Research Article
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Year 2024, Volume: 17 Issue: 1, 6 - 14, 23.04.2024

Adara M. Blaga , Gabriel Eduard Vilcu

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, Volume: 17 Issue: 1, 6 - 14, 23.04.2024

Adara M. Blaga , Gabriel Eduard Vilcu

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.

Keywords

Statistical structure affine connection gradient soliton curvature

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 Volume: 17 Issue: 1

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.
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