O(n)-invariant Finslerian Almost Ricci Solitons

Abstract

Finslerian almost Ricci solitons are natural extensions of Riemannian almost Ricci solitons. It not only contains Finslerian gradient almost Ricci soliton but also contains the Einstein Finsler metric. In this paper, we find equations that characterize O(n)-invariant Finslerian almost Ricci solitons. As its application, we construct explicitly infinitely many new Finslerian steady Ricci solitons. We also refine a conclusion regarding Randers cigar Ricci soliton in [J. Geom. Anal., 2025, 35(1): Paper No. 11, 41 pp.].

Keywords

Almost Ricci soliton / O(n)-invariant / Finsler metric / Killing field / steady Ricci soliton / 22E46 / 53C30

References

[1] Cao H-D. Recent progress on Ricci solitons. Recent Advances in Geometric Analysis, 2010, Somerville, MA: Int. Press, 1-3811

[2] Cao H-D, Zhou D. On complete gradient shrinking Ricci solitons. J. Differential Geom., 2010, 852175185.

[3] Fu Y, Chen C, Mo X. On spherically symmetric Finsler manifolds with conformal generalized radical fields. Sci. Sin. Math., 2020, 5079991006. in Chinese)

[4] Guo E, Mo X. The Geometry of Spherically Symmetric Finsler Manifolds. SpringerBriefs in Mathematics, 2018, Singapore: Springer

[5] Hamilton RS. The formation of singularities in the Ricci flow. Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1993), 1995, Cambridge, MA: Int. Press, 7-136

[6] Huang L, Liu H, Mo X. On the construction of dually flat Finsler metrics. Osaka J. Math., 2015, 522377391

[7] Huang L, Mo X. On conformal fields of a Randers metric with isotropic S-curvature. Illinois J. Math., 2013, 573685696.

[8] Huang L, Mo X. Projectively flat Finsler metrics with orthogonal invariance. Ann. Polon. Math., 2013, 1073259270.

[9] Li B. On the classification of projectively flat Finsler metrics with constant flag curvature. Adv. Math., 2014, 257: 266-284.

[10] Li Y, Mo X, Wang X. Navigation Finsler metrics on a gradient Ricci soliton. Appl. Math. J. Chinese Univ. Ser. B, 2024, 392266275.

[11] Liu H, Mo X. Examples of Finsler metrics with special curvature properties. Math. Nachr., 2015, 2881315271537.

[12] Mo X., Wang Z., On spherically symmetric Finsler gradient almost Ricci solitons. Differential Geom. Appl., 2026, 103: Paper No. 102375, 12 pp.

[13] Mo X., Zhu H., On a class of projectively flat Finsler metrics of negative constant flag curvature. Internat. J. Math., 2012, 23(8): Paper No. 1250084, 14 pp.

[14] Mo X, Zhu H, Zhu L. On a class of Finsler gradient Ricci solitons. Proc. Amer. Math. Soc., 2023, 151417631773

[15] Mu X, Miao H, Huang L. Isochronous centers and flat Finsler metrics (I). Canad. J. Math., 2025, 77412941314.

[16] Pigola S, Rigoli M, Rimoldi M, Setti AG. Ricci almost solitons. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2011, 104757799

[17] Rutz SF. Symmetry in Finsler spaces. Finsler Geometry (Seattle, WA, 1995), 1996, Providence, RI: Amer. Math. Soc., 289-300. 196

[18] Tan L., Xia Q., On a class of almost weighted Ricci solitons. Results Math., 2025, 80(2): Paper No. 49, 27 pp.

[19] Wang M., Xia Q., On almost square Ricci solitons. J. Geom. Anal., 2026, 36(1): Paper No. 9, 28 pp.

[20] Xia Q., Almost Ricci solitons on Finsler spaces. J. Geom. Anal., 2025, 35(1): Paper No. 11, 41 pp.

[21] Xia Q., Zhu J., Gradient Ricci solitons on Kropina measure spaces. J. Geom. Phys., 2024, 195: Paper No. 105037, 14 pp.

[22] Yin S., Mo X., Some results on complete Finsler measure spaces. J. Math. Anal. Appl., 2021, 497(1): Paper No. 124846, 16 pp.

[23] Zhu H., Rao P., Rigidity of Finsler gradient steady Ricci soliton. Calc. Var. Partial Differential Equations, 2023, 62(4): Paper No. 120, 23 pp.

Rights & permissions

Peking University