konuralp j. math. Konuralp Journal of Mathematics 2147-625X Mehmet Zeki SARIKAYA Applied Mathematics (Other) Uygulamalı Matematik (Diğer) $(h,\eta)$-Ricci-Bourguignon Soliton on the Poincare Disk $\mathbb{D}^2$ Diop Mafal Ndiaye Université Cheikh Anta Diop Bousso Abdou Université Cheikh Anta Diop Ndiaye Ameth Université Cheikh Anta Diop https://orcid.org/0000-0002-3979-8916 Mandal Abhıjıt Raiganj Surendranath Mahavidyalaya, Raiganj, West Bengal, India 04 30 2026 14 1 217 228 11 12 2025 03 14 2026 Copyright © 2013, Konuralp Journal of Mathematics 2013 Konuralp Journal of Mathematics

We present a new concept of $(h,\eta)$-Ricci-Bourguignon Soliton on a Riemannian manifold $(M,g)$ defined by \begin{equation}\label{eq1} \mathrm{Ric}+\frac{h}{2}\,\mathcal{L}_X g=(\lambda+\rho\,\mathrm{Scal})\,g + \omega\,\eta\otimes\eta, \end{equation} where $\eta$ is a $1$-form, $h$ is a non-zero smooth function, and $\lambda$, $\rho$ and $\omega$ are real constants, denoted by \textbf{$(M,g,X,\lambda,\rho,\omega)$}. We then explicitly write this equation on the Poincar\'e disk $\mathbb{D}^2$ equipped with the hyperbolic metric in polar coordinates.

 η)-Ricci-Bourguignon Soliton Hyperbolic Disk Beltrami-Laplacian. Poincaré Metric [1] A. Bousso, M. Traore, and A. Ndiaye, “Integral formulas in h-Almost Ricci-Bourguignon Solitons,” arXiv preprint arXiv:2501.12345, 2025. [2] Bourguignon, J.P., Ricci curvature and Einstein metrics, Global differential geometry and global analysis, (1981), 42-63. [3] B. B. Chaturvedi, P. Bhagat, and M. N. I. Khan, h-Ricci-Bourguignon solitons in an almost pseudo-W8 flat and M-projective flat symmetric Lorentzian Kahler space-time manifold, AIMS Mathematics, vol: 10, no. 1 (2025), 111-127. [4] S. K. Chaubey and A. Sharma, Almost h-Ricci-Bourguignon solitons on submersions, The European Mathematical Society Publishing House, 2022. [5] Besse,A.L., Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin 2008. [6] Cao, H-D., Zhou,D., On complete gradient shrinking Ricci solitons. J. Differential Geom. Vol:85, No 2 (2010), 175-185. [7] Fernandez-Lopez, M., Garcia-Rio, E., Rigidity of shrinking Ricci solitons. Math. Z. Vol:269, No 1-2 (2011), 461-466. [8] Hamilton, R. S.,Three-manifolds with positive Ricci curvature. J. Differential Geom. Vol:17, No 2 (1982), 255-306. [9] Hamilton, R.S., The Ricci flow on surfaces, Mathematics and general relativity. Contemp. Math. Vol:71 (1988), 237-262. [10] Manolo, E., Gabriele, L. N., Carlo, M. Ricci solitons - The equation point of view. https://doi.org/10.48550/arXiv.math/0607546 [11] Munteanu,O., Sesum, N., On Gradient Ricci Solitons. J. Geometric Analysis. Vol:23 (2013),539-561. [12] Petersen, P., Wylie, W., Rigidity of gradient Ricci solitons. Pacific Journal of Mathematics. Vol:241 (2009),329-345. [13] Petersen, P., Wylie,W., On gradient Ricci solitons with symmetry. Proc. Amer. Math. Soc. Vol:137 (2009), 2085-2092. [14] Petersen, P., Wylie,W., On the classification of gradient Ricci solitons. Geom. Topol. Vol:14 (2010), 2277-2300. [15] Traore, M., Tas¸tan, H. M. and Aydin, S. G., On almost h-Ricci-Bourguignon solitons, Periodica Polytechnica Ser. Mech. Eng., vol:68, No. 4 (2024) 319-325.