Duke Mathematical Journal

Strongly solvable spaces

Michael Jablonski

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Abstract

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

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

Dates
First available in Project Euclid: 30 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1422627051

Digital Object Identifier
doi:10.1215/00127094-2861277

Mathematical Reviews number (MathSciNet)
MR3306558

Zentralblatt MATH identifier
1323.53049

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

Citation

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


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