Curvature properties of \(\alpha\)-cosymplectic manifolds with \(\ast\)-\(\eta\)-Ricci-Yamabe solitons

Vandana; Rajeev Budhiraja; Aliya Naaz Siddiqui Diop

doi:10.56754/0719-0646.2601.091

DOI:

https://doi.org/10.56754/0719-0646.2601.091

Abstract

In this research article, we study \(\ast\)-\(\eta\)-Ricci-Yamabe solitons on an \(\alpha\)-cosymplectic manifold by giving an example in the support and also prove that it is an \(\eta\)-Einstein manifold. In addition, we investigate an \(\alpha\)-cosymplectic manifold admitting \(\ast\)-\(\eta\)-Ricci-Yamabe solitons under some conditions. Lastly, we discuss the concircular, conformal, conharmonic, and \(W_2\)-curvatures on the said manifold admitting \(\ast\)-\(\eta\)-Ricci-Yamabe solitons.

Keywords

∗-η-Ricci-Yamabe soliton , α-cosymplectic manifold , curvature , η-Einstein manifold

Mathematics Subject Classification:

53B20 , 53C21 , 53C44 , 53C25 , 53C50 , 53D35

Vandana Department of Mathematics and Humanities, Maharishi Markandeshwar (Deemed to be University), Mullana, 133207, Ambala-Haryana, India. https://orcid.org/0009-0005-6302-5533
Rajeev Budhiraja Department of Mathematics and Humanities, Maharishi Markandeshwar (Deemed to be University), Mullana, 133207, Ambala-Haryana, India. https://orcid.org/0009-0004-6974-5681
Aliya Naaz Siddiqui Diop Division of Mathematics, School of Basic Sciences, Galgotias University, Greater Noida, Uttar Pradesh 203201, India. https://orcid.org/0000-0003-3895-7548

Pages: 91–105
Date Published: 2024-04-06
Vol. 26 No. 1 (2024)

D. E. Blair, Contact manifolds in Riemannian geometry, ser. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1976, vol. 509.

J. T. Cho and M. Kimura, “Ricci solitons and real hypersurfaces in a complex space form,” Tohoku Math. J. (2), vol. 61, no. 2, pp. 205–212, 2009, doi: 10.2748/tmj/1245849443.

D. Dey, “Almost Kenmotsu metric as Ricci-Yamabe soliton,” 2020, arXiv:2005.02322.

S. Dey, P. L.-i. Laurian-ıoan, and S. Roy, “Geometry of ∗-k-ricci-yamabe soliton and gradi- ent ∗-k-ricci-yamabe soliton on kenmotsu manifolds,” Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 4, p. 907–922, 2023, doi: 10.15672/hujms.1074722.

S. Dey and S. Roy, “∗-η-Ricci soliton within the framework of Sasakian manifold,” J. Dyn. Syst. Geom. Theor., vol. 18, no. 2, pp. 163–181, 2020, doi: 10.1080/1726037X.2020.1856339.

L. P. Eisenhart, Riemannian Geometry. Princeton University Press, Princeton, NJ, 1949, 2d printing.

S. Güler and M. Crasmareanu, “Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy,” Turkish J. Math., vol. 43, no. 5, pp. 2631–2641, 2019, doi: 10.3906/mat-1902-38.

T. Hamada, “Real hypersurfaces of complex space forms in terms of Ricci ∗-tensor,” Tokyo J. Math., vol. 25, no. 2, pp. 473–483, 2002, doi: 10.3836/tjm/1244208866.

R. S. Hamilton, “Three-manifolds with positive Ricci curvature,” J. Differential Geometry, vol. 17, no. 2, pp. 255–306, 1982.

R. S. Hamilton, “The Ricci flow on surfaces,” in Mathematics and general relativity (Santa Cruz, CA, 1986), ser. Contemp. Math. Amer. Math. Soc., Providence, RI, 1988, vol. 71, pp. 237–262, doi: 10.1090/conm/071/954419.

A. Haseeb, D. G. Prakasha, and H. Harish, “∗-conformal η-Ricci solitons on α-cosymplectic manifolds,” IJAA, vol. 19, no. 2, pp. 165–179, 2021, Art. ID 5718736, doi: 10.28924/2291-8639.

A. Haseeb, R. Prasad, and F. Mofarreh, “Sasakian manifolds admitting ∗-η-Ricci-Yamabe solitons,” Adv. Math. Phys., 2022, Art. ID 5718736, doi: 10.1155/2022/5718736.

Y. Ishii, “On conharmonic transformations,” Tensor (N.S.), vol. 7, pp. 73–80, 1957.

Z. Olszak, “Locally conformal almost cosymplectic manifolds,” Colloq. Math., vol. 57, no. 1, pp. 73–87, 1989, doi: 10.4064/cm-57-1-73-87.

Z. Olszak and R. Roşca, “Normal locally conformal almost cosymplectic manifolds,” Publ. Math. Debrecen, vol. 39, no. 3-4, pp. 315–323, 1991, doi: 10.5486/pmd.1991.39.3-4.12.

G. P. Pokhariyal and R. S. Mishra, “Curvature tensors’ and their relativistics significance,” Yokohama Math. J., vol. 18, pp. 105–108, 1970.

S. Roy, S. Dey, and A. Bhattacharyya, “Some results on η-Yamabe solitons in 3-dimensional trans-Sasakian manifold,” Carpathian Math. Publ., vol. 14, no. 1, pp. 158–170, 2022, doi: 10.15330/cmp.14.1.158-170.

S. Roy, S. Dey, A. Bhattacharyya, and M. D. Siddiqi, “∗-η-Ricci-Yamabe solitons on α- cosymplectic manifolds with a quarter-symmetric metric connection,” 2021, arXiv:2109.04700.

R. Sharma, “Certain results on K-contact and (k,μ)-contact manifolds,” J. Geom., vol. 89, no. 1-2, pp. 138–147, 2008, doi: 10.1007/s00022-008-2004-5.

M. D. Siddiqi and M. A. Akyol, “η-Ricci-Yamabe soliton on Riemannian submersions from Riemannian manifolds,” 2021, arXiv:2004.14124.

A. N. Siddiqui and M. D. Siddiqi, “Almost Ricci-Bourguignon solitons and geometrical structure in a relativistic perfect fluid spacetime,” Balkan J. Geom. Appl., vol. 26, no. 2, pp. 126–138, 2021.

S.-i. Tachibana, “On almost-analytic vectors in almost-Kählerian manifolds,” Tohoku Math. J. (2), vol. 11, pp. 247–265, 1959, doi: 10.2748/tmj/1178244584.

H. I. Yoldaş, “On Kenmotsu manifolds admitting η-Ricci-Yamabe solitons,” Int. J. Geom. Methods Mod. Phys., vol. 18, no. 12, 2021, Art. ID 2150189, doi: 10.1142/S0219887821501899.

H. I. Yoldaş, “Some results on α-cosympletic manifolds,” Bull. Transilv. Univ. Braşov Ser. III. Math. Comput. Sci., vol. 1(63), no. 2, pp. 115–128, 2021, doi: 10.31926/but.mif.2021.1.63.2.10.

H. I. Yoldaş, “Some soliton types on Riemannian manifolds,” Rom. J. Math. Comput. Sci., vol. 11, no. 2, pp. 13–20, 2021.

H. I. Yoldaş, “Some classes of Ricci solitons on Lorentzian α-Sasakian manifolds,” Differ. Geom. Dyn. Syst., vol. 24, pp. 232–244, 2022, doi: 10.3390/e24020244.

P. Zhang, Y. Li, S. Roy, S. Dey, and A. Bhattacharyya, “Geometrical structure in a perfect fluid spacetime with conformal Ricci–Yamabe soliton,” Symmetry, vol. 14, no. 3, 2022, Art. ID 594, doi: 10.3390/sym14030594.