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DOI: 10.46698/s0610-2042-8837-a
Riemann Solitons on Lorentzian pr-Wave Manifolds
Azami, S.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 3.
Abstract: In this research, we delve into the study of Riemann solitons within the context of Lorentzian pr-wave manifolds. Our investigation aims to systematically classify all vector fields that are associated with Riemann solitons on these specific manifolds. A significant aspect of our work is to identify which of these vector fields can be categorized as the gradient type. Furthermore, we demonstrate that any vector field that has the potential to be linked with Riemann solitons can be classified into several categories, namely, Ricci bi-conformal, Killing, and Ricci collineation vector fields. This classification not only enhances our understanding of the geometric properties of these manifolds but also sheds light on the intricate relationships between the various types of vector fields and Riemann solitons. The analysis also gives explicit conditions for the steady and non-steady cases. Gradient Riemann solitons occur only in the steady case, while the potential vector fields may additionally be Killing, Ricci collineation, or Ricci bi-conformal vector fields.
For citation: Azami, S. Riemann Solitons on Lorentzian pr-Wave Manifolds, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp. 5-16. DOI 10.46698/s0610-2042-8837-a
1. Hamilton, R. S. The Ricci Flow on Surfaces, Contemporary Mathematics, vol. 71, Providence, RI, American Mathematical Society, 1988, pp. 237-262. DOI: 10.1090/conm/071/954419.
2. Calin, C. and Crasmareanu, M. From the Eisenhart Problem to Ricci Solitons in f-Kenmotsu Manifolds, Bulletin of the Malaysian Mathematical Sciences Society, 2010, vol. 33, no. 3, pp. 361-368.
3. Brozos-Vázquez, M., Calvaruso, G., García-Rio, E. and Gavino-Fernández, S. Three-Dimensional Lorentzian Homogeneous Ricci Solitons, Israel Journal of Mathematics, 2012, vol. 188, no. 1, pp. 385-403. DOI: 10.1007/s11856-011-0124-3.
4. Calvaruso, G. Homogeneous Structures on Three-Dimensional Lorentzian Manifolds, Journal of Geometry and Physics, 2007, vol. 57, no. 4, pp. 1279-1291. DOI: 10.1016/j.geomphys.2006.10.005.
5. Akbar, M. M. and Woolgar, E. Ricci Solitons and Einstein-Scalar Field Theory, Classical and Quantum Gravity, 2009, vol. 26, no. 5, article no. 055015, 14 pp. DOI: 10.1088/0264-9381/26/5/055015.
6. Chave, T. and Valent, G. Quasi-Einstein Metrics and Their Renormalizability Properties, Helvetica Physica Acta, 1996, vol. 69, no. 3, pp. 344-347. DOI: 10.5169/seals-116953.
7. Chave, T. and Valent, G. On a Class of Compact and Non-Compact Quasi-Einstein Metrics and Their Renormalizability Properties, Nuclear Physics B, 1996, vol. 478, no. 3, pp. 758-778. DOI: 10.1016/0550-3213(96)00341-0.
8. Friedan, D. H. Nonlinear Models in 2+ε Dimensions, Annals of Physics, 1985, vol. 163, no. 2, pp. 318-419. DOI: 10.1016/0003-4916(85)90384-7.
9. Kholodenko, A. L. Towards Physically Motivated Proofs of the Poincaré and Geometrization Conjectures, Journal of Geometry and Physics, 2008, vol. 58, no. 2, pp. 259-290. DOI: 10.1016/j.geomphys.2007.11.003.
10. Udrişte, C. Riemann Flow and Riemann Wave, Annals of West University of Timişoara. Series Mathematics and Informatics, 2010, vol. 48, no. 1-2, pp. 265-274.
11. Udrişte, C. Riemann Flow and Riemann Wave via Bialternate Product Riemannian Metric, arXiv:1112.4279 [math.AP], 2012. DOI: 10.48550/arXiv.1112.4279.
12. Hirică, I. E. and Udrişte, C. Ricci and Riemann Solitons, Balkan Journal of Geometry and Its Applications, 2016, vol. 21, no. 2, pp. 35-44.
13. Biswas, G. G., Chen, X. and De, U. C. Riemann Solitons on Almost Co-Kahler Manifolds, Filomat, 2022, vol. 36, no. 4, pp. 1403-1413. DOI: 10.2298/FIL2204403B.
14. Devaraja, M. N., Aruna Kumara, H. and Venkatesha, V. Riemann Soliton Within the Framework of Contact Geometry, Quaestiones Mathematicae, 2021, vol. 44, no. 5, pp. 637-651. DOI: 10.2989/16073606.2020.1732495.
15. Venkatesha, V., Aruna Kumara, H. and Naik, D. M. Riemann Solitons and Almost Riemann Solitons on Almost Kenmotsu Manifolds, International Journal of Geometric Methods in Modern Physics, 2020, vol. 17, no. 7, article no. 2050105. DOI: 10.1142/S0219887820501054.
16. Azami, S. and Jafari, M. Riemann Solitons on Relativistic Space-Times, Gravitation and Cosmology, 2024, vol. 30, no. 3, pp. 306-311. DOI: 10.1134/S020228932470021X.
17. Azami, S. and Jafari, M. Riemann Solitons on Perfect Fluid Spacetimes in f(r, T)-Gravity, Rendiconti del Circolo Matematico di Palermo. Series 2, 2025, vol. 74, no. 1, article no. 2, 13 pp. DOI: 10.1007/s12215-024-01116-1.
18. Bakshi, M. R. and Baishya, K. K. Four Classes of Riemann Solitons on α-Cosymplectic Manifolds, Afrika Matematika, 2021, vol. 32, no. 3-4, pp. 577-588. DOI: 10.1007/s13370-020-00846-6.
19. Blaga, A. M. Remarks on Almost Riemann Solitons with Gradient or Torse-Forming Vector Field, Bulletin of the Malaysian Mathematical Sciences Society, 2021, vol. 44, no. 5, pp. 3215-3227. DOI: 10.1007/s40840-021-01108-9.
20. Borges, V. On Complete Gradient Schouten Solitons, Nonlinear Analysis, 2022, vol. 221, article no. 112883, 15 pp. DOI: 10.1016/j.na.2022.112883.
21. De, K. and De, U. C. A Note on Almost Riemann Solitons and Gradient Almost Riemann Solitons, Afrika Matematika, 2022, vol. 33, no. 3, article no. 74, 10 pp. DOI: 10.1007/s13370-022-01010-y.
22. De, U. C., Sardar, A. and Sarkar, A. Some Conformal Vector Fields and Conformal Ricci Solitons on N(k)-Contact Metric Manifolds, AUT Journal of Mathematics and Computing, 2021, vol. 2, no. 1, pp. 61-71. DOI: 10.22060/ajmc.2021.19220.1043.
23. Azami, S. and De, U. C. Ricci Bi-Conformal Vector Fields on Lorentzian Four-Dimensional Generalized Symmetric Spaces, Filomat, 2024, vol. 38, no. 9, pp. 3157-3164. DOI: 10.2298/FIL2409157A.
24. Azami, S. and De, U. C. Ricci Bi-Conformal Vector Fields on Lorentzian Five-Dimensional Two-Step Nilpotent Lie Groups, Hacettepe Journal of Mathematics and Statistics, 2024, vol. 53, no. 4, pp. 1118-1129. DOI: 10.15672/hujms.1294973.
25. Azami, S. and Fasihi-Ramandi, G. Ricci Bi-Conformal Vector Fields on Siklos Spacetimes, Mathematics Interdisciplinary Research, 2024, vol. 9, no. 1, pp. 45-76. DOI: 10.22052/MIR.2023.252926.1408.
26. Azami, S. and Jafari, M. Ricci Bi-Conformal Vector Fields on Homogeneous Gödel-Type Spacetimes, Journal of Nonlinear Mathematical Physics, 2023, vol. 30, no. 4, pp. 1700-1718. DOI: 10.1007/s44198-023-00151-3.
27. Sohrabpour, M. and Azami, S. Ricci Bi-Conformal Vector Fields on Lorentzian Walker Manifolds of Low Dimension, Lobachevskii Journal of Mathematics, 2023, vol. 44, no. 12, pp. 5437-5443. DOI: 10.1134/S1995080223120338.
28. Brinkmann, H. W. Einstein Spaces Which Are Mapped Conformally on Each Other, Mathematische Annalen, 1925, vol. 94, no. 1, pp. 119-145. DOI: 10.1007/BF01208647.
29. Schimming, R. Riemannsche Raume mit ebenfrontiger und mit ebener Symmetrie, Mathematische Nachrichten, 1974, vol. 59, no. 1-6, pp. 129-162. DOI: 10.1002/mana.19740590111.
30. Leistner, T. Conformal Holonomy of C-Spaces, Ricci-Flat, and Lorentzian Manifolds, Differential Geometry and Its Applications, 2006, vol. 24, no. 5, pp. 458-478. DOI: 10.1016/j.difgeo.2006.04.008.
31. Batat, W. Curvature Properties and Ricci Solitons of Lorentzian pr-Waves Manifolds, Journal of Geometry and Physics, 2014, vol. 75, pp. 7-16. DOI: 10.1016/j.geomphys.2013.08.014.
32. Carroll, S. M. Spacetime and Geometry: An Introduction to General Relativity, San Francisco, Addison Wesley, 2004, xiv+513 p.
33. Deshmukh, S. Geometry of Conformal Vector Fields, Arab Journal of Mathematical Sciences, 2017, vol. 23, no. 1, pp. 44-73. DOI: 10.1016/j.ajmsc.2016.09.003.
34. Yano, K. The Theory of Lie Derivatives and Its Applications, Mineola, NY, Dover Publications, 2020, x+299 p.
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