Abstract
We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In Tamaru (Parabolic subgroups of semisimple Lie groups and einstein solvmanifolds. Math Ann 351(1):51–66, 2011) constructs new attached solvmanifolds, which are submanifolds of the solvmanifolds correspond to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to associated solvmanifolds, described in (Ann Global Anal Geom 19(1):75–101, 2001), obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geometrically distinct from noncompact symmetric spaces and their submanifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevskii, D.V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sb. 25, 87–109 (1975). (English transl, Math. USSR-Sb. 96, 93–117 (1975))
Besse, A.: Einstein manifolds. Springer, New York (1987)
Gordon, C.S., Kerr, M.M.: New homogeneous einstein metrics of negative ricci curvature. Ann. Glob. Anal. Geom. 19(1), 75–101 (2001)
Gordon, C.S., Wilson, E.N.: Isometry groups of riemannian solvmanifolds. Trans. Am. Math. Soc. 307 1, 245–269 (1988)
Helgason, S.: Differential geometry, Lie groups and symmetric spaces. Academic Press, New York (1978)
Jablonski, M.: Moduli of Einstein and non-Einstein nilradicals. Geom. Dedicata 152, 63–84 (2011)
Jablonski, M.: Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups. Geom. Topol. 15(2), 735–764 (2011)
Jablonski, M.: Distinguished orbits of reductive groups. Rocky Mt. J. Math. 42(5), 1521–1549 (2012)
Kaneyuki, S., Asano, H.: Graded Lie algebras and generalized Jordan triple systems. Nagoya Math. J. 112, 81–115 (1988)
Knapp, A.: Beyond an introduction. Birkhäuser, Basel (1996)
Lauret, J.: Einstein solvmanifolds and nilsolitons, New developments in Lie theory and geometry, Contemporary mathematics, 135. American Mathematical Society, Providence (2009)
Lauret, J.: Einstein solvmanifolds are standard. Ann. Math. 2(3), 1859–1877 (2010)
Lauret, J.: Ricci soliton solvmanifolds. J. Reine Angew. Math. 650, 1–21 (2011)
Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions. Math. Ann. 350(1), 199–225 (2011)
Nikolayevsky, Y.: Einstein solvmanifolds and the pre-Einstein derivation. Trans. Amer. Math. Soc. 363(8), 3935–3958 (2011)
Payne, T.: The existence of soliton metrics for nilpotent Lie groups. Geom. Dedicata 145, 71–88 (2010)
Tamaru, H.: Parabolic subgroups of semisimple Lie groups and einstein solvmanifolds. Math. Ann. 351(1), 51–66 (2011)
Wolter, T.H.: Einstein metrics on solvable groups. Math. Z. 206, 457–471 (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kerr, M.M. New examples of non-symmetric Einstein solvmanifolds of negative Ricci curvature. Ann Glob Anal Geom 46, 281–291 (2014). https://doi.org/10.1007/s10455-014-9423-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-014-9423-3