HyperKahler contact Distribution in 3-Sasakain Manifolds

TL;DR

This paper investigates the curvature properties of the HyperKahler contact distribution in 3-Sasakian manifolds, establishing conditions for constant holomorphic sectional curvatures and exploring relations between different curvature measures.

Contribution

It introduces a new metric connection determined by the HyperKahler contact distribution and characterizes when this distribution has constant holomorphic sectional curvature.

Findings

HyperKahler contact distribution has constant holomorphic sectional curvature iff the manifold has constant -sectional curvature.

The metric connection  is completely determined by the HyperKahler contact distribution.

A relation between sectional curvatures of -planes under  and Levi-Civita connection is established.

Abstract

In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection $\bar{\nabla}$ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler contact distribution is of constant holomorphic sectional curvatures if and only if its 3-Sasakian manifold is of constant $\varphi_{\alpha}$-sectional curvatures. Moreover, it is shown that there is an interesting relation between the sectional curvatures of $\varphi_{\alpha}$-planes on $TM$ of metric connection $\bar{\nabla}$ and the Levi-Civita connection.