Randers Ricci soliton homogeneous nilmanifolds

  • Hamid Reza Salimi Moghaddam


Let $F$ be a left-invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left-invariant Riemannian metric $\hat{\boldsymbol{a}}$ and a vector field $X$ which is $I_{\hat{\boldsymbol{a}}}(M)$-invariant. We show that if the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci soliton if and only if $(N,F)$ is a semialgebraic Ricci soliton.


Akbar-Zadeh, H., Generalized Einstein manifolds, J. Geom. Phys. 17 (1995), no. 4, 342–380. https://doi.org/10.1016/0393-0440(94)00052-2

Bao, D., On two curvature-driven problems in Riemann-Finsler geometry, in “Finsler geometry, Sapporo 2005—in memory of Makoto Matsumoto”, Adv. Stud. Pure Math., vol. 48, Math. Soc. Japan, Tokyo, 2007, pp. 19–71. https://doi.org/10.2969/aspm/04810019

Bao, D., Chern, S.-S., and Shen, Z., An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-1268-3

Bao, D., Robles, C., and Shen, Z., Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004), no. 3, 377–435.

Bidabad, B. and Yarahmadi, M., On quasi-Einstein Finsler spaces, Bull. Iranian Math. Soc. 40 (2014), no. 4, 921–930. https://doi.org/10.1162/coli_a_00204

Chern, S.-S. and Shen, Z., Riemann-Finsler geometry, Nankai Tracts in Mathematics, vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. https://doi.org/10.1142/5263

Chow, B. and Knopf, D., The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. https://doi.org/10.1090/surv/110

Deng, S., Homogeneous Finsler spaces, Springer Monographs in Mathematics, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-4244-8

Deng, S. and Hou, Z., The group of isometries of a Finsler space, Pacific J. Math. 207 (2002), no. 1, 149–155. https://doi.org/10.2140/pjm.2002.207.149

Deng, S. and Hou, Z., Invariant Randers metrics on homogeneous Riemannian manifolds, J. Phys. A 37 (2004), no. 15, 4353–4360. https://doi.org/10.1088/0305-4470/37/15/004

Figula, A. and Nagy, P. T., Isometry classes of simply connected nilmanifolds, J. Geom. Phys. 132 (2018), 370–381. https://doi.org/10.1016/j.geomphys.2018.06.014

Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306.

Hosseini, M. and Salimi Moghaddam, H. R., Classification of Douglas $(\alpha ,\beta )$-metrics on five-dimensional nilpotent Lie groups, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 8, 2050112, 15 pp. https://doi.org/10.1142/S0219887820501121

Jablonski, M., Homogeneous Ricci solitons are algebraic, Geom. Topol. 18 (2014), no. 4, 2477–2486. https://doi.org/10.2140/gt.2014.18.2477

Jablonski, M., Homogeneous Ricci solitons, J. Reine Angew. Math. 699 (2015), 159–182. https://doi.org/10.1515/crelle-2013-0044

Lauret, J., Ricci soliton homogeneous nilmanifolds, Math. Ann. 319 (2001), no. 4, 715–733. https://doi.org/10.1007/PL00004456

Randers, G., On an asymmetrical metric in the fourspace of general relativity, Phys. Rev. (2) 59 (1941), 195–199.

Salimi Moghaddam, H. R., The relation between automorphism group and isometry group of Randers Lie groups, Results Math. 61 (2012), no. 1-2, 137–142. https://doi.org/10.1007/s00025-010-0081-x

Wilson, E. N., Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982), no. 3, 337–346. https://doi.org/10.1007/BF00147318
How to Cite
Salimi Moghaddam, H. (2021). Randers Ricci soliton homogeneous nilmanifolds. MATHEMATICA SCANDINAVICA, 127(1), 100-110. https://doi.org/10.7146/math.scand.a-122610