HyperKahler Contact Distributions

Abstract

Let $({\phi }_{\alpha },{\xi }_{\alpha },g)$ for $\alpha =1,2$, and $3$ be a contact metric $3$-structure on the manifold $M^{4n+3}$. We show that the $3$-contact distribution of this structure admits a HyperKahler structure whenever $(M^{4n+3},{\phi }_{\alpha },{\xi }_{\alpha },g)$ is a $3$-Sasakian manifold. In this case, we call it HyperKahler contact distribution. To analyze the curvature properties of this distribution, we define a special metric connection that is completely determined by the HyperKahler contact distribution. We prove that the $3$-Sasakian manifold is of constant ${\phi }_{\alpha }$-sectional curvatures if and only if its HyperKahler contact distribution has constant holomorphic sectional curvatures.

arXiv admin note: substantial text overlap with arXiv:1204.3407

References in corpus (2)

• Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
• On Anosovity, divergence and bi-contact surgery