Pseudo-algebraic Ricci solitons on Einstein nilradicals
-
Zaili Yan
Abstract
We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups.As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.
Funding statement: The author is supported by NSFC (Nos. 11701300, 11626134) and by the K. C. Wong Magna Fund in Ningbo University
Communicated by: P. Eberlein
References
[1] D. V. Alekseevski˘ı, B. N. Kimel’ fel’d, Structure of homogeneous Riemannian spaces with zero Ricci curvature. (Russian) Funkcional. Anal. i PriloŽen. 9 (1975), 5–11. English translation: Functional Anal. Appl. 9 (1975), no. 2, 97–102. MR0402650 Zbl 0316.5304110.1007/BF01075445Search in Google Scholar
[2] W. Batat, K. Onda, Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. J. Geom. Phys. 114 (2017), 138–152. MR3610038 Zbl 1359.5303510.1016/j.geomphys.2016.11.018Search in Google Scholar
[3] A. L. Besse, Einstein manifolds. Springer 1987. MR867684 Zbl 0613.5300110.1007/978-3-540-74311-8Search in Google Scholar
[4] G. Calvaruso, A. Fino, Four-dimensional pseudo-Riemannian homogeneous Ricci solitons. Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550056, 21. MR3349925 Zbl 1405.5305410.1142/S0219887815500565Search in Google Scholar
[5] B. Chow, D. Knopf, The Ricci flow: an introduction, volume 110 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2004. MR2061425 Zbl 1086.5308510.1090/surv/110Search in Google Scholar
[6] D. Conti, F. A. Rossi, Einstein nilpotent Lie groups. J. Pure Appl. Algebra 223 (2019), 976–997. MR3862660 Zbl 1407.5307610.1016/j.jpaa.2018.05.010Search in Google Scholar
[7] J. Heber, Noncompact homogeneous Einstein spaces. Invent. Math. 133 (1998), 279–352. MR1632782 Zbl 0906.5303210.1007/s002220050247Search in Google Scholar
[8] J. Lauret, Ricci soliton homogeneous nilmanifolds. Math. Ann. 319 (2001), 715–733. MR1825405 Zbl 0987.5301910.1007/PL00004456Search in Google Scholar
[9] J. Lauret, Finding Einstein solvmanifolds by a variational method. Math. Z. 241 (2002), 83–99. MR1930986 Zbl 1015.5302810.1007/s002090100407Search in Google Scholar
[10] J. Lauret, A canonical compatible metric for geometric structures on nilmanifolds. Ann. Global Anal. Geom. 30 (2006), 107–138. MR2234091 Zbl 1102.5302110.1007/s10455-006-9015-ySearch in Google Scholar
[11] J. Lauret, Einstein solvmanifolds and nilsolitons. In: New developments in Lie theory and geometry, volume 491 of Contemp. Math., 1–35, Amer. Math. Soc. 2009. MR2537049 Zbl 1186.5305810.1090/conm/491/09607Search in Google Scholar
[12] J. Lauret, Ricci soliton solvmanifolds. J. Reine Angew. Math. 650 (2011), 1–21. MR2770554 Zbl 1210.5305110.1515/crelle.2011.001Search in Google Scholar
[13] J. Lauret, C. Will, Einstein solvmanifolds: existence and non-existence questions. Math. Ann. 350 (2011), 199–225. MR2785768 Zbl 1222.5304810.1007/s00208-010-0552-0Search in Google Scholar
[14] Y. Nikolayevsky, Einstein solvmanifolds and the pre-Einstein derivation. Trans. Amer. Math. Soc. 363 (2011), 3935–3958. MR2792974 Zbl 1225.5304910.1090/S0002-9947-2011-05045-2Search in Google Scholar
[15] K. Onda, Lorentz Ricci solitons on 3-dimensional Lie groups. Geom. Dedicata 147 (2010), 313–322. MR2660582 Zbl 1203.5304410.1007/s10711-009-9456-0Search in Google Scholar
[16] K. Onda, Examples of algebraic Ricci solitons in the pseudo-Riemannian case. Acta Math. Hungar. 144 (2014), 247–265. MR3267185 Zbl 1324.5306310.1007/s10474-014-0426-0Search in Google Scholar
[17] K. Onda, P. E. Parker, Nilsolitons of H-type in the Lorentzian setting. Houston J. Math. 41 (2015), 1137–1151. MR3455351 Zbl 1337.53086Search in Google Scholar
[18] B. O’Neill, Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press 1983. MR0719023 Zbl 0531.53051Search in Google Scholar
[19] S. Rahmani, Métriques de Lorentz sur les groupes de Lie unimodulaires, de dimension trois. J. Geom. Phys. 9 (1992), 295–302. MR1171140 Zbl 0752.5303610.1016/0393-0440(92)90033-WSearch in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- An 𝔽p2-maximal Wiman sextic and its automorphisms
- Pseudo-algebraic Ricci solitons on Einstein nilradicals
- The wobbly divisors of the moduli space of rank-2 vector bundles
- Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
- Betti numbers and pseudoeffective cones in 2-Fano varieties
- The generating rank of a polar Grassmannian
- The Beckman–Quarles theorem via the triangle inequality
- On Huisman’s conjectures about unramified real curves
- Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
- On the Segre invariant for rank two vector bundles on ℙ2
- Lifting coarse homotopies
- How to construct all metric f-K-contact manifolds
- An extremum problem for the power moment of a convex polygon contained in a disc
- Sharply transitive sets in PGL2(K)
Articles in the same Issue
- Frontmatter
- An 𝔽p2-maximal Wiman sextic and its automorphisms
- Pseudo-algebraic Ricci solitons on Einstein nilradicals
- The wobbly divisors of the moduli space of rank-2 vector bundles
- Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
- Betti numbers and pseudoeffective cones in 2-Fano varieties
- The generating rank of a polar Grassmannian
- The Beckman–Quarles theorem via the triangle inequality
- On Huisman’s conjectures about unramified real curves
- Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
- On the Segre invariant for rank two vector bundles on ℙ2
- Lifting coarse homotopies
- How to construct all metric f-K-contact manifolds
- An extremum problem for the power moment of a convex polygon contained in a disc
- Sharply transitive sets in PGL2(K)
