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Double Extensions on Riemannian Ricci Nilsolitons

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Abstract

In this paper, we study left invariant Lorentz algebraic Ricci solitons on nilpotent Lie groups. Based on the concept of Lorentz datum and the technique of double extensions, we are able to construct Lorentz Ricci nilsolitons from any Riemannian Ricci nilsoliton. Conversely, we prove that any Lorentz Ricci nilsoliton with degenerate center is a double extension of a Riemannian Ricci nilsoliton with respect to a Lorentz data. Moreover, we provide a strategy to classify Lorentz Ricci nilsolitons with degenerate center. As an application, we present a large number of Lorentz Ricci nilsolitons based on Abelian Riemannian Ricci solitons. Finally, a family of left invariant Lorentz Ricci nilsolitons on a 9-dimensional nilpotent Lie algebra is given.

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References

  1. Alekseevsky, D., Kimel’fel’d, B.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funct. Anal. Appl. 9, 97–102 (1975)

    MathSciNet  MATH  Google Scholar 

  2. Bajo, I., Benayadi, S., Medina, A.: Symplectic structures on quadratic Lie algebras. J. Algebra 316, 174–188 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Barco, V.D., Ovando, G.P.: Free nilpotent Lie algebras admitting ad-invariant metrics. J. Algebra 366, 205–216 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Batat, W., Onda, K.: Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. J. Geom. Phys. 114, 138–152 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Benayadi, S., Elduqe, A.: Classification of quadratic Lie algebras of low dimension. J. Math. Phys. 55, 0811703 (2014). https://doi.org/10.1063/1.4890646

    Article  MathSciNet  Google Scholar 

  6. Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)

    Book  Google Scholar 

  7. Bordemann, M.: Nondegenerate invariant bilinear forms on nonassociative algebras. Acta Math. Univ. Comen. 66(2), 151–201 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Boucetta, M., Tibssirte, O.: On Einstein Lorentzian nilpotent Lie groups. J. Pure Appl. Algebra 224(12), 106443(22 pages) (2020)

    MathSciNet  MATH  Google Scholar 

  9. Calvaruso, G., Fino, A.: Four-dimensional pseudo-Riemannian homogeneous Ricci solitons. Int. J. Geom Methods Mod. Phys. 12(05), 1550056 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Chen, S., Liang, K.: Left-invariant pseudo-Einstein metrics on Lie groups. J. Nonlinear Math. Phys. 19(2), 125001(11 pages)5 (2012)

    MathSciNet  Google Scholar 

  11. Chow, B., Knopf, D.: The Ricci flow: An Introduction. AMS, Providence (2004)

    MATH  Google Scholar 

  12. Conti, D., Rossi, F.A.: Einstein nilpotent Lie groups. J. Pure Appl. Algebra 222(3), 976–997 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Conti, D., Rossi, F.A.: Ricci-flat and Einstein pseudoriemannian nilmanifolds. Complex Manifolds 6(1), 170–193 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Derdzinski, A., Gal, ŚR.: Indefinite Einstein metrics on simple Lie group. Indiana Univ. Math. J. 63(1), 165–212 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Duong, M.T., Pinczon, G., Ushirobira, R.: A new invariant of quadratic Lie algebras. Algebr. Represent. Theory 15, 1163–1203 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Favre, G., Santharoubane, L.J.: Symmetric, invariant, non-degenerate bilinear form on a Lie algebra. J. Algebra 105, 451–464 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Figueroa-O’Farrill, J., Stanciu, S.: On the structure of symmetric self-dual Lie algebras. J. Math. Phys. 37(8), 4121–4134 (1996)

    MathSciNet  MATH  Google Scholar 

  18. Gordon, C.S., Kerr, M.M.: New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19(1), 75–101 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Heber, J.: Non-compact homogeneous Einstein manifolds. Invent. Math. 133, 279–352 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Jablonski, M.: Moduli of Einstein and non-Einstein nilradicals. Geom. Dedicata. 152, 63–84 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Kath, I.: Nilpotent metric Lie algebras of small dimension. J. Lie Theory 17(1), 41–61 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Kath, I., Olbrich, M.: Metric Lie algebras with maximal isotropic centre. Math. Z. 246(1–2), 23–53 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Kath, I., Olbrich, M.: Metric Lie algebras and quadratic extensions. Transform. Groups 11(1), 87–131 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Lauret, J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241, 83–99 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Lauret, J.: Ricci soliton solvmanifolds. J. Reine Angew. Math. 650, 1–21 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions. Math. Ann. 350(1), 199–225 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Medina, A.: Groupes de Lie munis de métriques bi-invariantes. Tohoku Math. J. 37, 405–421 (1985)

    MathSciNet  MATH  Google Scholar 

  29. Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant (French) [Lie algebras and invariant scalar products]. Ann. Sci. École Norm. Sup. (4) 18(3), 553–561 (1985)

    MathSciNet  MATH  Google Scholar 

  30. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    MathSciNet  MATH  Google Scholar 

  31. Nikolayevsky, Y.: Einstein solvmanifolds with a simple Einstein derivation. Geom. Dedicat 135, 87–102 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Nikolayevsky, Y.: Einstein solvmanifolds with free nilradical. Ann. Global Anal. Geom. 33(1), 71–87 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Nikolayevsky, Y.: Einstein solvmanifolds and the pre-Einstein derivation. Trans. Am. Math. Soc. 363(8), 3935–3958 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Nikolayevsky, Y.: Einstein solvmanifolds attached to two-step nilradicals. Math. Z. 272, 675–695 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Nomizu, K.: Left-invariant Lorentz metrics on Lie groups. Osaka J. Math. 16, 143–150 (1979)

    MathSciNet  MATH  Google Scholar 

  36. Onda, K.: Lorentz Ricci Solitons on 3-dimensional Lie groups. Geom. Dedicata 147, 313–322 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Onda, K.: Example of algebraic Ricci solitons in the pseudo-Riemannian case. Acta Math. Hungar. 144(1), 247–265 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Onda, K., Parker, P.: Nilsolitons of H-type in the Lorentzian setting. Houston J. Math. 41(4), 1137–1151 (2015)

    MathSciNet  MATH  Google Scholar 

  39. O’Neill, B.: Semi-Riemannian geometry with applications to relativity. In: Pure and Applied Mathematics, vol. 103. Academic Press, New York (1983)

  40. Ovando, G.P.: Two-step nilpotent Lie algebras with ad-invariant metrics and a special kind of skew-symmetric maps. J. Algebra Appl. 66, 897–917 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Payne, T.: The existence of soliton metrics for nilpotent Lie groups. Geom. Dedicata. 145, 71–88 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Rahmani, S.: Metriques de Lorentz sur les groupes de Lie unimodulaires, de dimension trois (French), [Lorentz metrics on three-dimensional unimodular Lie groups]. J. Geom. Phys. 9(3), 295–302 (1992)

    MathSciNet  MATH  Google Scholar 

  43. Tamaru, H.: Noncompact homogeneous Einstein manifolds attached to graded Lie algebras. Math. Z. 259, 171–186 (2008)

    MathSciNet  MATH  Google Scholar 

  44. Tamaru, H.: Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds. Math. Ann. 351, 51–66 (2011)

    MathSciNet  MATH  Google Scholar 

  45. Will, C.: Rank-one Einstein solvmanifolds of dimension 7. Differ. Geom. Appl. 19, 307–318 (2003)

    MathSciNet  MATH  Google Scholar 

  46. Wolter, T.H.: Einstein metrics on solvable groups. Math. Z. 206(3), 457–471 (1991)

    MathSciNet  MATH  Google Scholar 

  47. Yan, Z.: Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces. J. Geom. 111(1), Art. 4, 18 pp (2020)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are deeply grateful to the reviewers of this paper for very careful reading and useful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province, 315211, People’s Republic of China

    Zaili Yan

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Shaoqiang Deng

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  1. Zaili Yan
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  2. Shaoqiang Deng
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Correspondence to Shaoqiang Deng.

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Zaili Yan is supported by NSFC (No. 11701300) and K.C. Wong Magna Fund in Ningbo University. Shaoqiang Deng is supported by NSFC (Nos. 12071228), and the Fundamental Research Funds for the Central Universities.

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Yan, Z., Deng, S. Double Extensions on Riemannian Ricci Nilsolitons. J Geom Anal 31, 9996–10023 (2021). https://doi.org/10.1007/s12220-021-00636-x

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