Left-invariant Einstein metrics on $S^3 \times S^3$
Journal of Geometry and Physics(2018)
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)×SU(2)=S3×S3. 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 K≇Z2 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 K≅Z2 we present partial results.
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Key words
Einstein manifolds,Homogeneous compact spaces,Six dimensions,Left-invariant Einstein metrics,Product of two 3-spheres,
SU(2)×SU(2)
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