### 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 δCasorati curvatures and also consider the equality case of the inequality. Some immediate applications are discussed.

Keywords:

δCasorati curvatures, holomorphic statistical manifold, statistical hypersurfaces, normalized scalar curvature, dual connections.

References:

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

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

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

   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.

   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.

   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.

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

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

   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.

   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.

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

   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.

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

   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.

   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.

   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.

   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.

   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.

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

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

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

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

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

   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.

   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.

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

   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.

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

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