Four-dimensional Einstein manifolds with Heisenberg symmetry

2021/04/15 by Vicente Cortés, Cortés, Vicente, Arpan Saha +1
Mathematics · #53C26 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C26

paper · pdf · doi:10.48550/arxiv.2104.07280

22 pages, no figures. v2: minor corrections, new references added

arxiv created 2021/07/09 · arxiv updated 2021/07/12

Abstract

We classify Einstein metrics on ℝ⁴ 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-Kähler, 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.

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