Quaternionic-Hermitian manifold - Imported Knowlpedia Content

Definition

A quaternionic-Hermitian manifold is an almost-quaternionic manifold $(M,Q)$ of real dimension $4n$ together with a Riemannian metric $g$ such that

g(AX,AY)=g(X,Y)

for every local section $A$ of $Q$ satisfying $A^2=-\operatorname{id}_{TM}$ , and all tangent vectors $X,Y$ at the same point. Equivalently, for every local admissible frame $(I,J,K)$ of $Q$ , the metric is Hermitian with respect to $I$ , $J$ , and $K$ . This compatibility is pointwise: it imposes no integrability condition on $Q$ and no parallelism condition on $g$ .

Structure-group interpretation

An almost-quaternionic structure reduces the frame bundle to $GL(n,\mathbb H)Sp(1)$ . Choosing a compatible metric reduces it further to the compact group $Sp(n)Sp(1)$ ,

Sp(n)Sp(1)=(Sp(n)\times Sp(1))/\{\pm(1,1)\}.

This is the natural Riemannian structure group of quaternionic geometry. The formulation is independent of the chosen admissible frame because two such frames differ by an $SO(3)$ -valued change of basis in $Q$ .

Fundamental four-form

For a local admissible orthonormal frame $(I,J,K)$ , define 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).

The combination

\Omega=\omega_I\wedge\omega_I+\omega_J\wedge\omega_J+\omega_K\wedge\omega_K

is unchanged by rotating the admissible frame, so it defines a global differential four-form. Its normalization varies in the literature, but its stabilizer encodes the $Sp(n)Sp(1)$ -reduction.

Relationship to quaternion-Kähler geometry

The quaternionic-Hermitian condition is algebraic. A quaternion-Kähler metric additionally requires its Levi-Civita connection to preserve $Q$ , or equivalently that its holonomy lie in $Sp(n)Sp(1)$ ; see Besse, Chapter 14. Thus a compatible metric need not be quaternion-Kähler. A hyper-Hermitian manifold provides a quaternionic-Hermitian example by taking $Q=\operatorname{span}\{I,J,K\}$ , but the global triple is extra data invisible to $(Q,g)$ .

Conventions and scope

References

Arthur L. Besse, Einstein Manifolds, Springer, 1987. Springer DOI record. Relevant: Chapter 14, especially quaternionic-Hermitian structures and quaternion-Kähler metrics.
Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. Oxford DOI record. Relevant: quaternionic, hypercomplex, and hyperkähler structures.