Abstract
Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.
Highlights
The investigation of homogeneous Einstein manifolds and, more recently, homogeneous Ricci solitons constitutes an important area of research in differential geometry
Jablonski [18] proved the following remarkable result, which lies at the intersection of Ado’s and Nash’s embedding theorems: every connected Ricci soliton solvmanifold or 2-step nilpotent Lie group with left-invariant metric can be realized as a submanifold of a symmetric space
In view of the existence of general embedding results and some interesting families of examples, we propose to undertake the project of investigating homogeneous Ricci solitons from the perspective of extrinsic submanifold geometry
Summary
The investigation of homogeneous Einstein manifolds and, more recently, homogeneous Ricci solitons constitutes an important area of research in differential geometry. The reason is that the Levi-Civita connection of a symmetric space of non-compact type, despite admitting a Lie algebraic description, turns out to be quite hard to handle in full generality, since it involves Lie brackets which relate (positive) root spaces in a complicated way One sign of this difficulty is that, previously to this article, a classification as in Theorem A had only been obtained, apart from the well-known case of constant curvature M ∼= RHn, in the very specific setting of the complex hyperbolic space M ∼= CHn [15]. As already mentioned, the higher rank cases require a careful analysis of the geometry of Lie hypersurfaces in terms of restricted root systems, the case of rank one symmetric spaces is subsumed into the investigation of a broader family of Einstein solvmanifolds, namely, Damek-Ricci harmonic spaces [9] These are well-known solvable extensions of the so-called generalized Heisenberg groups (or H-type groups), which in turn constitute an important family of two-step nilpotent metric Lie groups. Each one of these three main sections contains a subsection with preliminaries on generalized Heisenberg groups, Damek-Ricci spaces and symmetric spaces of non-compact type, respectively
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