HyperKahler Contact Distributions

TL;DR

This paper demonstrates that the 3-contact distribution of a 3-Sasakian manifold admits a HyperKahler structure and explores its curvature properties using a specially defined metric connection.

Contribution

It establishes the existence of a HyperKahler structure on the 3-contact distribution of 3-Sasakian manifolds and analyzes its curvature characteristics.

Findings

HyperKahler structure exists on the 3-contact distribution of 3-Sasakian manifolds.

The manifold has constant i-sectional curvature iff its HyperKahler contact distribution has constant holomorphic sectional curvature.

A special metric connection is defined to study the curvature properties of the distribution.

Abstract

Let $(\varphi_\alpha,\xi_\alpha,g)$ 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 $(M^{4n+3},\varphi_\alpha,\xi_\alpha,g)$ 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 $\varphi_{\alpha}$-sectional curvatures if and only if its HyperKahler contact distribution has constant holomorphic sectional curvatures.