Obata connection - Knowlpedia

Let $(M,I,J,K)$ be a hypercomplex manifold. Its Obata connection is the unique torsion-free connection $\nabla^{\mathrm{Ob}}$ on $TM$ satisfying

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

Here $(\nabla_X I)Y=\nabla_X(IY)-I\nabla_XY$ , and similarly for $J,K$ ; torsion-free means $\nabla_XY-\nabla_YX=[X,Y]$ . Existence uses the integrability of the three complex structures, while preservation of the entire quaternionic triple forces uniqueness. Thus the connection is canonically determined by the hypercomplex structure and does not require a metric.

Characterization and holonomy

Obata's construction gives an equivalence: an almost-hypercomplex triple admits a torsion-free connection preserving $I,J,K$ exactly when the triple is hypercomplex. Parallel transport is therefore quaternionic-linear, and the holonomy of $\nabla^{\mathrm{Ob}}$ is contained in $GL(n,\mathbb H)$ when $\dim_{\mathbb R}M=4n$ .

Relation to a compatible metric

The Obata connection is defined without a metric and generally is not a Levi-Civita connection. If a hyper-Hermitian metric $g$ has $\nabla^{\mathrm{Ob}}g=0$ , then uniqueness of the torsion-free metric connection identifies $\nabla^{\mathrm{Ob}}$ with the Levi-Civita connection of $g$ ; the resulting metric is hyperkähler. On $\mathbb H^n$ with its constant hypercomplex triple, the Obata connection is the ordinary flat connection.

References

Morio Obata, “Affine Connections on Manifolds with Almost Complex, Quaternion or Hermitian Structure,” Japanese Journal of Mathematics 26 (1956), 43–77. J-STAGE DOI record and full text.
Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. Oxford DOI record. Relevant: hypercomplex and hyperkähler geometry.