Hypercomplex manifold - Imported Knowlpedia Content

Definition

A hypercomplex manifold is a smooth manifold $M$ equipped with three smooth bundle endomorphisms $I,J,K:TM\to TM$ such that

I^2=J^2=K^2=-\operatorname{id}_{TM},\qquad IJ=K=-JI,

and each of $I,J,K$ is an integrable almost-complex structure. Thus each member of the triple defines a complex-manifold structure, while the relations make every tangent space a left quaternionic vector space. In particular, the real dimension of $M$ is divisible by four. The ordered triple is part of the structure; merely specifying its rank-three span gives the weaker notion of an almost-quaternionic structure.

Canonical connection

Every hypercomplex manifold has a unique torsion-free connection $\nabla^{\mathrm{Ob}}$ on $TM$ satisfying

\nabla^{\mathrm{Ob}}I=\nabla^{\mathrm{Ob}}J=\nabla^{\mathrm{Ob}}K=0.

This is the Obata connection. Its existence and uniqueness convert the three integrability conditions into a useful differential-geometric structure: parallel transport is quaternionic-linear, and the holonomy lies in $GL(n,\mathbb H)$ . This characterization is developed in Joyce, §6.2.

Relationship to quaternionic geometry

The span

Q=\operatorname{span}_{\mathbb R}\{I,J,K\}\subseteq\operatorname{End}(TM)

is an almost-quaternionic structure. Because the Obata connection is torsion-free and preserves $Q$ , it makes $M$ a quaternionic manifold. The converse need not hold: a quaternionic manifold generally has only local admissible triples, related on overlaps by rotations, and need not admit any globally distinguished $I,J,K$ .

If $M$ also has a Riemannian metric that is Hermitian for $I,J,K$ and whose Levi-Civita connection preserves the triple, then $M$ is hyperkähler. Metric compatibility alone gives a hyper-Hermitian manifold and does not force the associated two-forms to be closed.

Examples and non-examples

The quaternionic vector space $\mathbb H^n$ , and every quotient of it by a lattice acting by translations, has the constant hypercomplex triple given by left multiplication by $i,j,k$ . More generally, a hyperkähler manifold is hypercomplex after forgetting its metric.

Three smooth endomorphisms satisfying the quaternion relations define only an almost-hypercomplex structure. If one of their Nijenhuis tensors is nonzero, the resulting structure is not hypercomplex because the required complex structure is not integrable.

References

Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. Oxford DOI record. Relevant: §6.2, hypercomplex structures and the Obata connection.
Simon Salamon, “Quaternionic Manifolds,” Symposia Mathematica 26, 1982, 139–151. Stable repository record. Relevant: quaternionic and hypercomplex structures.