Duke Mathematical Journal

Strongly solvable spaces

Michael Jablonski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This work builds on the foundation laid by Gordon and Wilson in their study of isometry groups of solvmanifolds—Riemannian manifolds that admit a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the completely solvable Lie groups. When the commutator subalgebra contains the center, we have a complete description of the isometry group of any left-invariant metric using only metric Lie algebra information.

Using our work on the isometry group of such spaces, we study quotients of solvmanifolds. Our first application is to the classification of homogeneous Ricci soliton metrics. We show that the verification of the generalized Alekseevsky conjecture reduces to the simply connected case. Our second application is a generalization of a result of Heintze on the rigidity of existence of compact quotients for certain homogeneous spaces. Heintze’s result applies to spaces with negative curvature. We remove all the geometric requirements, replacing them with algebraic requirements on the homogeneous structure.

Article information

Duke Math. J., Volume 164, Number 2 (2015), 361-402.

First available in Project Euclid: 30 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 22E25: Nilpotent and solvable Lie groups


Jablonski, Michael. Strongly solvable spaces. Duke Math. J. 164 (2015), no. 2, 361--402. doi:10.1215/00127094-2861277. https://projecteuclid.org/euclid.dmj/1422627051

Export citation


  • [1] D. V. Alekseevsky and V. Cortés, Isometry groups of homogeneous quaternionic Kähler manifolds, J. Geom. Anal. 9 (1999), 513–545.
  • [2] S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34.
  • [3] A. L. Besse, Einstein Manifolds, Classics Math., Springer, Berlin, 2008.
  • [4] P. Eberlein and M. Jablonski, “Closed orbits of semisimple group actions and the real Hilbert-Mumford function” in New Developments in Lie Theory and Geometry (Cruz Chica, 2007), Contemp. Math. 491, Amer. Math. Soc., Providence, 2009, 283–321.
  • [5] C. Gordon, Riemannian isometry groups containing transitive reductive subgroups, Math. Ann. 248 (1980), 185–192.
  • [6] C. S. Gordon and E. N. Wilson, Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc. 307, no. 1 (1988), 245–269.
  • [7] M. Gotô, Faithful representations of Lie groups, II, Nagoya Math. J. 1 (1950), 91–107.
  • [8] J. Heber, Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998), 279–352.
  • [9] E. Heintze, Compact quotients of homogeneous negatively curved riemannian manifolds, Math. Z. 140 (1974), 79–80.
  • [10] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23–34.
  • [11] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math. 34, Amer. Math. Soc., Providence, 2001.
  • [12] M. Jablonski, Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups, Geom. Topol. 15 (2011), 735–764.
  • [13] M. Jablonski, Homogeneous Ricci solitons, preprint, arXiv:1109.6556v2 [math.DG].
  • [14] N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics 10, Interscience, New York, 1962.
  • [15] R. Lafuente and J. Lauret, On homogeneous Ricci solitons, Q. J. Math. 65 (2014), 399–419.
  • [16] J. Lauret, Einstein solvmanifolds are standard, Ann. of Math. (2) 172 (2010), 1859–1877.
  • [17] J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011), 1–21.
  • [18] A. Malcev, On linear Lie groups, C. R. (Doklady) Acad. Sci. URSS (N. S.) 40 (1943), 87–89.
  • [19] G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221.
  • [20] R. W. Richardson and P. J. Slodowy, Minimum vectors for real reductive algebraic groups, J. Lond. Math. Soc. (2) 42 (1990), 409–429.
  • [21] È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Lie Groups and Lie Algebras, III, Encyclopaedia Math. Sci. 41, Springer, Berlin, 1994.