# Holonomy and 3-Sasakian homogeneous manifolds versus symplectic triple systems

## TL;DR

This paper investigates special connections with torsion on 3-Sasakian manifolds, showing their holonomy groups are smaller than the Levi-Civita connection's, using symplectic triple systems as a key algebraic tool.

## Contribution

It demonstrates the holonomy reduction for certain connections on 3-Sasakian manifolds via symplectic triple systems, providing a unified computational approach.

## Key findings

- Holonomy groups reduce to proper subgroups in homogeneous cases.
- Unified computation method using symplectic triple systems.
- Supports specific connection choices with torsion in 3-Sasakian manifolds.

## Abstract

Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the special orthogonal group, of dimension considerably smaller. We realize the computations of the holonomies in a unified way, by using as a main algebraic tool a nonassociative structure, that one of symplectic triple system.