Research Article
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Year 2024, , 6 - 14, 23.04.2024
https://doi.org/10.36890/iejg.1388147

Abstract

References

  • [1] Amari, S.-I.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. 28. Springer-Verlag, New York (1985). https://doi.org/10.1007/978-1-4612-5056-2
  • [2] Amari, S.-I., Nagaoka, H.: Method of Information Geometry. American Mathematical Society: Providence, RI, USA (2000).
  • [3] Antonelli, P.L.: Non-Euclidean allometry and the growth of forests and corals. In: P.L. Antonelli (Eds.), Mathematical Essays on Growth and the Emergence of Form. The University of Alberta Press, Edmonton, AB, 45–57 (1985).
  • [4] Aquib, M., Boyom, M.N., Alkhaldi, A.H., Shahid, M.H.: B.-Y. Chen inequalities for statistical submanifolds in Sasakian statistical manifolds. Lecture Notes in Comput. Sci., 11712 Springer, Cham, 398–406 (2019).
  • [5] Aydin, M.E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29 (3), 465–477 (2015). https://doi.org/10.2298/FIL1503465A
  • [6] Aydin, M.E., Mihai, A., Mihai, I.: Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sc. 7, 155–166 (2017). https://doi.org/10.1007/s13373-016-0086-1
  • [7] Besse, A.L.: Einstein manifolds. Classics in Mathematics. Springer (1987). https://doi.org/10.1007/978-3-540-74311-8
  • [8] Blaga, A.M.: On solitons in statistical geometry. Int. J. Appl. Math. Stat. 58 (4) (2019).
  • [9] Blaga, A.M., Chen, B.-Y.: Gradient solitons on statistical manifolds. J. Geom. Phys. 164, 104195 (2021). https://doi.org/10.1016/j.geomphys.2021.104195
  • [10] Chaki, M.R., Maity, R.K.: On quasi-Einstein manifolds. Publ. Math. Debrecen. 57 (3-4), 297–306 (2000). https://doi.org/10.1023/B:MAHU.0000038977.94711.ab
  • [11] 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), 1399 (2021). https://doi.org/10.3390/e23111399
  • [12] 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). https://doi.org/10.1007/s00025-019-1091-y
  • [13] Crasmareanu, M.: A new approach to gradient Ricci solitons and generalizations. Filomat. 32 (9), 3337–3346 (2018). https://doi.org/10.2298/FIL1809337C
  • [14] Crasmareanu, M.: General adapted linear connections in almost paracontact and contact geometries. Balkan J. Geom. Appl. 25 (2), 12–29 (2020).
  • [15] Deshmukh, S., Al-Sodais, H., Vîlcu, G.-E.: A note on some remarkable differential equations on a Riemannian manifold. J. Math. Anal. Appl. 519 (1), 126778 (2023). https://doi.org/10.1016/j.jmaa.2022.126778
  • [16] Dragomir, S., Ornea, L.: Locally Conformal Kähler Geometry. Progr. in Math. 155. Birkhäuser, Boston (1998). https://doi.org/10.1007/978- 1-4612-2026-8
  • [17] Fischer, A.E., Marsden, J.E.: Manifolds of Riemannian metrics with prescribed scalar curvature. Bull. Amer. Math. Soc. 80, 479–484 (1974).
  • [18] Furuhata, H., Hasegawa, I.: Submanifold theory in holomorphic statistical manifolds. In: Geometry of Cauchy–Riemann Submanifolds. Springer, Singapore, 179–215 (2016).
  • [19] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, K., Shahid, M.H.: Sasakian statistical manifolds. J. Geom. Phys. 117, 179–186 (2017). https://doi.org/10.1016/j.geomphys.2017.03.010
  • [20] Hitchin, N.: The moduli space of special Lagrangian submanifolds. Ann. Scuola Norm. Sup. Pisa. 25 (3-4), 503–515 (1997).
  • [21] Kazan, A.: Conformally-projectively flat trans-Sasakian statistical manifolds. Physica A Stat. Mech. Appl. 535, 122441 (2019). https://doi.org/10.1016/j.physa.2019.122441
  • [22] Kazan, S., Takano, K.: Anti-invariant holomorphic statistical submersions. Results Math. 78, 128 (2023). https://doi.org/10.1007/s00025-023- 01904-8
  • [23] Lauritzen, S.: Statistical manifolds. In: Differential geometry in statistical inference. IMS lecture notes monograph series 1987 (10). Institute of mathematical statistics: Hyward, CA, USA: 96–163. http://www.jstor.org/stable/4355557
  • [24] Lone, M.S., Lone, M.A., Mihai, A.: A characterization of totally real statistical submanifolds in quaternion Kaehler-like statistical manifolds. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 116, 55 (2022). https://doi.org/10.1007/s13398-021-01200-6
  • [25] Matsuzoe, H.: Statistical manifolds and affine differential geometry. Advanced Studies in Pure Mathematics. 57, 303–321 (2010). https://doi.org/10.2969/aspm/05710303
  • [26] Miao, P., Tam, L.-F.: On the volume functional of compact manifolds with boundary with constant scalar curvature. Calc. Var. PDE. 36, 141–171 (2009). https://doi.org/10.1007/s00526-008-0221-2
  • [27] Mihai, A., Mihai, I.: The δ(2, 2)-invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Entropy. 22 (2), 164 (2020). https://doi.org/10.3390/e22020164
  • [28] Mihai, I.: Statistical manifolds and their submanifolds. Results on Chen-like invariants, Contemp. Math. 756, American Mathematical Society, Providence, RI, 163–172 (2020).
  • [29] Murathan, C., ¸Sahin, B.: A study of Wintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109, 30 (2018). https://doi.org/10.1007/s00022-018-0436-0
  • [30] Neac¸su, C.D.: On some optimal inequalities for statistical submanifolds of statistical space forms. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 85 (1), 107–118 (2023).
  • [31] Noda, T.: Symplectic structures on statistical manifolds. J. Aust. Math. Soc. 90 (3), 371–384 (2011). https://doi.org/10.1017/S1446788711001285
  • [32] Peyghan, E., Gezer, A., Nourmohammadifar, L.: Kähler–Norden structures on statistical manifolds. Filomat. 36 (17), 5691–5706 (2022). https://doi.org/10.2298/FIL2217691P
  • [33] 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). https://doi.org/10.2298/FIL2111571S
  • [34] Siddiqui, A.N., Chen, B.-Y., Bahadir, O.: Statistical solitons and inequalities for statistical warped product submanifolds. Mathematics. 7 (9), 797 (2019). https://doi.org/10.3390/math7090797
  • [35] Slesar, V., Vîlcu, G.-E.: Vaisman manifolds and transversally Kähler–Einstein metrics. Ann. Mat. Pura Appl. 202 (4), 1855–1876 (2023). https://doi.org/10.1007/s10231-023-01304-3
  • [36] Takano, K.: Statistical manifolds with almost complex structures and its statistical submersions. Tensor. N.S. 65, 128–142 (2004).
  • [37] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85, 171–187 (2006). https://doi.org/10.1007/s00022-006-0052-2
  • [38] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (4), 757–799 (2011).
  • [39] Vîlcu, A.-D., Vîlcu, G.-E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy. 17 (9), 6213–6228 (2015). https://doi.org/10.3390/e17096213
  • [40] Vîlcu, G.-E.: Almost product structures on statistical manifolds and para-Kähler-like statistical submersions. Bull. Sc. Math. 171, 103018 (2021). https://doi.org/10.1016/j.bulsci.2021.103018
  • [41] Wan, J., Xie, Z.: Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Ann. Mat. Pura Appl. 202 (3), 1369–1380 (2023). https://doi.org/10.1007/s10231-022-01284-w

Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons

Year 2024, , 6 - 14, 23.04.2024
https://doi.org/10.36890/iejg.1388147

Abstract

We put into light some properties of statistical structures with Ricci and Hessian metrics and provide some examples, relating them to Miao-Tam and Fischer-Marsden equations, and to gradient solitons.

References

  • [1] Amari, S.-I.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. 28. Springer-Verlag, New York (1985). https://doi.org/10.1007/978-1-4612-5056-2
  • [2] Amari, S.-I., Nagaoka, H.: Method of Information Geometry. American Mathematical Society: Providence, RI, USA (2000).
  • [3] Antonelli, P.L.: Non-Euclidean allometry and the growth of forests and corals. In: P.L. Antonelli (Eds.), Mathematical Essays on Growth and the Emergence of Form. The University of Alberta Press, Edmonton, AB, 45–57 (1985).
  • [4] Aquib, M., Boyom, M.N., Alkhaldi, A.H., Shahid, M.H.: B.-Y. Chen inequalities for statistical submanifolds in Sasakian statistical manifolds. Lecture Notes in Comput. Sci., 11712 Springer, Cham, 398–406 (2019).
  • [5] Aydin, M.E., Mihai, A., Mihai, I.: Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29 (3), 465–477 (2015). https://doi.org/10.2298/FIL1503465A
  • [6] Aydin, M.E., Mihai, A., Mihai, I.: Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sc. 7, 155–166 (2017). https://doi.org/10.1007/s13373-016-0086-1
  • [7] Besse, A.L.: Einstein manifolds. Classics in Mathematics. Springer (1987). https://doi.org/10.1007/978-3-540-74311-8
  • [8] Blaga, A.M.: On solitons in statistical geometry. Int. J. Appl. Math. Stat. 58 (4) (2019).
  • [9] Blaga, A.M., Chen, B.-Y.: Gradient solitons on statistical manifolds. J. Geom. Phys. 164, 104195 (2021). https://doi.org/10.1016/j.geomphys.2021.104195
  • [10] Chaki, M.R., Maity, R.K.: On quasi-Einstein manifolds. Publ. Math. Debrecen. 57 (3-4), 297–306 (2000). https://doi.org/10.1023/B:MAHU.0000038977.94711.ab
  • [11] 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), 1399 (2021). https://doi.org/10.3390/e23111399
  • [12] 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). https://doi.org/10.1007/s00025-019-1091-y
  • [13] Crasmareanu, M.: A new approach to gradient Ricci solitons and generalizations. Filomat. 32 (9), 3337–3346 (2018). https://doi.org/10.2298/FIL1809337C
  • [14] Crasmareanu, M.: General adapted linear connections in almost paracontact and contact geometries. Balkan J. Geom. Appl. 25 (2), 12–29 (2020).
  • [15] Deshmukh, S., Al-Sodais, H., Vîlcu, G.-E.: A note on some remarkable differential equations on a Riemannian manifold. J. Math. Anal. Appl. 519 (1), 126778 (2023). https://doi.org/10.1016/j.jmaa.2022.126778
  • [16] Dragomir, S., Ornea, L.: Locally Conformal Kähler Geometry. Progr. in Math. 155. Birkhäuser, Boston (1998). https://doi.org/10.1007/978- 1-4612-2026-8
  • [17] Fischer, A.E., Marsden, J.E.: Manifolds of Riemannian metrics with prescribed scalar curvature. Bull. Amer. Math. Soc. 80, 479–484 (1974).
  • [18] Furuhata, H., Hasegawa, I.: Submanifold theory in holomorphic statistical manifolds. In: Geometry of Cauchy–Riemann Submanifolds. Springer, Singapore, 179–215 (2016).
  • [19] Furuhata, H., Hasegawa, I., Okuyama, Y., Sato, K., Shahid, M.H.: Sasakian statistical manifolds. J. Geom. Phys. 117, 179–186 (2017). https://doi.org/10.1016/j.geomphys.2017.03.010
  • [20] Hitchin, N.: The moduli space of special Lagrangian submanifolds. Ann. Scuola Norm. Sup. Pisa. 25 (3-4), 503–515 (1997).
  • [21] Kazan, A.: Conformally-projectively flat trans-Sasakian statistical manifolds. Physica A Stat. Mech. Appl. 535, 122441 (2019). https://doi.org/10.1016/j.physa.2019.122441
  • [22] Kazan, S., Takano, K.: Anti-invariant holomorphic statistical submersions. Results Math. 78, 128 (2023). https://doi.org/10.1007/s00025-023- 01904-8
  • [23] Lauritzen, S.: Statistical manifolds. In: Differential geometry in statistical inference. IMS lecture notes monograph series 1987 (10). Institute of mathematical statistics: Hyward, CA, USA: 96–163. http://www.jstor.org/stable/4355557
  • [24] Lone, M.S., Lone, M.A., Mihai, A.: A characterization of totally real statistical submanifolds in quaternion Kaehler-like statistical manifolds. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 116, 55 (2022). https://doi.org/10.1007/s13398-021-01200-6
  • [25] Matsuzoe, H.: Statistical manifolds and affine differential geometry. Advanced Studies in Pure Mathematics. 57, 303–321 (2010). https://doi.org/10.2969/aspm/05710303
  • [26] Miao, P., Tam, L.-F.: On the volume functional of compact manifolds with boundary with constant scalar curvature. Calc. Var. PDE. 36, 141–171 (2009). https://doi.org/10.1007/s00526-008-0221-2
  • [27] Mihai, A., Mihai, I.: The δ(2, 2)-invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Entropy. 22 (2), 164 (2020). https://doi.org/10.3390/e22020164
  • [28] Mihai, I.: Statistical manifolds and their submanifolds. Results on Chen-like invariants, Contemp. Math. 756, American Mathematical Society, Providence, RI, 163–172 (2020).
  • [29] Murathan, C., ¸Sahin, B.: A study of Wintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109, 30 (2018). https://doi.org/10.1007/s00022-018-0436-0
  • [30] Neac¸su, C.D.: On some optimal inequalities for statistical submanifolds of statistical space forms. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 85 (1), 107–118 (2023).
  • [31] Noda, T.: Symplectic structures on statistical manifolds. J. Aust. Math. Soc. 90 (3), 371–384 (2011). https://doi.org/10.1017/S1446788711001285
  • [32] Peyghan, E., Gezer, A., Nourmohammadifar, L.: Kähler–Norden structures on statistical manifolds. Filomat. 36 (17), 5691–5706 (2022). https://doi.org/10.2298/FIL2217691P
  • [33] 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). https://doi.org/10.2298/FIL2111571S
  • [34] Siddiqui, A.N., Chen, B.-Y., Bahadir, O.: Statistical solitons and inequalities for statistical warped product submanifolds. Mathematics. 7 (9), 797 (2019). https://doi.org/10.3390/math7090797
  • [35] Slesar, V., Vîlcu, G.-E.: Vaisman manifolds and transversally Kähler–Einstein metrics. Ann. Mat. Pura Appl. 202 (4), 1855–1876 (2023). https://doi.org/10.1007/s10231-023-01304-3
  • [36] Takano, K.: Statistical manifolds with almost complex structures and its statistical submersions. Tensor. N.S. 65, 128–142 (2004).
  • [37] Takano, K.: Statistical manifolds with almost contact structures and its statistical submersions. J. Geom. 85, 171–187 (2006). https://doi.org/10.1007/s00022-006-0052-2
  • [38] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (4), 757–799 (2011).
  • [39] Vîlcu, A.-D., Vîlcu, G.-E.: Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions. Entropy. 17 (9), 6213–6228 (2015). https://doi.org/10.3390/e17096213
  • [40] Vîlcu, G.-E.: Almost product structures on statistical manifolds and para-Kähler-like statistical submersions. Bull. Sc. Math. 171, 103018 (2021). https://doi.org/10.1016/j.bulsci.2021.103018
  • [41] Wan, J., Xie, Z.: Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Ann. Mat. Pura Appl. 202 (3), 1369–1380 (2023). https://doi.org/10.1007/s10231-022-01284-w
There are 41 citations in total.

Details

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

Adara M. Blaga 0000-0003-0237-3866

Gabriel Eduard Vilcu This is me 0000-0001-6922-756X

Early Pub Date April 5, 2024
Publication Date April 23, 2024
Submission Date November 9, 2023
Acceptance Date April 1, 2024
Published in Issue Year 2024

Cite

APA Blaga, A. M., & Vilcu, G. E. (2024). Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons. International Electronic Journal of Geometry, 17(1), 6-14. https://doi.org/10.36890/iejg.1388147
AMA Blaga AM, Vilcu GE. Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons. Int. Electron. J. Geom. April 2024;17(1):6-14. doi:10.36890/iejg.1388147
Chicago Blaga, Adara M., and Gabriel Eduard Vilcu. “Statistical Structures With Ricci and Hessian Metrics and Gradient Solitons”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 6-14. https://doi.org/10.36890/iejg.1388147.
EndNote Blaga AM, Vilcu GE (April 1, 2024) Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons. International Electronic Journal of Geometry 17 1 6–14.
IEEE A. M. Blaga and G. E. Vilcu, “Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 6–14, 2024, doi: 10.36890/iejg.1388147.
ISNAD Blaga, Adara M. - Vilcu, Gabriel Eduard. “Statistical Structures With Ricci and Hessian Metrics and Gradient Solitons”. International Electronic Journal of Geometry 17/1 (April 2024), 6-14. https://doi.org/10.36890/iejg.1388147.
JAMA Blaga AM, Vilcu GE. Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons. Int. Electron. J. Geom. 2024;17:6–14.
MLA Blaga, Adara M. and Gabriel Eduard Vilcu. “Statistical Structures With Ricci and Hessian Metrics and Gradient Solitons”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 6-14, doi:10.36890/iejg.1388147.
Vancouver Blaga AM, Vilcu GE. Statistical Structures with Ricci and Hessian Metrics and Gradient Solitons. Int. Electron. J. Geom. 2024;17(1):6-14.