Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8255589 | Journal of Geometry and Physics | 2018 | 12 Pages |
Abstract
The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics g on G=SU(2)ÃSU(2)=S3ÃS3. Einstein metrics are critical points of the total scalar curvature functional for fixed volume. The scalar curvature S of a left-invariant metric g is constant and can be expressed as a rational function in the parameters determining the metric. The critical points of S, subject to the volume constraint, are given by the zero locus of a system of polynomials in the parameters. In general, however, the determination of the zero locus is apparently out of reach. Instead, we consider the case where the isotropy group K of g in the group of motions is non-trivial. When KâZ2 we prove that the Einstein metrics on G are given by (up to homothety) either the standard metric or the nearly Kähler metric, based on representation-theoretic arguments and computer algebra. For the remaining case Kâ
Z2 we present partial results.
Keywords
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Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
Florin Belgun, Vicente Cortés, Alexander S. Haupt, David Lindemann,