Pseudo-Riemannian Sasaki solvmanifolds

TL;DR

This paper investigates a specific class of pseudo-Riemannian Sasaki metrics on solvable Lie groups, characterizing their geometry via Sasaki reduction and classifying certain low-dimensional examples.

Contribution

It introduces a new class of pseudo-Riemannian Sasaki solvmanifolds, characterizes their geometry through reduction techniques, and classifies low-dimensional cases.

Findings

Classification of 5-dimensional pseudo-Riemannian Sasaki solvmanifolds.

Classification of 7-dimensional cases with abelian Kähler reduction.

Characterization of these geometries via moment map and reduction methods.

Abstract

We study a class of left-invariant pseudo-Riemannian Sasaki metrics on solvable Lie groups, which can be characterized by the property that the zero level set of the moment map relative to the action of some one-parameter subgroup $\{\exp tX\}$ is a normal nilpotent subgroup commuting with $\{\exp tX\}$, and $X$ is not lightlike. We characterize this geometry in terms of the Sasaki reduction and its pseudo-K\"ahler quotient under the action generated by the Reeb vector field. We classify pseudo-Riemannian Sasaki solvmanifolds of this type in dimension $5$ and those of dimension $7$ whose K\"ahler reduction in the above sense is abelian.