Quaternionic K\"ahler manifolds fibered by solvsolitons

TL;DR

This paper studies the geometry of principal orbits in quaternionic K"ahler manifolds of cohomogeneity one, showing they form a solvsoliton fibration in certain symmetric space deformations, with specific group structures and soliton properties.

Contribution

It demonstrates that principal orbits in deformed quaternionic K"ahler symmetric spaces form a solvsoliton fibration, detailing the group structures and soliton characteristics.

Findings

Principal orbits form a solvsoliton fibration at zero deformation.

The solvable group is non-unimodular for n>1 and Heisenberg for n=1.

Deformation preserves solvmanifold structure but not Ricci solitons.

Abstract

This paper is concerned with the geometry of principal orbits in quaternionic K\"ahler manifolds $M$ of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic K\"ahler 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.