In the present study, the concept of Sasakian statistical manifold has been generalized to indefinite Sasakian statistical manifolds. We also introduce lightlike hypersurfaces of an indefinite Sasakian statistical manifold and establish relations between induced geometrical objects with respect to dual connections. Finally, invariant lightlike submanifold of indefinite Sasakian statistical manifold is proved to be an indefinite Sasakian statistical manifold.
Citation: Oğuzhan Bahadır. On lightlike geometry of indefinite Sasakian statistical manifolds[J]. AIMS Mathematics, 2021, 6(11): 12845-12862. doi: 10.3934/math.2021741
In the present study, the concept of Sasakian statistical manifold has been generalized to indefinite Sasakian statistical manifolds. We also introduce lightlike hypersurfaces of an indefinite Sasakian statistical manifold and establish relations between induced geometrical objects with respect to dual connections. Finally, invariant lightlike submanifold of indefinite Sasakian statistical manifold is proved to be an indefinite Sasakian statistical manifold.
| [1] | S. Amari, Differential geometry of curved exponential families-curvature and information loss, Ann. Statist., 10 (1982), 357–385. |
| [2] | S. Amari, Differential-geometrical methods in statistics, In: Lecture notes in statistics, Vol. 28, New York: Springer, 1985. |
| [3] | S. Amari, H. Nagaoka, Methods of information geometry, Vol. 191, Oxford, U.K.: AMS/Oxford University Press, 2000. |
| [4] | C. Atindogbe, J. P. Ezin, J. Tossa, Lightlike Einstein hypersurfaces in Lorentzian manifolds with constant curvature, Kodai Math. J., 29 (2006), 58–71. |
| [5] |
M. E. Aydin, A. Mihai, I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat, 29 (2015), 465–476. doi: 10.2298/FIL1503465A
|
| [6] | O. Bahadir, M. M. Tripathi, Geometry of lightlike hypersurfaces of a statistical manifold, 2019. Available from: https://arXiv.org/abs/1901.09251. |
| [7] | B. Bartlett, A "generative" model for computing electromagnetic field solutions, 2018. Available from: http://cs229.stanford.edu/proj2018/report/233.pdf. |
| [8] | J. K. Beem, P. E. Ehrlich, K. L. Easley, Global Lorentzian geometry, 2Eds., New York: CRC Press, 1996. |
| [9] | O. Calin, C. Udriste, Geometric modeling in probability and statistics, Springer, 2014. |
| [10] | K. L. Duggal, Foliations of lightlike hypersurfaces and their physical interpretation, Open Math., 10 (2012), 1789–1800. |
| [11] | K. L. Duggal, A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Dordrecht: Kluwer Academic Publishers Group, 1996. |
| [12] | K. L. Duggal, D. H. Jin, Null curves and hypersurfaces of semi-Riemannian manifolds, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2007. |
| [13] | K. L. Duggal, B. Sahin, Differential geometry of lightlike submanifolds, Basel: Birkhauser Verlag, 2010. |
| [14] | B. Efron, Defining the curvature of a statistical problem (with applications to second order efficiency), Ann. Statist., 3 (1975), 1189–1242. |
| [15] |
H. Furuhata, Hypersurfaces in statistical manifolds, Differential Geom. Appl., 27 (2009), 420–429. doi: 10.1016/j.difgeo.2008.10.019
|
| [16] |
H. Furuhata, Statistical hypersurfaces in the space of Hessian curvature zero, Differ. Geom. Appl., 29 (2011), S86–S90. doi: 10.1016/j.difgeo.2011.04.012
|
| [17] | H. Furuhata, I. Hasegawa, Submanifold theory in holomorphic statistical manifolds, Geometry of Cauchy-Riemann submanifolds, Singapore: Springer, 2016,179–215. |
| [18] |
H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato, M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys., 117 (2017), 179–186. doi: 10.1016/j.geomphys.2017.03.010
|
| [19] | J. V. D. Gucht, J. Davelaar, L. Hendriks, O. Porth, H. Olivares, Y. Mizuno, et al., Deep Horizon: A machine learning network that recovers accreting black hole parameters, Astronomy Astrophysics, 636 (2020), 1–12. |
| [20] |
V. Jain, A. P. Singh, R. Kumar, On the geometry of lightlike submanifolds of indefinite statistical manifolds, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050099. doi: 10.1142/S0219887820500991
|
| [21] |
A. Kazan, Conformally-projectively flat trans-Sasakian statistical manifolds, Phys. A, 535 (2019), 122441. doi: 10.1016/j.physa.2019.122441
|
| [22] | E. Kilic, O. Bahadir, Lightlike hypersurfaces of a semi-Riemannian product manifold and quarter-symmetric nonmetric connections, Int. J. Math. Math. Sci., 2012 (2012), 178390. |
| [23] | T. Kurose, Conformal-projective geometry of statistical manifolds, Interdiscip. Inform. Sci., 8 (2002), 89–100. |
| [24] |
İ. Erken, C. Murathan, A. Yazla, Almost cosympletic statistical manifolds, Quaest. Math., 43 (2020), 265–282. doi: 10.2989/16073606.2019.1576069
|
| [25] | F. Massamba, Lightlike hypersurfaces of indefinite Sasakian manifolds with parallel symmetric bilinear forms, Differ. Geom. Dyn. Syst., 10 (2008), 226–234. |
| [26] | F. Massamba, Killing and geodesic lightlike hypersurfaces of indefinite Sasakian manifolds, Turkish J. Math., 32 (2008), 325–347. |
| [27] | S. Ssekajja, Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold, Arab J. Math. Sci., 2021. DOI: 10.1108/AJMS-10-2020-0097. |
| [28] |
K. Takano, Statistical manifolds with almost contact structures and its statistical submersions, J. Geom., 85 (2006), 171–187. doi: 10.1007/s00022-006-0052-2
|
| [29] |
A. D. Vilcu, G. E. Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kahler-like statistical submersions, Entropy, 17 (2015), 6213–6228. doi: 10.3390/e17096213
|
| [30] |
P. W. Vos, Fundamental equations for statistical submanifolds with applications to the Bartlett correction, Ann. Inst. Statist. Math., 41 (1989), 429–450. doi: 10.1007/BF00050660
|
Oğuzhan Bahadır. On lightlike geometry of indefinite Sasakian statistical manifolds[J]. AIMS Mathematics, 2021, 6(11): 12845-12862. doi: 10.3934/math.2021741
DownLoad: