lee, A pinching theorem for statistical manifolds with Casorati curvatures, J. Nonlinear Sci. Appl. 10 (2017), 4908–4914.

[18] C. W. Lee, D. W. Yoon and J. W. Lee, Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections, J. Inequal. Appl. 2014 (2014), 9 pages.

[19] C. W. Lee, J. W. Lee and G. E. Vilcu, A new proof for some optimal inequalities involving generalized normalized δ−Casorati curvatures, J. Inequal. Appl. 2015 (2015), 9 pages.

[20] C. W. Lee and J. W. Lee, Inequalities on Sasakian statistical manifolds in terms of casorati curvatures, Mathematics 6(259) (2018), 10 pages.

[21] B. Opozda, Bochner’s technique for statistical structures, Ann. Global Anal. Geom. 48(4) (2015), 357–395.

[22] B. Opozda, A sectional curvature for statistical structures, Linear Algebra Appl. 497 (2016), 134–161.

[23] T. Oprea, Optimization methods on Riemannian submanifolds, An. Univ. Bucur. Mat. 54(1) (2005), 127–136.

[24] A. N. Siddiqui and M. H. Shahid, 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) (2018), 127–140.

[25] A. N. Siddiqui, Upper bound inequalities for δ-Casorati curvatures of submanifolds in generalized Sasakian space forms admitting a semi-symmetric metric connection, Int. Electron. J. Geom. 11(1) (2018), 57–67.

[26] A. N. Siddiqui and M. H. Shahid, On totally real statistical submanifold, Filomat 32(13) (2018), 11 pages.

[27] V. Slesar, B. Sahin and G. E. Vilcu, Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms, J. Inequal. Appl. 2014 (2014), 10 pages.

[28] A. D. Vilcu and G. E. Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions, Entropy 17 (2015), 6213–6228.

[29] K. Yano and M. Kon, Structures on Manifolds, Worlds Scientiﬁc, Singapore, 1984.