$\mathcal{Z}$-solitons on statistical submersions
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Abstract
In this research article, we study $\mathcal{Z}$-solitons on statistical submersions with parallel vertical or horizontal distribution. Finally, we study $\mathcal{Z}$-solitons on statistical submersions with conformal or gradient potential vector field.
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References
[1] N. Abe and K. Hasegawa, An affine submersion with horizontal distribution and its applications, Differential Geom. Appl. 14 (3) (2001), 235–250. https://doi.org/10.1016/S0926-2245(01)00034-1 Google Scholar
[2] T. Aubin, Métriques riemanniennes et courbure, J. Differential Geometry 4 (1970), 383–424. http://projecteuclid.org/euclid.jdg/1214429638 Google Scholar
[3] H. Aytimur and C. Özgür, On cosymplectic-like statistical submersions, Mediterr. J. Math. 16 (3) (2019), Paper No. 70, 14 pp. https://doi.org/10.1007/s00009-019-1332-z Google Scholar
[4] O. E. Barndorff-Nielsen and P. E. Jupp, Differential geometry, profile likelihood, sufficiency and composite transformation models, Ann. Statist. 16 (3) (1988), 1009–1043. https://doi.org/10.1214/aos/1176350946 Google Scholar
[5] A. M. Blaga and B.-Y. Chen, Gradient solitons on statistical manifolds, J. Geom. Phys. 164 (2021), Paper No. 104195, 10 pp. https://doi.org/10.1016/j.geomphys.2021.104195 Google Scholar
[6] A. M. Blaga and B.-Y. Chen, Harmonic forms and generalized solitons, Results Math. 79 (1) (2024), Paper No. 16, 18 pp. https://doi.org/10.1007/s00025-023-02041-y Google Scholar
[7] J. P. Bourguignon, Ricci curvature and Einstein metrics, Lecture Notes in Math., 838 (1981), 42–63, Springer, Berlin. Google Scholar
[8] B. B. Chaturvedi and P. Pandey, Study on Special Type of a Weakly Symmetric Kahler Manifold, Differential Geometry-Dynamical System, 17 (2015), 32–37. Google Scholar
[9] B.-Y. Chen, Riemannian submersions, minimal immersions and cohomology class, Proc. Japan Acad. Ser. A Math. Sci. 81 (10) (2005), 162–167. http://projecteuclid.org/euclid.pja/1135791768 Google Scholar
[10] B.-Y. Chen, M. D. Siddqi and A. N. Siddiqui, On Ricci-Bourguignon solitons for statical submersions, Bull. Korean Math. Soc. 62 (1) (2025), 91–110. https://doi.org/10.4134/BKMS.b240063 Google Scholar
[11] B. Chow and D. Knopf, The Ricci flow: An Introduction, Mathematical Surveys and Monographs, vol. 110, AMS, 2004. Google Scholar
[12] U. C. De, C. A. Mantica, L. G. Molinari and Y. J. Suh, On weakly cyclic Z symmetry manifolds, Acta Math. Hunar. 149 (2016), 462–477. Google Scholar
[13] U. C. De, C. A. Mantica and Y. J. Suh, On weakly cyclic Z symmetry manifolds, Acta Math. Hunar. 146 (2015), 153–167. Google Scholar
[14] D. DeTurck, Deforming metrics in direction of their Ricci tensors, J. Diff. Geom. 18 (1983), 157–162. Google Scholar
[15] Ş. Eken Meriç and E. Kılıç, Riemannian submersions whose total manifolds admit a Ricci soliton, Int. J. Geom. Methods Mod. Phys. 16 (12) (2019), 1950196, 12 pp. Google Scholar
[16] M. Falcitelli, S. Ianuş, and A. M. Pastore, Riemannian submersions and related topics, World Sci. Publishing, Inc., River Edge, NJ, 2004. Google Scholar
[17] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715–737. Google Scholar
[18] M. Gülbahar, Ş. Eken Meriç, and E. Kılıç, Sharp inequalities involving the Ricci curvature for Riemannian submersions, Kragujevac J. Math. 41 (2) (2017), 279–293. https://doi.org/10.5937/kgjmath1702279g Google Scholar
[19] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255–306. Google Scholar
[20] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity (Santa Cruz, CA, 1986), 237–262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988. https://doi.org/10.1090/conm/071/954419 Google Scholar
[21] S. Kazan and K. Takano, Anti-invariant holomorphic statistical submersions, Results Math. 78 (4) (2023), Paper No. 128, 18 pp. https://doi.org/10.1007/s00025-023-01904-8 Google Scholar
[22] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. http://projecteuclid.org/euclid.mmj/1028999604 Google Scholar
[23] C. A. Mantica and L. G. Molinari, Weakly Z-symmetric manifolds, Acta Math. Hung. 135 (1) (2012), 80–96. Google Scholar
[24] C. A. Mantica and Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensor, Int. J. Geom. Methods Mod. Phys. 9 (1) (2012), 1250004. Google Scholar
[25] C. A. Mantica and Y. J. Suh, Pseudo Z symmetric spacetimes, J. Math. Phys. 55 (2014), 042502. Google Scholar
[26] P. Pandey, On weakly cyclic generalized Z-symmetric manifolds, Natl. Acad. Sci. Lett. 43 (2020), 347–350. Google Scholar
[27] B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Cent. Eur. J. Math. 8 (3) (2010), 437–447. https://doi.org/10.2478/s11533-010-0023-6 Google Scholar
[28] B. Sahin, Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and Their Applications, Elsevier/Academic Press, London, 2017. Google Scholar
[29] M. D. Siddiqi, A. H. Alkhaldi, M. A. Khan, and A. N. Siddiqui, Conformal η-Ricci solitons on Riemannian submersions under canonical variation, Axioms 11 (2022), 594. Google Scholar
[30] M. D. Siddiqi, F. Mofarreh, M. A. Akyol, and A. H. Hakami, η-Ricci-Yamabe Solitons along Riemannian Submersions, Axioms 12 (2023), 796. https://doi.org/10.3390/axioms12080796 Google Scholar
[31] M. D. Siddiqi, A. N. Siddiqui, F. Mofarreh, and H. Aytimur, A study of Kenmotsu-like statistical submersions, Symmetry 14 (8) (2022), 1–13. Google Scholar
[32] A. N. Siddiqui, B.-Y. Chen, and O. Bahadir, Statistical solitons and inequalities for statistical warped product submanifolds, Mathematics 7 (9) (2019), Paper No. 797, 19 pp. https://doi.org/10.3390/math7090797 Google Scholar
[33] A. N. Siddiqui Diop, M. D. Siddiqi, A. H. Alkhaldi, and A. Ali, Lower bounds on statistical submersions with vertical Casorati curvatures, Int. J. Geom. Methods Mod. Phys. 19 (3) (2022), Paper No. 2250044, 25 pp. https://doi.org/10.1142/S021988782250044X Google Scholar
[34] K. Takano, Statistical manifolds with almost complex structures and its statistical submersions, Tensor (N.S.) 65 (2) (2004), 123–137. Google Scholar
[35] K. Takano, Examples of the statistical submersion on the statistical model, Tensor (N.S.) 65 (2) (2004), 170–178. Google Scholar
[36] K. Takano, Statistical manifolds with almost contact structures and its statistical submersions, J. Geom. 85 (1-2) (2006), 171–187. https://doi.org/10.1007/s00022-006-0052-2 Google Scholar
[37] K. Takano, E. Erkan, and M. Gülbahar, Locally product-like statistical submersions, Turkish J. Math. 47 (2) (2023), 846–869. https://doi.org/10.55730/1300-0098-3397 Google Scholar
[38] K. Takano and S. Kazan, Statistical submersions with parallel almost complex structures, Mediterr. J. Math. 21 (3) (2024), Paper No. 109, 26 pp. https://doi.org/10.1007/s00009-024-02621-4 Google Scholar
[39] G.-E. Vîlcu, Almost product structures on statistical manifolds and para-Kähler-like statistical submersions, Bull. Sci. Math. 171 (2021), Paper No. 103018, 21 pp. https://doi.org/10.1016/j.bulsci.2021.103018 Google Scholar
[40] A.-D. Vîlcu and G.-E. Vîlcu, Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions, Entropy 17 (9) (2015), 6213–6228. https://doi.org/10.3390/entropy17096213 Google Scholar
[41] B. Watson, G,G′-Riemannian submersions and nonlinear gauge field equations of general relativity, Global Analysis–Analysis on Manifolds, 324–349, Teubner-Texte Math., vol. 57, Teubner, Leipzig, 1983. Google Scholar