Half-flat structures inducing Einstein metrics on homogeneous spaces

2014/10/31 by Alberto Raffero · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Curvature #Differential geometry #Einstein #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Homogeneous #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Pure mathematics #math.DG #msc:53C10 #msc:53C25 #msc:53C30

paper · pdf · doi:10.1007/s10455-015-9457-1

17 pages, to appear in Annals of Global Analysis and Geometry

openalex publication_date 2015/03/11 · arxiv created 2015/04/09 · arxiv updated 2015/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we consider half-flat SU(3)-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form w₁^- is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left invariant half-flat (non coupled and non double) structure inducing an Einstein metric on S³× S³ and we show there does not exist any left invariant coupled structure inducing an \rm Ad(S¹)-invariant Einstein metric on it. Finally, we show that there are no coupled structures inducing the Einstein metric on Einstein solvmanifolds and on homogeneous Einstein manifolds of nonpositive sectional curvature.

Citations

• Classification des varietes approximativement kahleriennes homogenes
• Half-flat structures on products of three-dimensional Lie groups
• A step towards the Alekseevskii Conjecture
• Einstein Manifolds

Cited by

• Special Types of Locally Conformal Closed G2-Structures