Left-invariant Einstein metrics on $S^3\times S^3$

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 $G=\mathrm{SU}(2)\times \mathrm{SU}(2)=S^3\times S^3$. 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 $K\ncong {\mathbb{Z}}_2$ 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 $K\cong {\mathbb{Z}}_2$ we present partial results.

1+19 pages. v2: minor changes and references added. Final version accepted for publication. v3: minor correction in section 2

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Cited by in corpus (1)

• Homogeneous Einstein metrics on non-Kähler C-spaces