Esra ERKAN, Kazuhiko TAKANO, Mehmet GÜLBAHAR

Lokal Çarpım benzeri İstatistiksel Manifoldlar ve Hiperyüzeyleri

Bu çalışmada, hemen hemen çarpım benzeri Riemann manifoldları araştırılmıştır. Hemen hemen çarpım benzeri Riemann manifoldlarının teğet hiperyüzeyleri üzerindeki temel özellikler elde edilmiştir. Teğet hiperyüzeylerin bazı örnekleri sunulmuştur. Bir teğet hiperyüzeyin Riemann eğrilik tensörünü içeren bazı bağıntılar hesaplanmıştır.

Anahtar Kelimeler:

Riemann çarpım manifoldu, hiperyüzey, istatistiksel manifold, eğrilik

Locally Product-like Statistical Manifolds and Their Hypersurfaces

In this paper, almost product-like Riemannian manifolds are investigated. Basic properties on tangential hypersurfaces of almost product-like Riemannian manifolds are obtained. Some examples of tangential hypersurfaces are presented. Some relations involving the Riemannian curvature tensor of a tangential hypersurface are computed.

Keywords:

Riemannian product manifold, hypersurface, statistical manifold, curvature,

PDF

___

[1] Adachi, T., Kimura, M., Maeda, S.: Real hypersurfaces some of whose geodesics are plane curves in non flat complex space form. Tohoku Math. J. 57 (2), 223-230 (2005).
[2] Adati, T.: Submanifolds of an almost product Riemannian manifold. Kodai Math. J. 4 (2), 327-343 (1981).
[3] Akbari, H., Malek, F.: On the hypersurfaces of almost hermitian statistical manifolds. Bull. Iran. Math. Soc. 48, 2669–2684 (2022).
[4] Amari, S.: Differential-geometrical methods in statistics, Lecture Notes in Statistics. vol. 28. Springer-Verlag, New York (1985).
[5] Aytimur, H. B., Özgür, C.: Inequalities for submanifolds of Sasaki-like statistical manifolds. Turk. J. Math. 42 (6), 3149-3163 (2018).
[6] Blair, D. E., Ludden, G. D.: Hypersurfaces in almost contact manifolds. Tohoku Math. J. 21 (3), 354-362 (1969).
[7] Calin, O., Udriste, C.: Geometric modeling in probability and statistics. Springer, Berlin, Germany (2014).
[8] Chen, B.-Y.: Geometry of Submanifolds. Marcel Dekker, New York-Basel (1973).
[9] Chen, B.-Y., Maeda, S.: Hopf hypersurfaces with constant principal curvatures in complex projective or complex hyperbolic spaces. Tokyo J. Math. 24 (1), 133-152 (2001).
[10] Deshmukh, S., Sharfuddin, A., Husain, S.I.: Hypersurfaces of almost product manifolds. Tamkang J. Math. 10, 169-181 (1979).
[11] Efron, B.: Defining the curvature of a statistical problem (with applications to second order efficiency). Ann. Statist. 3, 1189–1242 (1975).
[12] Erken, I. K., Murathan, C., Yazla, A.: Almost cosympletic statistical manifolds. Quaest. Math. 43 (2), 265-282 (2020).
[13] Eum, S. S.: On complex hypersurface in normal almost contact spaces. Tensor N. S. 19, 45-50 (1968).
[14] Feng, W. U., Jiang, Y., Zhang, L.: Some results on statistical hypersurfaces of Sasakian statistical manifolds and holomorphic statistical manifolds.
Int. Electron. J. Geom. 14 (1), 46-58 (2021).
[15] Furuhata, H.: Hypersurfaces in statistical manifolds. Differ. Geom. Appl. 27 (3), 420-429 (2009).
[16] Furuhata, H.: Statistical hypersurfaces in the space of Hessian curvature zero. Differ. Geom. Appl. 29 Suppl.1, S86-S90 (2011).
[17] Furuhata, H., Hasegawa, I.: Submanifold theory in holomorphic statistical manifolds, in: S. Dragomir, M.H. Shahid, F.R. Al-Solamy (Eds.),Geometry of Cauchy-Riemann Submanifolds. Springer, Singapore, 179-215 (2016).
[18] Furuhata, H., Hu, N., Vrancken, L.: Statistical hypersurfaces in the space of Hessian curvature zero II. Pure and Applied DifferentialGeometry-PADGE 2012 In Memory of Franki Dillen, Shaker Verlag GmbH, Germany, 136-142 (2013).
[19] Gucht, J.V.D., Davelaar, J., Hendriks, L., Porth, O., Olivares, H., Mizuno, Y., Fromm, C.M., Falcke, H.: Deep Horizon; a machine learningnetwork that recovers accreting black hole parameters. Astronomy-Astrophysics 636, A94 (2020).
[20] Kon, M.: A characterization of totally η-umbilical real hypersurfaces and ruled hypersurfaces of a complex space form. Czech. Math. J. 58 (4),1279-1287 (2008).
[21] Murathan, C., ¸Sahin, B.: A study ofWintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109 (2), 1-18 (2018).
[22] Sato, I., Matsumoto, K.: On P-Sasakian manifolds satisfying certain conditions. Tensor, New Ser. 33, 173-178 (1979).
[23] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85, 171-187 (2006).
[24] Takano, K.: Statistical manifolds with almost complex structures. Tensor, New Ser. 72 (3), 225-231 (2010).
[25] Vilcu, A.D., Vilcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy 17 (9), 6213-6228 (2015).
[26] Vilcu, G.-E.: Almost product structures on statistical manifolds and para-Kähler-like statistical submersions. Bull. Sci. Math. 171, 21 (2021).
[27] Vos, P.W.: Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Ann. Inst. Statist. Math. 41 (3), 429–450 (1989).
[28] Yano, K., Kon, M.: Structures on manifolds. Ser. Pure Math. 3, World Scientific Publishing Co., Singapore (1984).