Cohomogeneity-one Einstein-Weyl structures : a local approach

TL;DR

This paper systematically analyzes cohomogeneity-one Einstein-Weyl spaces, revealing conditions for their existence and linking certain subclasses to locally conformally Kähler metrics with symmetric Kähler coset spaces.

Contribution

It provides a classification of Einstein-Weyl structures under cohomogeneity-one conditions, identifying when non-exact structures can exist and their geometric properties.

Findings

Non-exact Einstein-Weyl structures require specific subgroup conditions.

Subclass with a one-dimensional subgroup corresponds to locally conformally Kähler metrics.

The (n-2)-dimensional space is an arbitrary compact symmetric Kähler coset space.

Abstract

We analyse in a systematic way the (non-)compact n-dimensional Einstein Weyl spaces equipped with a cohomogeneity-one metric. With no compactness hypothesis, we prove that, as soon as the (n-1)-dimensional space is an homogeneous reductive Riemannian space with an unimodular group of left-acting isometries G 1)a non-exact Einstein-Weyl stucture may exist only if the (n-1)-dimensional homogeneous space G/H contains a non trivial subgroup H' that commutes with the isotropy subgroup H, 2) the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H'. We also prove that the subclass with a one-dimensional subgroup H' corresponds to n-dimensional Riemannian locally conformally K\"ahler metrics, the (n-2)-dimensional space G/(HxH') being an arbitrary compact symmetric K\"ahler coset space.