Research Article
BibTex RIS Cite

Year 2024, Volume: 17 Issue: 1, 44 - 62, 23.04.2024

Serap Sevim , Cengizhan Murathan

https://doi.org/10.36890/iejg.1378844

Abstract

References

  • [1] Akbari, H., Malek, F.: On contact metric statistical manifolds. Differential Geom. Appl. 75, Paper No. 101735, 16 pp (2021).
  • [2] Akbari, H., Malek, F.: On the hypersurfaces of almost Hermitian statistical manifolds. Bull. Iranian Math. Soc. 48 (5), 2669–2684 (2022).
  • [3] Amari, S.: Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics,vol. 28. Springer, Berlin (1985).
  • [4] Arslan, K., Bayram, B., Bulca,B., Öztürk, G.: On translation surfaces in 4-dimensional Euclidean space. Acta Comment. Univ. Tartu. Math. 20(2), 123–133 (2016).
  • [5] Al-Solamy, F.R., Bansal, P., Chen, B-Y, Murathan, C., Shahid, M.H.: Geometry of Chen invariants in statistical warped product manifolds. Int. J.Geom. Methods Mod. Phys. 17(6), 2050081, 22 pp (2020).
  • [6] Aydın, M.E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29(3), 465-477 (2015).
  • [7] Aydın, M.: A generalization of translation surfaces with constant curvature in the isotropic space. J. Geom. 107(3), 603–615 (2016).
  • [8] Aydın, M.E., Mihai,A., Mihai, I.: Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sci.,7 (1), 155-166 (2017).
  • [9] Aydın, M. E., Ergut, M.: Affine translation surfaces in the isotropic 3-space. Int. Electron. J. Geom. 10(1), 21–30 (2017).
  • [10] Aydın,M. E., Lopez, E., Vilcu, G-E.: Classification of seperable hypersurfaces with constant sectional curvature. https://arxiv.org/pdf/2309.06025.pdf
  • [11] Aytimur, H., Özgür, C.: On cosymplectic-like statistical submersions. Mediterr. J. Math. 16(3), Paper No. 70, 14 pp (2019).
  • [12] Bansal, P., Uddin, S., Shahid, M.H.: Optimal inequalities for submanifolds in statistical manifolds of quasi constant curvature. Filomat 35(10), 3319–3330 (2021).
  • [13] Blaga, A.M., Crasmareanu, C.: Statistical Structures in Almost Paracontact Geometry. Bulletin of the Iranian Mathematical Society. 44, 1407–1413 (2018).
  • [14] Cai, D.D., Liu, X., Zhang, L.: Inequalities on generalized normalized δ-Casorati curvatures for submanifolds in statistical manifolds of constant curvatures. (Chinese) J. Jilin Univ. Sci. 57(2), 206–212 (2019).
  • [15] Chen, B-Y., Mihai,A., Mihai, I.: A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Results Math. 74(4), 165 (2019).
  • [16] Chen, B-Y., Decu, S., Vîlcu, G-E.: Inequalities for the Casorati curvature of totally real spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Entropy. 23(11), Paper No. 1399, 13 pp (2021).
  • [17] Decu, S., Vîlcu, G-E.: Casorati inequalities for statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ-sectional curvature with semi-symmetric metric connection. Entropy. 24(6),(2022).
  • [18] Dillen, F., Verstraelen, L., Zafindratafa, G.: A generalization of the translation surfaces of Scherk Differential Geometry in honor of Radu Rosca, K.U.L. (1991), 107–109.
  • [19] Erkan, E., Gülbahar, M.: Chen’s basic inequalities for hypersurfaces of statistical Riemannian manifolds. Int. J. Maps Math. 6 (1), 37–53(2023).
  • [20] Furuhata, H.: Hypersurfaces in statistical manifolds. Diff . Geom. Appl. 27, 420-429 (2009).
  • [21] Furuhata, H., Hasegawa, I.: Submanifold theory in holomorphic statistical manifolds, in: Geometry of Cauchy-Riemann Submanifolds, Springer, 2016, pp. 179–215
  • [22] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, K., Shahid, M.H.: Sasakian statistical manifolds. J. Geom. Phys. 117, 179–186 (2017).
  • [23] Furuhata, H., Inoguchi, J., Kobayashi, S.: A characterization of the alpha-connections on the statistical manifold of normal distributions. Inf. Geom. 4(1), 177–188 (2021).
  • [24] Furuhata, H., Hasegawa, I., Satoh, N.: Chen invariants and statistical submanifolds. Commun. Korean Math. Soc. 37(3), 851–864 (2022).
  • [25] Görünü¸s, R., Erken Küpeli,İ. ,Yazla, A., Murathan, C.: A generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds. Int. Electron. J. Geom. 12(1), 43–56 (2019).
  • [26] Malek, F., Akbari, H.: Casorati curvatures of submanifolds in cosymplectic statistical space forms. Bull. Iranian Math. Soc. 46(5), 1389–1403 (2020).
  • [27] Hasanis, T., Lopez, R.: Translation surfaces in Euclidean space with constant Gaussian curvature. Comm. Anal. Geom. 29(6), 1415–1447 (2021).
  • [28] Hasanis, T., Lopez, R.: Classification of separable surfaces with constant Gaussian curvature. Manuscripta Math. 166, 403-417 (2021).
  • [29] Hirohiko, S.: The geometry of Hessian Structures World Scientific Publishing Co. Pte. Ltd. 2007.
  • [30] Humby, C.: https://en.wikipedia.org/wiki/Clive_Humby#cite_note-10
  • [31] Humby, C.:https://medium.com/project-2030/data-is-the-new-oil-a-ludicrous-proposition-1d91bba4f294
  • [32] Inoguchi, J.I, López, R., Munteanu, M.I., Minimal translation surfaces in the Heisenberg group Nil3. Geom. Dedicata. 161 , 221–231 (2012).
  • [33] Jeffreys,H.: An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. Lond. Ser. A. 196,453–461 (1946).
  • [34] Jiang, Y., Wu, F., Zhang, L.: Some results on Kenmotsu statistical manifolds. Hacet. J. Math. Stat. 51(3), 800–816 (2022).
  • [35] Küpeli Erken, İ., Murathan, C., Yazla, A.: Almost cosympletic statistical manifolds. Quaest. Math. 43(2), 265–282 (2020).
  • [36] Kazan, S., Takano, K.: Anti-invariant holomorphic statistical submersions. Results Math. 78(4), Paper No. 128, 18 pp (2023).
  • [37] Lauritzen, S.: In statistical manifolds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, vol. 10, pp. 163–216. IMS Lecture NotesInstitute of Mathematical Statistics, Hayward (1987).
  • [38] Lee, C.W., Yoon, D.W., Lee, J.W.: A pinching theorem for statistical manifolds with Casorati curvatures. J. Nonlinear Sci. Appl. 10 (9), 4908–4914 (2017).
  • [39] Lopez,R.: Minimal translation surfaces in hyperbolic space. Beitr. Algebra Geom. 52, 105–112 (2011).
  • [40] Lopez, R., Munteanu,M. I.: Surfaces with constant mean curvature in Sol geometry. Differential Geom. Appl. 29, S238–S245 (2011).
  • [41] Malek, F., Akbari, H.: Casorati curvatures of submanifolds in cosymplectic statistical space forms. Bull. Iranian Math. Soc. 46 (5), (2020).
  • [42] Milijevic, M.: Totally real statistical submanifolds. Interdiscip. Inf. Sci. 21 , 87-96 (2015).
  • [43] Milijevic, M.: CR statistical submanifolds. Kyushu J. Math. 73(1) , 89–101 (2019).
  • [44] Murathan, C., ¸Sahin, B.: A study of Wintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109(2) ,Paper No. 30, 18 pp.(2018).
  • [45] Nomizu, K., Sasaki, T.: Affine Differential Geometry,Cambridge Univ. Press (1994)1989, KU Leuven, Departement Wiskunde (1991), pp. 107–109.
  • [46] Nomizu, K., Simon, U.: Notes on conjugate connections. Geometry and topology of submanifolds, IV (Leuven, 1991), 152–173,World Sci. Publ., River Edge, NJ, (1992).
  • [47] Bahadır, O.: On lightlike geometry of indefinite Sasakian statistical manifolds. AIMS Math. 6(11) , 12845–12862 (2021).
  • [48] Opozda, B.: A sectional curvature for statistical structures. Linear Alg. Appl. 497, 134–161 (2016).
  • [49] Palmer, M., https://ana.blogs.com/maestros/2006/11/data_is_the_new.html
  • [50] Rao, C.R.: Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 , 81–91 (1945).
  • [51] Samereh, L., Peyghan, E., Mihai, I.: On almost Norden statistical manifolds. Entropy 24(6), 758. 10pg.(2022).
  • [52] Seo, K.: Translation hypersurfaces with constant curvature in space forms. Osaka J. Math. 50 (3), 631–641 (2013).
  • [53] Siddiqui, A. N., Murathan, C., Siddiqi, M. D.: The Chen’s first inequality for submanifolds of statistical warped product manifolds. J. Geom. Phys. 169, Paper No. 104344, 13 pp (2021).
  • [54] Siddiqui, A.N., Shahid, M.H.: Optimizations on statistical hypersurfaces with Casorati curvatures. Kragujevac J. Math. 45(3) , 449–463 (2021).
  • [55] Siddiqui, A.N., Al-Solamy, F.R., Shahid, M.H.,Mihai, I.: On CR-statistical submanifolds of holomorphic statistical manifolds. Filomat 35(11) , 3571–3584 (2021).
  • [56] Siddiqui, A. N., Chen, B.Y., Siddiqi, M. D.: Chen inequalities for statistical submersions between statistical manifolds. Int. J. Geom. Methods Mod. Phys. 18 (4), Paper No. 2150049, 17 pp (2021).
  • [57] Siddiqui, Al.N., Uddin, S., Shahid, M.H.: B.-Y. Chen’s inequality for Kähler-like statistical submersions. Int. Electron. J. Geom. 15 (2), 277–286 (2022).
  • [58] Siddiqui, A.N, Siddiqi, M.D., Alkhaldi, A. H., Akram, A.: Lower bounds on statistical submersions with vertical Casorati curvatures. Int. J. Geom. Methods Mod. Phys. 19 (3), Paper No. 2250044, 25 pp (2022).
  • [59] Simon, U. Affine Differential Geometry. In Handbook of Differential Geometry, Dillen, F., Verstraelen, L., Eds.; North-Holland: Amsterdam, The Netherlands, 2000; Volume 1, pp. 905–961;
  • [60] Takano, K., Erkan, E., Gülbahar, M.: Locally product-like statistical submersions. Turkish J. Math. 47(2) , 846–869 (2023).
  • [61] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85(1-2) ,171–187 (2006).
  • [62] Takano, K: Statistical manifolds with almost complex structures and its statistical submersions. Tensor (N.S.). 65(2) ,128–142 (2004).
  • [63] Verstraelen, V., Walrave, J., Yaprak, ¸S.: The Minimal Translation Surface in Euclidean Space, Soochow J. Math.textbf20 , 77-82 (1994).
  • [64] Vîlcu, G.E.: Almost product structures on statistical manifolds and para-Kähler-like statistical submersions. Bull. Sci. Math. 171 , Paper No. 103018 (2021).
  • [65] Vîlcu, A.D., Vîlcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy. 17 (9) , 6213–6228 (2015).
  • [66] Vos, P.W.: Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Ann. Inst. Statist. Math. 41 (3), (1989).
  • [67] Yang, D., Fu, Y.: On affine translation surfaces in affine space. J. Math. Anal. Appl. 440 (2), 437-450 (2016).
  • [68] Yoon, D., Lee, C.W., Karacan, M.K.: Some translation surfaces in the 3-dimensional Heisenberg group. Bull. Korean Math. Soc. 50(4) , 1329– 1343(2013).
  • [69] Wan, J., Xie, Z.: Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Ann. Mat. Pura Appl. 4 (3), 1369– 1380 (2023).

The Translation Surfaces on Statistical Manifolds with Normal Distribution

Year 2024, Volume: 17 Issue: 1, 44 - 62, 23.04.2024

Serap Sevim , Cengizhan Murathan

https://doi.org/10.36890/iejg.1378844

Abstract

In this paper, we will investigate translation surfaces on statistical manifolds. Statistical manifolds are mathematical structures that describe the geometric properties of statistical models. We will focus on minimal statistical translation surfaces and then classify statistical translation surfaces of null sectional curvature in three-dimensional hyperbolic statistical manifolds

Keywords

Statistical manifolds statistical submanifolds normal distribution translation surfaces hyperbolic models

References

  • [1] Akbari, H., Malek, F.: On contact metric statistical manifolds. Differential Geom. Appl. 75, Paper No. 101735, 16 pp (2021).
  • [2] Akbari, H., Malek, F.: On the hypersurfaces of almost Hermitian statistical manifolds. Bull. Iranian Math. Soc. 48 (5), 2669–2684 (2022).
  • [3] Amari, S.: Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics,vol. 28. Springer, Berlin (1985).
  • [4] Arslan, K., Bayram, B., Bulca,B., Öztürk, G.: On translation surfaces in 4-dimensional Euclidean space. Acta Comment. Univ. Tartu. Math. 20(2), 123–133 (2016).
  • [5] Al-Solamy, F.R., Bansal, P., Chen, B-Y, Murathan, C., Shahid, M.H.: Geometry of Chen invariants in statistical warped product manifolds. Int. J.Geom. Methods Mod. Phys. 17(6), 2050081, 22 pp (2020).
  • [6] Aydın, M.E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29(3), 465-477 (2015).
  • [7] Aydın, M.: A generalization of translation surfaces with constant curvature in the isotropic space. J. Geom. 107(3), 603–615 (2016).
  • [8] Aydın, M.E., Mihai,A., Mihai, I.: Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sci.,7 (1), 155-166 (2017).
  • [9] Aydın, M. E., Ergut, M.: Affine translation surfaces in the isotropic 3-space. Int. Electron. J. Geom. 10(1), 21–30 (2017).
  • [10] Aydın,M. E., Lopez, E., Vilcu, G-E.: Classification of seperable hypersurfaces with constant sectional curvature. https://arxiv.org/pdf/2309.06025.pdf
  • [11] Aytimur, H., Özgür, C.: On cosymplectic-like statistical submersions. Mediterr. J. Math. 16(3), Paper No. 70, 14 pp (2019).
  • [12] Bansal, P., Uddin, S., Shahid, M.H.: Optimal inequalities for submanifolds in statistical manifolds of quasi constant curvature. Filomat 35(10), 3319–3330 (2021).
  • [13] Blaga, A.M., Crasmareanu, C.: Statistical Structures in Almost Paracontact Geometry. Bulletin of the Iranian Mathematical Society. 44, 1407–1413 (2018).
  • [14] Cai, D.D., Liu, X., Zhang, L.: Inequalities on generalized normalized δ-Casorati curvatures for submanifolds in statistical manifolds of constant curvatures. (Chinese) J. Jilin Univ. Sci. 57(2), 206–212 (2019).
  • [15] Chen, B-Y., Mihai,A., Mihai, I.: A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Results Math. 74(4), 165 (2019).
  • [16] Chen, B-Y., Decu, S., Vîlcu, G-E.: Inequalities for the Casorati curvature of totally real spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Entropy. 23(11), Paper No. 1399, 13 pp (2021).
  • [17] Decu, S., Vîlcu, G-E.: Casorati inequalities for statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ-sectional curvature with semi-symmetric metric connection. Entropy. 24(6),(2022).
  • [18] Dillen, F., Verstraelen, L., Zafindratafa, G.: A generalization of the translation surfaces of Scherk Differential Geometry in honor of Radu Rosca, K.U.L. (1991), 107–109.
  • [19] Erkan, E., Gülbahar, M.: Chen’s basic inequalities for hypersurfaces of statistical Riemannian manifolds. Int. J. Maps Math. 6 (1), 37–53(2023).
  • [20] Furuhata, H.: Hypersurfaces in statistical manifolds. Diff . Geom. Appl. 27, 420-429 (2009).
  • [21] Furuhata, H., Hasegawa, I.: Submanifold theory in holomorphic statistical manifolds, in: Geometry of Cauchy-Riemann Submanifolds, Springer, 2016, pp. 179–215
  • [22] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, K., Shahid, M.H.: Sasakian statistical manifolds. J. Geom. Phys. 117, 179–186 (2017).
  • [23] Furuhata, H., Inoguchi, J., Kobayashi, S.: A characterization of the alpha-connections on the statistical manifold of normal distributions. Inf. Geom. 4(1), 177–188 (2021).
  • [24] Furuhata, H., Hasegawa, I., Satoh, N.: Chen invariants and statistical submanifolds. Commun. Korean Math. Soc. 37(3), 851–864 (2022).
  • [25] Görünü¸s, R., Erken Küpeli,İ. ,Yazla, A., Murathan, C.: A generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds. Int. Electron. J. Geom. 12(1), 43–56 (2019).
  • [26] Malek, F., Akbari, H.: Casorati curvatures of submanifolds in cosymplectic statistical space forms. Bull. Iranian Math. Soc. 46(5), 1389–1403 (2020).
  • [27] Hasanis, T., Lopez, R.: Translation surfaces in Euclidean space with constant Gaussian curvature. Comm. Anal. Geom. 29(6), 1415–1447 (2021).
  • [28] Hasanis, T., Lopez, R.: Classification of separable surfaces with constant Gaussian curvature. Manuscripta Math. 166, 403-417 (2021).
  • [29] Hirohiko, S.: The geometry of Hessian Structures World Scientific Publishing Co. Pte. Ltd. 2007.
  • [30] Humby, C.: https://en.wikipedia.org/wiki/Clive_Humby#cite_note-10
  • [31] Humby, C.:https://medium.com/project-2030/data-is-the-new-oil-a-ludicrous-proposition-1d91bba4f294
  • [32] Inoguchi, J.I, López, R., Munteanu, M.I., Minimal translation surfaces in the Heisenberg group Nil3. Geom. Dedicata. 161 , 221–231 (2012).
  • [33] Jeffreys,H.: An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. Lond. Ser. A. 196,453–461 (1946).
  • [34] Jiang, Y., Wu, F., Zhang, L.: Some results on Kenmotsu statistical manifolds. Hacet. J. Math. Stat. 51(3), 800–816 (2022).
  • [35] Küpeli Erken, İ., Murathan, C., Yazla, A.: Almost cosympletic statistical manifolds. Quaest. Math. 43(2), 265–282 (2020).
  • [36] Kazan, S., Takano, K.: Anti-invariant holomorphic statistical submersions. Results Math. 78(4), Paper No. 128, 18 pp (2023).
  • [37] Lauritzen, S.: In statistical manifolds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, vol. 10, pp. 163–216. IMS Lecture NotesInstitute of Mathematical Statistics, Hayward (1987).
  • [38] Lee, C.W., Yoon, D.W., Lee, J.W.: A pinching theorem for statistical manifolds with Casorati curvatures. J. Nonlinear Sci. Appl. 10 (9), 4908–4914 (2017).
  • [39] Lopez,R.: Minimal translation surfaces in hyperbolic space. Beitr. Algebra Geom. 52, 105–112 (2011).
  • [40] Lopez, R., Munteanu,M. I.: Surfaces with constant mean curvature in Sol geometry. Differential Geom. Appl. 29, S238–S245 (2011).
  • [41] Malek, F., Akbari, H.: Casorati curvatures of submanifolds in cosymplectic statistical space forms. Bull. Iranian Math. Soc. 46 (5), (2020).
  • [42] Milijevic, M.: Totally real statistical submanifolds. Interdiscip. Inf. Sci. 21 , 87-96 (2015).
  • [43] Milijevic, M.: CR statistical submanifolds. Kyushu J. Math. 73(1) , 89–101 (2019).
  • [44] Murathan, C., ¸Sahin, B.: A study of Wintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109(2) ,Paper No. 30, 18 pp.(2018).
  • [45] Nomizu, K., Sasaki, T.: Affine Differential Geometry,Cambridge Univ. Press (1994)1989, KU Leuven, Departement Wiskunde (1991), pp. 107–109.
  • [46] Nomizu, K., Simon, U.: Notes on conjugate connections. Geometry and topology of submanifolds, IV (Leuven, 1991), 152–173,World Sci. Publ., River Edge, NJ, (1992).
  • [47] Bahadır, O.: On lightlike geometry of indefinite Sasakian statistical manifolds. AIMS Math. 6(11) , 12845–12862 (2021).
  • [48] Opozda, B.: A sectional curvature for statistical structures. Linear Alg. Appl. 497, 134–161 (2016).
  • [49] Palmer, M., https://ana.blogs.com/maestros/2006/11/data_is_the_new.html
  • [50] Rao, C.R.: Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 , 81–91 (1945).
  • [51] Samereh, L., Peyghan, E., Mihai, I.: On almost Norden statistical manifolds. Entropy 24(6), 758. 10pg.(2022).
  • [52] Seo, K.: Translation hypersurfaces with constant curvature in space forms. Osaka J. Math. 50 (3), 631–641 (2013).
  • [53] Siddiqui, A. N., Murathan, C., Siddiqi, M. D.: The Chen’s first inequality for submanifolds of statistical warped product manifolds. J. Geom. Phys. 169, Paper No. 104344, 13 pp (2021).
  • [54] Siddiqui, A.N., Shahid, M.H.: Optimizations on statistical hypersurfaces with Casorati curvatures. Kragujevac J. Math. 45(3) , 449–463 (2021).
  • [55] Siddiqui, A.N., Al-Solamy, F.R., Shahid, M.H.,Mihai, I.: On CR-statistical submanifolds of holomorphic statistical manifolds. Filomat 35(11) , 3571–3584 (2021).
  • [56] Siddiqui, A. N., Chen, B.Y., Siddiqi, M. D.: Chen inequalities for statistical submersions between statistical manifolds. Int. J. Geom. Methods Mod. Phys. 18 (4), Paper No. 2150049, 17 pp (2021).
  • [57] Siddiqui, Al.N., Uddin, S., Shahid, M.H.: B.-Y. Chen’s inequality for Kähler-like statistical submersions. Int. Electron. J. Geom. 15 (2), 277–286 (2022).
  • [58] Siddiqui, A.N, Siddiqi, M.D., Alkhaldi, A. H., Akram, A.: Lower bounds on statistical submersions with vertical Casorati curvatures. Int. J. Geom. Methods Mod. Phys. 19 (3), Paper No. 2250044, 25 pp (2022).
  • [59] Simon, U. Affine Differential Geometry. In Handbook of Differential Geometry, Dillen, F., Verstraelen, L., Eds.; North-Holland: Amsterdam, The Netherlands, 2000; Volume 1, pp. 905–961;
  • [60] Takano, K., Erkan, E., Gülbahar, M.: Locally product-like statistical submersions. Turkish J. Math. 47(2) , 846–869 (2023).
  • [61] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85(1-2) ,171–187 (2006).
  • [62] Takano, K: Statistical manifolds with almost complex structures and its statistical submersions. Tensor (N.S.). 65(2) ,128–142 (2004).
  • [63] Verstraelen, V., Walrave, J., Yaprak, ¸S.: The Minimal Translation Surface in Euclidean Space, Soochow J. Math.textbf20 , 77-82 (1994).
  • [64] Vîlcu, G.E.: Almost product structures on statistical manifolds and para-Kähler-like statistical submersions. Bull. Sci. Math. 171 , Paper No. 103018 (2021).
  • [65] Vîlcu, A.D., Vîlcu, G.E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy. 17 (9) , 6213–6228 (2015).
  • [66] Vos, P.W.: Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Ann. Inst. Statist. Math. 41 (3), (1989).
  • [67] Yang, D., Fu, Y.: On affine translation surfaces in affine space. J. Math. Anal. Appl. 440 (2), 437-450 (2016).
  • [68] Yoon, D., Lee, C.W., Karacan, M.K.: Some translation surfaces in the 3-dimensional Heisenberg group. Bull. Korean Math. Soc. 50(4) , 1329– 1343(2013).
  • [69] Wan, J., Xie, Z.: Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Ann. Mat. Pura Appl. 4 (3), 1369– 1380 (2023).
There are 69 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Serap Sevim 0009-0000-5545-7946

Cengizhan Murathan 0000-0002-2273-3243

Early Pub Date April 5, 2024
Publication Date April 23, 2024
Submission Date October 20, 2023
Acceptance Date January 8, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Sevim, S., & Murathan, C. (2024). The Translation Surfaces on Statistical Manifolds with Normal Distribution. International Electronic Journal of Geometry, 17(1), 44-62. https://doi.org/10.36890/iejg.1378844
AMA Sevim S, Murathan C. The Translation Surfaces on Statistical Manifolds with Normal Distribution. Int. Electron. J. Geom. April 2024;17(1):44-62. doi:10.36890/iejg.1378844
Chicago Sevim, Serap, and Cengizhan Murathan. “The Translation Surfaces on Statistical Manifolds With Normal Distribution”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 44-62. https://doi.org/10.36890/iejg.1378844.
EndNote Sevim S, Murathan C (April 1, 2024) The Translation Surfaces on Statistical Manifolds with Normal Distribution. International Electronic Journal of Geometry 17 1 44–62.
IEEE S. Sevim and C. Murathan, “The Translation Surfaces on Statistical Manifolds with Normal Distribution”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 44–62, 2024, doi: 10.36890/iejg.1378844.
ISNAD Sevim, Serap - Murathan, Cengizhan. “The Translation Surfaces on Statistical Manifolds With Normal Distribution”. International Electronic Journal of Geometry 17/1 (April 2024), 44-62. https://doi.org/10.36890/iejg.1378844.
JAMA Sevim S, Murathan C. The Translation Surfaces on Statistical Manifolds with Normal Distribution. Int. Electron. J. Geom. 2024;17:44–62.
MLA Sevim, Serap and Cengizhan Murathan. “The Translation Surfaces on Statistical Manifolds With Normal Distribution”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 44-62, doi:10.36890/iejg.1378844.
Vancouver Sevim S, Murathan C. The Translation Surfaces on Statistical Manifolds with Normal Distribution. Int. Electron. J. Geom. 2024;17(1):44-62.
 Download Cover Image