Paper · Oct 25, 2018
Einstein solvmanifolds as submanifolds of symmetric spaces
As on arXiv
Abstract
Lying at the intersection of Ado's theorem and the Nash embedding theorem, we consider the problem of finding faithful representations of Lie groups which are simultaneously isometric embeddings. Such special maps are found for a certain class of solvable Lie groups which includes all Einstein and Ricci soliton solvmanifolds, as well as all Riemannian 2-step nilpotent Lie groups. As a consequence, we extend work of Tamaru by showing that all Einstein solvmanifolds can be realized as submanifolds (in the submanifold geometry) of a symmetric space.
- Submitted
- Oct 25, 2018
- Last revised
- Jun 24, 2025 (v2)
- Subjects
- math.DG
Comments: 13 pages, exposition significantly revised, final version to appear in Bulletin of the London Mathematical Society
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Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.