Almost Ricci soliton in $Q^{m^{\ast}}$

Document Type : Original Article

Authors

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Abstract

In this paper, we will focus our attention on the structure of $h$-almost Ricci solitons on complex hyperbolic quadric. We will prove non-existence a contact real hypersurface in the complex hyperbolic quadric $Q^{m^*}, m\geq 3$, admitting the gradient almost Ricci soliton. Moreover, the gradient almost Ricci soliton function $f$ is trivial.

Keywords

Main Subjects


[1] J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, Internat. J.
Math., 24 (2013), pp. 1350050, 18.
[2] J. Berndt and Y. J. Suh, Contact hypersurfaces in K¨ahler manifolds, Proc. Amer. Math. Soc., 143 (2015),
pp. 2637–2649.
[3] H.-D. Cao, Recent progress on Ricci solitons, in Recent advances in geometric analysis, vol. 11 of Adv. Lect.
Math. (ALM), Int. Press, Somerville, MA, 2010, pp. 1–38.
[4] B.-Y. Chen, Pseudo-Riemannian geometry, δ-invariants and applications, World Scientific Publishing Co.
Pte. Ltd., Hackensack, NJ, 2011. With a foreword by Leopold Verstraelen.
[5] S. Deshmukh, Almost Ricci solitons isometric to spheres, Int. J. Geom. Methods Mod. Phys., 16 (2019),
pp. 1950073, 9.
[6] S. Deshmukh and H. Al-Sodais, A note on almost Ricci solitons, Anal. Math. Phys., 10 (2020), pp. Paper
No. 76, 11.
[7] H. Faraji, S. Azami, and G. Fasihi-Ramandi, h-almost Ricci solitons with concurrent potential fields,
AIMS Math., 5 (2020), pp. 4220–4228.
[8] J. Gasqui and H. Goldschmidt, On the geometry of the complex quadric, Hokkaido Math. J., 20 (1991),
pp. 279–312.
[9] J. N. Gomes, Q. Wang, and C. Xia, On the h-almost Ricci soliton, J. Geom. Phys., 114 (2017), pp. 216–222.
[10] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982), pp. 255–
306.
[11] , The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), vol. 71 of
Contemp. Math., Amer. Math. Soc., Providence, RI, 1988, pp. 237–262.
[12] , The formation of singularities in the Ricci flow, in Surveys in differential geometry, Vol. II (Cambridge,
MA, 1993), Int. Press, Cambridge, MA, 1995, pp. 7–136.
[13] S. K. Hui and D. Chakraborty, Ricci almost solitons on concircular Ricci pseudosymmetric β-Kenmotsu
manifolds, Hacet. J. Math. Stat., 47 (2018), pp. 579–587.
[14] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Wiley Classics Library, John
Wiley & Sons, Inc., New York, 1996. Reprint of the 1969 original, A Wiley-Interscience Publication.
[15] N. Sesum, Convergence of the Ricci flow toward a soliton, Comm. Anal. Geom., 14 (2006), pp. 283–343.
[16] Y. J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Adv. in
Appl. Math., 50 (2013), pp. 645–659.
[17] , Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow, Commun. Contemp.
Math., 20 (2018), pp. 1750031, 20.
[18] , Pseudo-anti commuting Ricci tensor for real hypersurfaces in the complex hyperbolic quadric, Sci. China
Math., 62 (2019), pp. 679–698.
[19] , Real hypersurfaces in the complex quadric with Reeb parallel Ricci tensor, J. Geom. Anal., 29 (2019),
pp. 3248–3269.
[20] , Ricci soliton and pseudo-Einstein real hypersurfaces in the complex hyperbolic quadric, J. Geom. Phys.,
162 (2021), pp. Paper No. 103888, 16.
[21] , Ricci-Bourguignon solitons on real hypersurfaces in the complex hyperbolic quadric, Rev. R. Acad. Cienc.
Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116 (2022), pp. Paper No. 110, 23.