Hyper-Hermitian manifold

A hyper-Hermitian manifold is a hypercomplex manifold $(M,I,J,K)$ together with a Riemannian metric $g$ that is Hermitian for each member of the hypercomplex triple:

g(IX,IY)=g(JX,JY)=g(KX,KY)=g(X,Y)

for all tangent vectors $X,Y$ at the same point. Equivalently, every complex structure $aI+bJ+cK$ with $a^2+b^2+c^2=1$ is orthogonal for $g$ . The triple $(I,J,K)$ and metric $g$ are chosen data. No closedness or parallelism condition is imposed on the associated fundamental two-forms, so hyper-Hermitian is strictly weaker than hyperkähler.

Associated two-forms

The metric determines three real two-forms

\omega_I(X,Y)=g(IX,Y),\qquad \omega_J(X,Y)=g(JX,Y),\qquad \omega_K(X,Y)=g(KX,Y).

They encode the same pointwise compatibility as the metric, but they need not be closed. The structure is hyperkähler precisely when all three forms are closed; equivalently, the Levi-Civita connection preserves $I,J,K$ .

HKT condition

An HKT metric is a hyper-Hermitian metric whose three Hermitian Bismut connections coincide. Equivalently, after fixing one complex structure, the corresponding complex two-form satisfies a $\partial$ -closedness condition. Thus HKT adds a differential condition to the pointwise hyper-Hermitian data but is weaker than the hyperkähler condition.

Examples and scope

Quaternionic Euclidean space $\mathbb H^n$ , with its flat metric and complex structures given by multiplication by $i,j,k$ , is hyper-Hermitian and in fact hyperkähler. More generally, every hyperkähler manifold is hyper-Hermitian after forgetting the closedness of its three Kähler forms.

A hypercomplex manifold equipped with a Riemannian metric that is Hermitian only for $I$ is a near miss: invariance under $J$ and $K$ is part of the definition and does not follow from invariance under one complex structure.

References

Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. Oxford DOI record. Relevant: Chapter 7, hyperkähler structures and their underlying hyper-Hermitian data.