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Quaternionic Kähler manifolds fibered by solvsolitons

Vicente Cortés, Alejandro Gil-García, Markus Röser

TL;DR

The paper investigates cohomogeneity-one actions on one-loop deformed quaternionic Kähler spaces, focusing on the geometry of the level-set hypersurfaces N_ ho. By deriving a general Ricci-curvature formula for hypersurfaces in Einstein manifolds and applying it to the one-loop deformed c-map metric, it computes the Ricci endomorphism on N_ ho and reveals a solvsoliton structure in the $c=0$ case. The level sets are realized as left-invariant solvmanifolds on a solvable Lie group L = $ rak{b} times ext{Heis}_{2n+1}$, with an explicit Iwasawa-based description of the Lie algebra and its inner product, enabling a thorough solvsoliton test. The key results are: (i) for $n=1$ all fibers are nilsolitons, (ii) for $n>1$ and $c=0$ fibers are solvsolitons, and (iii) for $n>1$ and $c>0$ the fibers are not solvsolitons, which supports a broader conjecture that one-loop deformations of symmetric supergravity c-map spaces yield non-soliton hypersurfaces in higher dimensions. These findings illuminate how algebraic Ricci solitons interact with quaternionic Kähler cohomogeneity-one geometries and provide a concrete framework to study curvature in solvmanifold fibers.

Abstract

This paper is concerned with the geometry of principal orbits in quaternionic Kähler manifolds $M$ of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic Kähler symmetric spaces associated with the simple Lie groups of type A by the one-loop deformation. We prove that for zero deformation parameter the principal orbits form a fibration by solvsolitons (nilsolitons if $4n=\dim M=4$). The underlying solvable group is non-unimodular if $n>1$ and is the Heisenberg group if $n=1$. We show that under the deformation, the hypersurfaces remain solvmanifolds but cease to be Ricci solitons.

Quaternionic Kähler manifolds fibered by solvsolitons

TL;DR

The paper investigates cohomogeneity-one actions on one-loop deformed quaternionic Kähler spaces, focusing on the geometry of the level-set hypersurfaces N_ ho. By deriving a general Ricci-curvature formula for hypersurfaces in Einstein manifolds and applying it to the one-loop deformed c-map metric, it computes the Ricci endomorphism on N_ ho and reveals a solvsoliton structure in the case. The level sets are realized as left-invariant solvmanifolds on a solvable Lie group L = , with an explicit Iwasawa-based description of the Lie algebra and its inner product, enabling a thorough solvsoliton test. The key results are: (i) for all fibers are nilsolitons, (ii) for and fibers are solvsolitons, and (iii) for and the fibers are not solvsolitons, which supports a broader conjecture that one-loop deformations of symmetric supergravity c-map spaces yield non-soliton hypersurfaces in higher dimensions. These findings illuminate how algebraic Ricci solitons interact with quaternionic Kähler cohomogeneity-one geometries and provide a concrete framework to study curvature in solvmanifold fibers.

Abstract

This paper is concerned with the geometry of principal orbits in quaternionic Kähler manifolds of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic Kähler symmetric spaces associated with the simple Lie groups of type A by the one-loop deformation. We prove that for zero deformation parameter the principal orbits form a fibration by solvsolitons (nilsolitons if ). The underlying solvable group is non-unimodular if and is the Heisenberg group if . We show that under the deformation, the hypersurfaces remain solvmanifolds but cease to be Ricci solitons.
Paper Structure (10 sections, 22 theorems, 120 equations)

This paper contains 10 sections, 22 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on solvsolitons
  3. The Ricci curvature of a hypersurface in an Einstein manifold
  4. Ricci tensor of N in general
  5. Application to one-loop deformed hypermultiplet manifolds
  6. The level sets (N,g) as Riemannian solvmanifolds
  7. The Iwasawa decomposition of su(1,n-1)
  8. The Lie algebra l
  9. The metric g as a left-invariant metric on L
  10. Existence of solvsolitons on N

Key Result

Theorem 1.1

Let $(\bar{N},g_{\bar{N}}^c)$ be the one-loop deformation of the symmetric space eq:SU_sym_space and let $(\bar{N}_\rho,g_\rho^c)$ be the fiber of $\rho\colon\bar{N}\to\mathbb{R}_{>0}$ considered as a solvmanifold. Then:

Theorems & Definitions (48)

  • Theorem 1.1
  • Conjecture 1
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Lau01
  • Theorem 2.4: Lau11
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 38 more