Knowlpedia

Theorem

Quaternion-Kähler manifolds are Einstein

Every quaternion-Kähler manifold of real dimension at least eight has Ricci tensor proportional to its metric.

Statement

Let (M4n,g)(M^{4n},g) be a connected with n≥2n\geq2. Then gg is an : there is a constant λ∈R\lambda\in\mathbb R such that

Ric⁡g=λg.\operatorname{Ric}_g=\lambda g.

Equivalently, its scalar curvature is constant and equals 4nλ4n\lambda. The conclusion includes the case λ=0\lambda=0, which occurs when the holonomy reduces to the hyperkähler subgroup Sp⁡(n)\operatorname{Sp}(n). The restriction to real dimension at least eight is essential: in dimension four the inclusion Hol⁡(g)⊆Sp⁡(1)Sp⁡(1)=SO⁡(4)\operatorname{Hol}(g)\subseteq\operatorname{Sp}(1)\operatorname{Sp}(1)=\operatorname{SO}(4) alone is automatic and does not imply the Einstein equation.

Geometric mechanism

The of Sp⁡(n)Sp⁡(1)\operatorname{Sp}(n)\operatorname{Sp}(1) sharply restricts the curvature tensor. The Ricci contraction of the allowed curvature components has only the metric as an invariant symmetric two-tensor, forcing proportionality to gg. Constancy of the proportionality factor then follows from the contracted . Besse states the result as Theorem 14.39.

Sign and examples

All three signs occur. Quaternionic projective space with its standard metric has positive Einstein constant, quaternionic hyperbolic space has negative Einstein constant, and have zero Einstein constant. Thus the theorem does not determine the sign of the scalar curvature.

Under the usual four-dimensional convention, “quaternion-Kähler” is redefined to include the Einstein condition together with self-duality. The theorem then becomes definitional in dimension four rather than a consequence of the unrestricted holonomy condition.

References
  1. Arthur L. Besse, Einstein Manifolds, Springer, 1987. Springer DOI record. Relevant: Chapter 14, especially Theorem 14.39.
  2. Simon Salamon, “Quaternionic Kähler Manifolds,” Inventiones Mathematicae 67 (1982), 143–171. DOI record. Relevant: the curvature and Einstein properties of quaternion-Kähler metrics.