On Sasakian manifolds with special transverse holonomy

TL;DR

This paper investigates special holonomy conditions in compact Sasakian manifolds, linking trivial holonomy to Heisenberg quotients and Sp(n) holonomy to transverse hypercomplex structures, extending Verbitsky's theorem.

Contribution

It characterizes Sasakian manifolds with specific Tondeur connection holonomy, revealing geometric structures and providing a Sasakian analogue of Verbitsky's theorem.

Findings

Trivial Tondeur holonomy implies the manifold is a quotient of the Heisenberg group.

Holonomy contained in Sp(n) corresponds to the existence of a transverse hypercomplex structure.

Results extend the understanding of holonomy in Sasakian geometry.

Abstract

We study compact Sasakian manifolds whose Tondeur connection has holonomy group either trivial or contained in Sp(n). We show that the first condition forces the manifold to be a compact quotient of the Heisenberg Lie group, while in the simply-connected case a Sasakian structure has the holonomy of the Tondeur connection contained in Sp(n) if and only if there exists a transverse hypercomplex structure. This latter result is the "Sasakian version" of a theorem of Verbitsky.