Four-dimensional Einstein manifolds with Heisenberg symmetry

TL;DR

This paper classifies certain Einstein metrics on four-dimensional Euclidean space with Heisenberg symmetry, revealing that Ricci-flat examples are incomplete while two negatively curved metrics are complete.

Contribution

It provides a classification of Einstein metrics with Heisenberg symmetry on , identifying the completeness and curvature properties of these metrics.

Findings

Ricci-flat metrics are incomplete, including hyper-Ka4hler examples.

Two complete negatively curved metrics: complex hyperbolic and one-loop deformed universal hypermultiplet.

All metrics with this symmetry are either Ricci-flat or negatively curved.

Abstract

We classify Einstein metrics on $\mathbb{R}^4$ invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. The metrics are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-K\"ahler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.